diff --git a/.gitignore b/.gitignore index b13f4e6..007baf1 100644 --- a/.gitignore +++ b/.gitignore @@ -13,3 +13,5 @@ /docs/site/ /.vscode +.ipynb_checkpoints +.ipynb_checkpoints/* diff --git a/Project.toml b/Project.toml index 09c0fea..2d51538 100644 --- a/Project.toml +++ b/Project.toml @@ -1,13 +1,14 @@ name = "SimpleHypergraphs" uuid = "aa4a32ff-dd5d-5357-90e3-e7a9512f0501" +version = "0.3.3" authors = ["Przemysław Szufel ", "Bogumił Kamiński ", "Carmine Spagnuolo ", "Alessia Antelmi ", "Evan Walter Clark Spotte-Smith "] -version = "0.3.2" [deps] Conda = "8f4d0f93-b110-5947-807f-2305c1781a2d" +DataFrames = "a93c6f00-e57d-5684-b7b6-d8193f3e46c0" DataStructures = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8" Graphs = "86223c79-3864-5bf0-83f7-82e725a168b6" -JSON3 = "0f8b85d8-7281-11e9-16c2-39a750bddbf1" +JSON = "682c06a0-de6a-54ab-a142-c8b1cf79cde6" LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" PyCall = "438e738f-606a-5dbb-bf0a-cddfbfd45ab0" PyPlot = "d330b81b-6aea-500a-939a-2ce795aea3ee" @@ -19,9 +20,10 @@ StructTypes = "856f2bd8-1eba-4b0a-8007-ebc267875bd4" [compat] Conda = "^1.5.0" -DataStructures = "^0.18.11" +DataFrames = "^1.7.0" +DataStructures = "^0.18.11, ^0.19.1" Graphs = "^1.4.1" -JSON3 = "^1.0.1" +JSON = "^1.0.0" PyCall = "^1.91.2" PyPlot = "^2.8.2" SimpleTraits = "^0.9.4" diff --git a/REQUIRE b/REQUIRE index 5ed57c9..cafd82c 100644 --- a/REQUIRE +++ b/REQUIRE @@ -3,8 +3,7 @@ LightGraphs 1.2.0 StatsBase 0.32 DataStructures 0.17.7 Conda 1.3.0 -JSON 0.21.0 -JSON3 0.1.13 +JSON 1.0.0 PyCall 1.91.2 PyPlot 2.8.2 StructTypes 1.0.1 diff --git a/src/SimpleHypergraphs.jl b/src/SimpleHypergraphs.jl index 234d743..c276e4c 100644 --- a/src/SimpleHypergraphs.jl +++ b/src/SimpleHypergraphs.jl @@ -3,10 +3,11 @@ module SimpleHypergraphs using Graphs using StatsBase using DataStructures +using DataFrames using PyCall using Conda using PyPlot -using JSON3 +using JSON using Random using LinearAlgebra using SparseArrays @@ -27,7 +28,7 @@ export get_twosection_adjacency_mx, get_twosection_weighted_adjacency_mx export dual export random_model, random_kuniform_model, random_dregular_model, random_preferential_model -export Abstract_HG_format, HGF_Format, JSON_Format +export Abstract_HG_format, HGF_Format, JSON_Format, HIF_Format export hg_load, hg_save export modularity @@ -81,6 +82,9 @@ include("abstracttypes.jl") include("hypergraph.jl") include("io.jl") +# support for HIF standard +include("io_hif.jl") + include("models/bipartite.jl") include("models/twosection.jl") include("models/random-models.jl") diff --git a/src/io.jl b/src/io.jl index 3fbc9df..adb388a 100644 --- a/src/io.jl +++ b/src/io.jl @@ -1,19 +1,47 @@ -# TODO: maybe more fancy file format and correctness checking should be done abstract type Abstract_HG_format end +""" + HGF_Format +Simple text serialization format. +This format only stores the incidence structure of the hypergraph and ignores any metadata. +""" struct HGF_Format <: Abstract_HG_format end + +""" + JSON_Format + + Implementation of the `JSON` format for hypergraph input/output. +Note that more advanced features are supported in `HIF_Format`. +""" struct JSON_Format <: Abstract_HG_format end +""" + HIF_Format + + Implementation of the `HIF` (Hypergraph Interchange Format) format for hypergraph input/output. +See https://github.com/pszufe/HIF-standard for more details about the format. +See also the paper https://doi.org/10.1017/nws.2025.10018 +""" +struct HIF_Format <: Abstract_HG_format end +# note that the HIF format support is implemented in a separate file io_hif.jl + +""" + elemtypes(v::AbstractVector) + +Returns the union of the element types of the elements of vector `v`. +""" +elemtypes(v::AbstractVector) = foldl((T, x) -> Union{T, typeof(x)}, v; init=Union{}) -"""`` - hg_save(io::IO, h::H, format::HGF_Format) where {H <: AbstractSimpleHypergraph} +""" + hg_save(io::IO, h::H, format::HGF_Format; pretty::Bool=false) where {H <: AbstractSimpleHypergraph} Saves an undirected hypergraph `h` to an output stream `io` in `hgf` format. +This format only stores the incidence structure of the hypergraph and ignores any metadata. -TODO: what to do about metadata? +TODO: pretty option currently ignored """ -function hg_save(io::IO, h::H, format::HGF_Format) where {H <: AbstractSimpleHypergraph} +function hg_save(io::IO, h::H, ::HGF_Format; pretty::Bool=false) where {H <: AbstractSimpleHypergraph} println(io, length(h.v2he), " ", length(h.he2v)) for he in h.he2v skeys = sort(collect(keys(he))) @@ -23,16 +51,13 @@ end """ - hg_save(io::IO, h::Hypergraph, format::JSON_Format) + hg_save(io::IO, h::Hypergraph, ::JSON_Format; pretty::Bool=false) Saves an undirected hypergraph `h` to an output stream `io` in `json` format. If `h` has `Composite Types` either for vertex metadata or hyperedges metadata, -the user has to explicit tell the JSON3 package about it, for instance using: - -`JSON3.StructType(::Type{MyType}) = JSON3.Struct()`. +...TODO: complete this part -See the (JSON3.jl documentation)[https://github.com/quinnj/JSON3.jl] for more details. The `json` in output contains the following information (keys): @@ -44,36 +69,42 @@ The `json` in output contains the following information (keys): * `he_meta` : hyperedges metadata """ -function hg_save(io::IO, h::Hypergraph, format::JSON_Format) +function hg_save(io::IO, h::Hypergraph, ::JSON_Format; pretty::Bool=false) json_hg = Dict{Symbol, Any}() json_hg[:n] = nhv(h) json_hg[:k] = nhe(h) - json_hg[:m] = JSON3.write(Matrix(h)) - json_hg[:v2he] = JSON3.write(h.v2he) + #vec was addded when upgrading to JSON.jl from JSON3.jl + #to ensure proper serialization - JSON3 serialized to vec + json_hg[:m] = vec(Matrix(h)) - json_hg[:v_meta] = JSON3.write(h.v_meta) - json_hg[:he_meta] = JSON3.write(h.he_meta) + json_hg[:v2he] = h.v2he - JSON3.write(io, json_hg) + json_hg[:v_meta] = h.v_meta + json_hg[:he_meta] = h.he_meta + JSON.json(io, json_hg; pretty) end """ hg_save( fname::AbstractString, h::AbstractHypergraph; - format::Abstract_HG_format=HGF_Format() + format::Abstract_HG_format=HGF_Format(), pretty::Bool=false ) Saves a hypergraph `h` to a file `fname` in the specified `format`. The default saving format is `hgf`. - """ -hg_save( +function hg_save( fname::AbstractString, h::AbstractHypergraph; - format::Abstract_HG_format=HGF_Format() - ) = open(io -> hg_save(io, h, format), fname, "w") + format::Abstract_HG_format = HGF_Format(), pretty::Bool=false + ) + open(io -> hg_save(io, h, format; pretty=pretty), fname, "w") +end + + + """ @@ -97,10 +128,12 @@ Skips a single initial comment. """ function hg_load( io::IO, - format::HGF_Format; + ::HGF_Format; HType::Type{H} = Hypergraph, T::Type{U} = Bool, D::Type{<:AbstractDict{Int, U}} = Dict{Int, T}, + V = Nothing, + E = Nothing ) where {U <: Real, H <: AbstractSimpleHypergraph} line = readline(io) @@ -124,7 +157,7 @@ function hg_load( l = split(line) length(l) == 2 || throw(ArgumentError("expected two integers")) n, k = parse.(Int, l) - h = HType{T, D}(n, k) + h = HType{T, V, E, D}(n, k) for i in 1:k lastv = 0 @@ -167,34 +200,37 @@ Loads a hypergraph from a stream `io` from `json` format. * `V` : type of values stored in the vertices of the hypergraph * `E` : type of values stored in the edges of the hypergraph + + """ function hg_load( io::IO, - format::JSON_Format; + ::JSON_Format; HType::Type{H} = Hypergraph, T::Type{U} = Bool, D::Type{<:AbstractDict{Int, U}} = Dict{Int, T}, V = Nothing, E = Nothing ) where {H <: AbstractSimpleHypergraph, U <: Real} - json_hg = JSON3.read(readline(io)) - - m = reshape(JSON3.read(json_hg.m, Array{Union{T, Nothing}}), json_hg.n, json_hg.k) - - if V != Nothing && E != Nothing && hasvertexmeta(HType) && hashyperedgemeta(HType) - v_meta = JSON3.read(json_hg.v_meta, Array{Union{V, Nothing}}) - he_meta = JSON3.read(json_hg.he_meta, Array{Union{E, Nothing}}) - h = HType{T, V, E, D}(m; v_meta=v_meta, he_meta=he_meta) - elseif V != Nothing && hasvertexmeta(HType) - v_meta = JSON3.read(json_hg.v_meta, Array{Union{V, Nothing}}) - h = HType{T, V, D}(m; v_meta=v_meta) - elseif E != Nothing && hashyperedgemeta(HType) - he_meta = JSON3.read(json_hg.he_meta, Array{Union{E, Nothing}}) - h = HType{T, E, D}(m; he_meta=he_meta) + json_hg = JSON.parse(read(io, String)) + m = reshape(Vector{Union{T, Nothing}}(json_hg.m), json_hg.n, json_hg.k) + + V2 = (V == :auto) ? ("v_meta" ∈ keys(json_hg) && length(json_hg.v_meta) > 0 ? elemtypes(json_hg.v_meta) : Nothing) : V + E2 = (E == :auto) ? ("he_meta" ∈ keys(json_hg) && length(json_hg.he_meta) > 0 ? elemtypes(json_hg.he_meta) : Nothing) : E + + if V2 != Nothing && E2 != Nothing && hasvertexmeta(HType) && hashyperedgemeta(HType) + v_meta = Vector{Union{V2, Nothing}}(json_hg.v_meta) + he_meta = Vector{Union{E2, Nothing}}(json_hg.he_meta) + h = HType{T, V2, E2, D}(m; v_meta, he_meta) + elseif V2 != Nothing && hasvertexmeta(HType) + v_meta = Vector{Union{V2, Nothing}}(json_hg.v_meta) + h = HType{T, V2, D}(m; v_meta=v_meta) + elseif E2 != Nothing && hashyperedgemeta(HType) + he_meta = Vector{Union{E2, Nothing}}(json_hg.he_meta) + h = HType{T, E2, D}(m; he_meta=he_meta) else - h = HType{T, D}(m) + h = HType{T, V2, E2, D}(m) end - h end @@ -217,28 +253,37 @@ The default saving format is `hgf`. * `HType`: type of hypergraph to store data in * `T` : type of weight values stored in the hypergraph's adjacency matrix -* `D` : dictionary for storing values the default is `Dict{Int, T}` * `V` : type of values stored in the vertices of the hypergraph * `E` : type of values stored in the edges of the hypergraph - +* `D` : dictionary for storing values the default is `Dict{Int, T}` +* `show_warning` : whether to show warnings during loading +* `sort_by_id` : whether to sort vertices and hyperedges by their original ids (only supported for HIF_Format) +* `add_original_id_to_meta` : if a `Symbol` is provided, the original ids are added to the vertex and hyperedge metadata under that key (only supported for HIF_Format) """ function hg_load( fname::AbstractString; format::Abstract_HG_format = HGF_Format(), HType::Type{H} = Hypergraph, T::Type{U} = Bool, + V = :auto, + E = :auto, D::Type{<:AbstractDict{Int, U}} = Dict{Int, T}, - V = Nothing, - E = Nothing + show_warning::Bool=true, + sort_by_id::Bool=false, + add_original_id_to_meta::Union{Symbol, Nothing}=nothing ) where {U <: Real, H <: AbstractSimpleHypergraph} - - if format == HGF_Format() + @assert format isa HGF_Format || !sort_by_id "sort_by_id only supported for HIF_Format" + if format isa HGF_Format if HType == Hypergraph - open(io -> hg_load(io, format; HType=HType, T=T, D=D), fname, "r") + open(io -> hg_load(io, format; HType, T, D), fname, "r") else error("HGF loading only implemented for Hypergraph") end else - open(io -> hg_load(io, format; HType=HType, T=T, D=D, V=V, E=E), fname, "r") + if format isa HIF_Format + open(io -> hg_load(io, format; HType, T, D, V, E, show_warning, sort_by_id, add_original_id_to_meta), fname, "r") + else + open(io -> hg_load(io, format; HType, T, D, V, E), fname, "r") + end end end diff --git a/src/io_hif.jl b/src/io_hif.jl new file mode 100644 index 0000000..d9332ef --- /dev/null +++ b/src/io_hif.jl @@ -0,0 +1,295 @@ +""" + hg_load( + io::IO, + format::HIF_Format; + HType::Type{H} = Hypergraph, + T::Type{U} = Bool, + D::Type{<:AbstractDict{Int, U}} = Dict{Int, T}, + V::Union{Type, Symbol} = :auto, + E::Union{Type, Symbol} = :auto, + sort_by_id::Bool=false, + add_original_id_to_meta::Union{Symbol, Nothing}=nothing, + show_warning::Bool=true + ) where {U<:Real, H <: AbstractSimpleHypergraph} + +Loads a hypergraph from an input stream `io` in `HIF` format where + `T` is type of weight values stored in the hypergraph's adjacency matrix +and `D` is the type of the dictionary used to store weights for each hyperedge. + +If the hypergraph has vertex metadata or hyperedge metadata, their types can be specified +using the `V` and `E` parameters respectively. +If `V` or `E` is set to `:auto`, the types will be inferred from the data. + +SimpleHypergraphs.jl uses 1-based indexing so node and edge ids are regenerated accordingly. +The original ids of vertices and hyperedges can be preserved by setting `add_original_id_to_meta` +to a Symbol representing the key under which the original id will be stored in the metadata dictionary. +""" +function hg_load( + io::IO, + ::HIF_Format; + HType::Type{H} = Hypergraph, + T::Type{U} = Bool, + D::Type{<:AbstractDict{Int, U}} = Dict{Int, T}, + V::Union{Type, Symbol} = :auto, + E::Union{Type, Symbol} = :auto, + sort_by_id::Bool=false, + add_original_id_to_meta::Union{Symbol, Nothing}=nothing, + show_warning::Bool=true, +) where {U<:Real, H <: AbstractSimpleHypergraph} + data = JSON.parse(io; dicttype=Dict{Symbol, Any}) + + haskey(data, :incidences) || throw(ArgumentError("Missing required attribute 'incidences'")) + + if isempty(data[:incidences]) + if isempty(get(data, :edges, [])) && isempty(get(data, :nodes, [])) + return Hypergraph{ + T, + V == :auto ? Nothing : V, + E == :auto ? Nothing : E, + D, + }(0, 0) + end + end + + nodesdf = _build_attr_dataframe(data, :nodes, V, add_original_id_to_meta) + edgesdf = _build_attr_dataframe(data, :edges, E, add_original_id_to_meta) + + attr_nodes_N = nrow(nodesdf) + attr_edges_N = nrow(edgesdf) + if attr_nodes_N == 0 && isnothing(add_original_id_to_meta) + # no node attributes found so all attrs set to Nothing + nodesdf.attrs = Nothing[] + end + if attr_edges_N == 0 && isnothing(add_original_id_to_meta) + # no edge attributes found so all attrs set to Nothing + edgesdf.attrs = Nothing[] + end + + _add_nodes_and_edges_from_incidences!(data, nodesdf, edgesdf, add_original_id_to_meta) + + # narrow types for attrs if V or E is :auto + if V == :auto + _sanitize_types_items!(nodesdf) + end + if E == :auto + _sanitize_types_items!(edgesdf) + end + + _add_id_sort_column!(nodesdf) + _add_id_sort_column!(edgesdf) + # if all nodes or edges were discovered from incidences, sort by id to have consistent ordering + if attr_nodes_N == 0 + sort!(nodesdf, :id_sort) + end + if attr_edges_N == 0 + sort!(edgesdf, :id_sort) + end + + if sort_by_id + sort!(nodesdf, :id_sort) + sort!(edgesdf, :id_sort) + end + + if show_warning + if nrow(nodesdf) > 0 && nodesdf.id != 1:nrow(nodesdf) + @warn "Nodes in the source file were not sorted or not consistent - their order will change" + end + + if nrow(edgesdf) > 0 && edgesdf.id != 1:nrow(edgesdf) + @warn "Edges in the source file were not sorted or not consistent - their order will change" + end + end + + hg = HType{ + T, + eltype(nodesdf.attrs), + eltype(edgesdf.attrs), + D, + }(nrow(nodesdf), nrow(edgesdf), nodesdf.attrs, edgesdf.attrs) + + _add_weights_from_incidences!(data, hg, edgesdf, nodesdf) + hg +end + + +""" + _add_weights_from_incidences!(data::Dict{String, Any}, hg::AbstractSimpleHypergraph, edges::DataFrame, nodes::DataFrame) + +THIS FUNCTION IS INTERNAL AND SHOULD NOT BE CALLED DIRECTLY. +Adds weights to the hypergraph `hg` based on the incidences provided in `data`. +The `edges` and `nodes` DataFrames are used to map edge and node identifiers to +their respective indices in the hypergraph. +""" +function _add_weights_from_incidences!(data::Dict{Symbol, Any}, + hg::AbstractSimpleHypergraph{Union{Nothing, T}}, + edges::DataFrame, nodes::DataFrame) where {T <: Real} + node_dict = Dict{Union{String, Int}, Int}(id => idx for (id, idx) in zip(nodes.id, 1:nrow(nodes))) + edge_dict = Dict{Union{String, Int}, Int}(id => idx for (id, idx) in zip(edges.id, 1:nrow(edges))) + incidences = data[:incidences] + for incidence in incidences + node_idx = node_dict[incidence[:node]] + edge_idx = edge_dict[incidence[:edge]] + weight = get(incidence, :weight, one(T)) + hg[node_idx, edge_idx] = T(weight) + end +end + +function _build_attr_dataframe(data::Dict{Symbol, Any}, field::Symbol, V::Union{Type, Symbol}, + add_original_id_to_meta::Union{Symbol, Nothing}) + @assert field ∈ (:nodes, :edges) + fid = Symbol(string(field)[1:end-1]) # :node or :edge + + target_attr_type = Union{Nothing, Any} + if V != :auto + if isnothing(add_original_id_to_meta) + target_attr_type = Union{Nothing, V} + else + target_attr_type = Union{Nothing, Dict{Symbol, Union{V, Int, String}}} + end + end + items = DataFrame(; + id=Union{String, Int}[], + attrs= target_attr_type[] + ) + if !haskey(data, field) + return items + end + seen = Set{Union{Int, String}}() + for item in data[field] + id = item[fid] + if id ∈ seen + continue + end + val = get(item, :attrs, nothing) + if V == String && val !== nothing && !(isa(val, String)) + val = JSON.json(val) + end + if isnothing(add_original_id_to_meta) + if val !== nothing && V != :auto + val = convert(V, val) + end + else + if isnothing(val) + val = Dict(add_original_id_to_meta => id) + elseif val isa AbstractDict && ismutable(val) + val[add_original_id_to_meta] = id + else + val = Dict{Symbol, Union{typeof(val), typeof(id)}}( + add_original_id_to_meta => id, :value => val + ) + end + end + push!(items, [id, val]) + push!(seen, id) + end + items +end + +function _add_id_sort_column!(items::DataFrame) + if any(x -> x isa String, items.id) + items.id_sort = string.(items.id) + else + items.id_sort = Int.(items.id) + end +end + +""" + _sanitize_types_items!(items::DataFrame) +Narrow the type of the `attrs` column in the provided `items` DataFrame +THIS FUNCTION IS INTERNAL AND SHOULD NOT BE CALLED DIRECTLY. +""" +function _sanitize_types_items!(items::DataFrame) + types = unique!(typeof.(items.attrs)) + if length(types) <= 5 + vals = Union{Nothing, types...}[] + items.attrs = append!(vals, items.attrs) + end +end + +""" + _add_nodes_and_edges_from_incidences!(data::Dict{String, Any}, nodes::DataFrame, edges::DataFrame) + +THIS FUNCTION IS INTERNAL AND SHOULD NOT BE CALLED DIRECTLY. +Adds nodes and edges to the provided `nodes` and `edges` DataFrames +based on the incidences provided in `data`. +""" +function _add_nodes_and_edges_from_incidences!(data::Dict{Symbol, Any}, nodes::DataFrame, edges::DataFrame, add_original_id_to_meta::Union{Symbol, Nothing}) + seen_node_ids = Set{Union{String, Int}}(nodes.id) + seen_edge_ids = Set{Union{String, Int}}(edges.id) + for incidence in data[:incidences] + node = incidence[:node] + edge = incidence[:edge] + if node ∉ seen_node_ids + if isnothing(add_original_id_to_meta) + push!(nodes, [node, nothing]) + else + push!(nodes, [node, Dict(add_original_id_to_meta => node)]) + end + push!(seen_node_ids, node) + end + if edge ∉ seen_edge_ids + if isnothing(add_original_id_to_meta) + push!(edges, [edge, nothing]) + else + push!(edges, [edge, Dict(add_original_id_to_meta => edge)]) + end + push!(seen_edge_ids, edge) + end + end +end + + +""" + hg_save(io::IO, h::Hypergraph, format::HIF_Format; pretty::Bool=false) + +Saves a hypergraph `h` to an output stream `io` in `HIF` format. + +If `h` has `Composite Types` either for vertex metadata or hyperedges metadata, +TODO: complete this part + +""" +function hg_save(io::IO, h::Hypergraph{T, V, E, D}, ::HIF_Format; pretty::Bool=false) where {T, V, E, D} + incidences = Vector{OrderedDict{Symbol, Union{Int, T}}}() + for i in 1:nhv(h) + for j in sort!(collect(keys(gethyperedges(h, i)))) + push!(incidences, OrderedDict{Symbol, Union{Int, T}}(:node => i, :edge => j, :weight => T(h[i, j]))) + end + end + #decide whether to include metadata for nodes and edges + #there are two poossible reasons to include metadata: + #1. there is at least one metadata entry + #2. there is at least one node or edge with no connections (isolated vertex or empty hyperedge) + node_meta_included = any(x -> !(isnothing(x)), h.v_meta ) || any(v -> isempty(gethyperedges(h, v)), 1:nhv(h)) + edge_meta_included = any(x -> !(isnothing(x)), h.he_meta) || any(e -> isempty(getvertices(h, e)), 1:nhe(h)) + + json_node_meta = Vector{OrderedDict{Symbol, Any}}() + json_edge_meta = Vector{OrderedDict{Symbol, Any}}() + + if node_meta_included + for i in 1:nhv(h) + node_entry = OrderedDict{Symbol, Union{Int, typeof(h.v_meta[i])}}(:node => i) + if !(isnothing(h.v_meta[i])) + node_entry[:attrs] = h.v_meta[i] + end + push!(json_node_meta, node_entry) + end + end + if edge_meta_included + for j in 1:nhe(h) + edge_entry = OrderedDict{Symbol, Union{Int, typeof(h.he_meta[j])}}(:edge => j) + if !(isnothing(h.he_meta[j])) + edge_entry[:attrs] = h.he_meta[j] + end + push!(json_edge_meta, edge_entry) + end + end + json_hg = OrderedDict{Symbol, Union{typeof(incidences), typeof(json_node_meta), typeof(json_edge_meta)}}() + json_hg[:incidences] = incidences + if length(json_node_meta) > 0 + json_hg[:nodes] = json_node_meta + end + if length(json_edge_meta) > 0 + json_hg[:edges] = json_edge_meta + end + JSON.json(io, json_hg; pretty) +end diff --git a/src/viz/drawing.jl b/src/viz/drawing.jl index 607710f..0e47108 100644 --- a/src/viz/drawing.jl +++ b/src/viz/drawing.jl @@ -106,8 +106,11 @@ function draw( prune_hypergraph!(_h) w = widget_graph( - JSON3.write(_h.v2he), - JSON3.write(_h.he2v), + # JSON.json(_h.v2he), + # JSON.json(_h.he2v), + # TODO: check if upgrade to JSON.jl works + JSON.json(_h.v2he), + JSON.json(_h.he2v), element; v_meta=_h.v_meta, he_meta=_h.he_meta, diff --git a/src/viz/widget.jl b/src/viz/widget.jl index 9917736..fff712f 100644 --- a/src/viz/widget.jl +++ b/src/viz/widget.jl @@ -51,36 +51,36 @@ function widget_graph( $(v2he), $(he2v), "div$(div_id)", - vmeta=$(JSON3.write(v_meta)), - hemeta=$(JSON3.write(he_meta)), - width=$(JSON3.write(width)), - height=$(JSON3.write(height)), + vmeta=$(JSON.json(v_meta)), + hemeta=$(JSON.json(he_meta)), + width=$(JSON.json(width)), + height=$(JSON.json(height)), strength=-60, linkDistance=40, linkStrength=1, theta=0.8, - radius=$(JSON3.write(radius)), - nodeRadii=$(JSON3.write(node_radii)), - nodeColor=$(JSON3.write(node_color)), - nodeColors=$(JSON3.write(node_colors)), - nodeStroke=$(JSON3.write(node_stroke)), - nodeStrokes=$(JSON3.write(node_strokes)), - strokeWidth=$(JSON3.write(stroke_width)), - strokeWidths=$(JSON3.write(stroke_widths)), - nodeOpacity=$(JSON3.write(node_opacity)), - nodeOpacities=$(JSON3.write(node_opacities)), - strokeOpacity=$(JSON3.write(stroke_opacity)), - strokeOpacities=$(JSON3.write(stroke_opacities)), - withNodeLabels=$(JSON3.write(with_node_labels)), - nodeLabels=$(JSON3.write(node_labels)), + radius=$(JSON.json(radius)), + nodeRadii=$(JSON.json(node_radii)), + nodeColor=$(JSON.json(node_color)), + nodeColors=$(JSON.json(node_colors)), + nodeStroke=$(JSON.json(node_stroke)), + nodeStrokes=$(JSON.json(node_strokes)), + strokeWidth=$(JSON.json(stroke_width)), + strokeWidths=$(JSON.json(stroke_widths)), + nodeOpacity=$(JSON.json(node_opacity)), + nodeOpacities=$(JSON.json(node_opacities)), + strokeOpacity=$(JSON.json(stroke_opacity)), + strokeOpacities=$(JSON.json(stroke_opacities)), + withNodeLabels=$(JSON.json(with_node_labels)), + nodeLabels=$(JSON.json(node_labels)), nodeLabelsAttr=null, nodeLabelsStyle=null, - withNodeMetadataOnHover=$(JSON3.write(with_node_metadata_hover)), - withNodeWeight=$(JSON3.write(with_node_weight)), - edgeColors=$(JSON3.write(he_colors)), - withEdgeLabels=$(JSON3.write(with_he_labels)), - edgeLabels=$(JSON3.write(he_labels)), - withHyperedgesMetadataOnHover=$(JSON3.write(with_he_metadata_hover)) + withNodeMetadataOnHover=$(JSON.json(with_node_metadata_hover)), + withNodeWeight=$(JSON.json(with_node_weight)), + edgeColors=$(JSON.json(he_colors)), + withEdgeLabels=$(JSON.json(with_he_labels)), + edgeLabels=$(JSON.json(he_labels)), + withHyperedgesMetadataOnHover=$(JSON.json(with_he_metadata_hover)) ); diff --git a/test/data/HIF-standard/duplicated_nodes_edges.json b/test/data/HIF-standard/duplicated_nodes_edges.json new file mode 100644 index 0000000..a02124e --- /dev/null +++ b/test/data/HIF-standard/duplicated_nodes_edges.json @@ -0,0 +1,7 @@ +{ + "network-type": "undirected", + "metadata": {}, + "nodes": [{"node": "n1"}, {"node": "n1"}], + "edges": [{"edge": "e1"}, {"edge": "e1"}], + "incidences": [{"edge": "e1", "node": "n1"}, {"edge": "e1", "node": "n1"}] +} \ No newline at end of file diff --git a/test/data/HIF-standard/empty_arrays.json b/test/data/HIF-standard/empty_arrays.json new file mode 100644 index 0000000..7b0ce2d --- /dev/null +++ b/test/data/HIF-standard/empty_arrays.json @@ -0,0 +1,7 @@ +{ + "network-type": "undirected", + "metadata": {}, + "incidences": [], + "nodes": [], + "edges": [] +} \ No newline at end of file diff --git a/test/data/HIF-standard/empty_hypergraph.json b/test/data/HIF-standard/empty_hypergraph.json new file mode 100644 index 0000000..7a65310 --- /dev/null +++ b/test/data/HIF-standard/empty_hypergraph.json @@ -0,0 +1,3 @@ +{ + "incidences": [] +} \ No newline at end of file diff --git a/test/data/HIF-standard/metadata_with_deeply_nested_attributes.json b/test/data/HIF-standard/metadata_with_deeply_nested_attributes.json new file mode 100644 index 0000000..bd78510 --- /dev/null +++ b/test/data/HIF-standard/metadata_with_deeply_nested_attributes.json @@ -0,0 +1,15 @@ +{ + "network-type": "asc", + "metadata": { + "level1": { + "level2": { + "level3": { + "key": "value" + } + } + } + }, + "incidences": [{"edge": 1, "node": 2}], + "nodes": [{"node": "n1", "attrs": {"nested_attr": {"key1": "value1"}}}], + "edges": [{"edge": "e1", "attrs": {"nested_attr": {"key2": "value2"}}}] +} \ No newline at end of file diff --git a/test/data/HIF-standard/metadata_with_nested_attributes.json b/test/data/HIF-standard/metadata_with_nested_attributes.json new file mode 100644 index 0000000..697be84 --- /dev/null +++ b/test/data/HIF-standard/metadata_with_nested_attributes.json @@ -0,0 +1,13 @@ +{ + "network-type": "asc", + "metadata": { + "creator": "nested_test", + "extra_info": { + "key1": "value1", + "key2": "value2" + } + }, + "incidences": [{"edge": 10, "node": 20}], + "nodes": [{"node": 20, "attrs": {"color": "blue", "size": "large"}}], + "edges": [{"edge": 10, "attrs": {"priority": "high"}}] +} \ No newline at end of file diff --git a/test/data/HIF-standard/missing_direction.json b/test/data/HIF-standard/missing_direction.json new file mode 100644 index 0000000..f0a4be6 --- /dev/null +++ b/test/data/HIF-standard/missing_direction.json @@ -0,0 +1,5 @@ +{ + "network-type": "directed", + "metadata": {}, + "incidences": [{"edge": 1, "node": 2}] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_edge.json b/test/data/HIF-standard/single_edge.json new file mode 100644 index 0000000..6688947 --- /dev/null +++ b/test/data/HIF-standard/single_edge.json @@ -0,0 +1,8 @@ +{ + "incidences": [], + "edges": [ + { + "edge": 3 + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_edge_with_attrs.json b/test/data/HIF-standard/single_edge_with_attrs.json new file mode 100644 index 0000000..4e5871e --- /dev/null +++ b/test/data/HIF-standard/single_edge_with_attrs.json @@ -0,0 +1,12 @@ +{ + "incidences": [], + "edges": [ + { + "edge": 3, + "attrs": { + "timestamp": "2020-04-01", + "weight": 2.0 + } + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_incidence.json b/test/data/HIF-standard/single_incidence.json new file mode 100644 index 0000000..1453273 --- /dev/null +++ b/test/data/HIF-standard/single_incidence.json @@ -0,0 +1,8 @@ +{ + "incidences": [ + { + "edge": "abcd", + "node": 42 + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_incidence_with_attrs.json b/test/data/HIF-standard/single_incidence_with_attrs.json new file mode 100644 index 0000000..5d5dece --- /dev/null +++ b/test/data/HIF-standard/single_incidence_with_attrs.json @@ -0,0 +1,12 @@ +{ + "incidences": [ + { + "edge": "abcd", + "node": 42, + "attrs": { + "role": "PI", + "age": 42 + } + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_incidence_with_weights.json b/test/data/HIF-standard/single_incidence_with_weights.json new file mode 100644 index 0000000..52cb6a4 --- /dev/null +++ b/test/data/HIF-standard/single_incidence_with_weights.json @@ -0,0 +1,9 @@ +{ + "incidences": [ + { + "edge": "abcd", + "node": 42, + "weight": -2 + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_node.json b/test/data/HIF-standard/single_node.json new file mode 100644 index 0000000..54987b2 --- /dev/null +++ b/test/data/HIF-standard/single_node.json @@ -0,0 +1,8 @@ +{ + "incidences": [], + "nodes": [ + { + "node": 42 + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/single_node_with_attrs.json b/test/data/HIF-standard/single_node_with_attrs.json new file mode 100644 index 0000000..efb03ef --- /dev/null +++ b/test/data/HIF-standard/single_node_with_attrs.json @@ -0,0 +1,13 @@ +{ + "incidences": [], + "nodes": [ + { + "node": 42, + "attrs": { + "weight": 2, + "color": "blue", + "online": true + } + } + ] +} \ No newline at end of file diff --git a/test/data/HIF-standard/valid_incidence_head.json b/test/data/HIF-standard/valid_incidence_head.json new file mode 100644 index 0000000..ed554cc --- /dev/null +++ b/test/data/HIF-standard/valid_incidence_head.json @@ -0,0 +1,5 @@ +{ + "network-type": "directed", + "metadata": {}, + "incidences": [{"edge": 1, "node": 2, "direction": "head"}] +} \ No newline at end of file diff --git a/test/data/HIF-standard/valid_incidence_tail.json b/test/data/HIF-standard/valid_incidence_tail.json new file mode 100644 index 0000000..3aa3bb2 --- /dev/null +++ b/test/data/HIF-standard/valid_incidence_tail.json @@ -0,0 +1,5 @@ +{ + "network-type": "directed", + "metadata": {}, + "incidences": [{"edge": 1, "node": 2, "direction": "tail"}] +} \ No newline at end of file diff --git a/test/data/test1.hgf b/test/data/hgf/test1.hgf similarity index 100% rename from test/data/test1.hgf rename to test/data/hgf/test1.hgf diff --git a/test/data/test_argumenterror.hgf b/test/data/hgf/test_argumenterror.hgf similarity index 100% rename from test/data/test_argumenterror.hgf rename to test/data/hgf/test_argumenterror.hgf diff --git a/test/data/test_malformedcomment.hgf b/test/data/hgf/test_malformedcomment.hgf similarity index 100% rename from test/data/test_malformedcomment.hgf rename to test/data/hgf/test_malformedcomment.hgf diff --git a/test/data/test_multiplelinescomment.hgf b/test/data/hgf/test_multiplelinescomment.hgf similarity index 100% rename from test/data/test_multiplelinescomment.hgf rename to test/data/hgf/test_multiplelinescomment.hgf diff --git a/test/data/test_singlelinecomment.hgf b/test/data/hgf/test_singlelinecomment.hgf similarity index 100% rename from test/data/test_singlelinecomment.hgf rename to test/data/hgf/test_singlelinecomment.hgf diff --git a/test/hif-standard-tests.jl b/test/hif-standard-tests.jl new file mode 100644 index 0000000..4953fb6 --- /dev/null +++ b/test/hif-standard-tests.jl @@ -0,0 +1,86 @@ +using Test +using StatsBase +using Random +using DataStructures +import Graphs +using SimpleHypergraphs + + +@testset "HIF load-save basic tests" begin + h1 = Hypergraph{Float64, Int, String}(5,4) + h1[1:3,1] .= 1.5 + h1[3,4] = 2.5 + h1[2,3] = 3.5 + h1[4,3:4] .= 4.5 + h1[5,4] = 5.5 + h1[5,2] = 6.5 + + path1, _ = mktemp() + hg_save(path1, h1; format=HIF_Format(), pretty=true) + + loaded_hg = hg_load(path1; format=HIF_Format(), HType=Hypergraph, T=Float64, V=String, E=String) + + @test loaded_hg == h1 + + hg = Hypergraph{Int, String, String}( + [1 nothing 2 nothing; + 3 1 nothing 4]) + + set_vertex_meta!(hg, "vertex 1", 1) + set_hyperedge_meta!(hg, "h-edge 2", 2) + hg_save(path1, hg; format=HIF_Format(), pretty=true) + loaded_hg = hg_load( + path1; + format=HIF_Format(), + HType=Hypergraph, + T=Int, V=String, E=String + ) + loaded_hg_auto = hg_load( + path1; + format=HIF_Format(), + HType=Hypergraph, + T=Int, V=:auto, E=:auto + ) + @test hg == loaded_hg == loaded_hg_auto + @test hg.v_meta == loaded_hg.v_meta == loaded_hg_auto.v_meta + @test hg.he_meta == loaded_hg.he_meta == loaded_hg_auto.he_meta +end +@testset "HIF load-save on HIF standard files" begin + # note SimpleHypergraphs.jl uses 1-based indexing so node and edge ids are regenerated accordingly + dir = "data/HIF-standard" + files = filter!(endswith(".json"), readdir(dir)) + for file in files + full_path = joinpath(dir, file) + println("Testing HIF file: $full_path") + flush(stdout) + flush(stderr) + h = hg_load(full_path; format=HIF_Format(), T=Int, show_warning=false); + io_h = IOBuffer() + hg_save(io_h, h, HIF_Format(); pretty=true) + h_loaded = hg_load(seekstart(io_h), HIF_Format(); T=Int, show_warning=true) + @test h == h_loaded + h2 = hg_load(full_path; format=HIF_Format(), T=Int, show_warning=false, add_original_id_to_meta=:id); + io_h2 = IOBuffer() + hg_save(io_h2, h2, HIF_Format(); pretty=true) + h_loaded2 = hg_load(seekstart(io_h2), HIF_Format(); T=Int, show_warning=true, add_original_id_to_meta=nothing) + @test h2 == h_loaded2 == h + @test h2.v_meta == h_loaded2.v_meta + @test h2.he_meta == h_loaded2.he_meta + end +end + +#== +using Revise +using SimpleHypergraphs +cd("test") +full_path = "data/HIF-standard/missing_direction.json" +run(`cat $full_path`); +h2 = hg_load(full_path; format=HIF_Format(), T=Int, show_warning=false, add_original_id_to_meta=:id); +io_h2 = IOBuffer() +hg_save(io_h2, h2, HIF_Format(); pretty=true) +println(String(read(seekstart(io_h2)))) +h_loaded2 = hg_load(seekstart(io_h2), HIF_Format(); T=Int, show_warning=true, add_original_id_to_meta=nothing) +h2 == h_loaded2 +h2.v_meta == h_loaded2.v_meta +h2.he_meta == h_loaded2.he_meta +==# \ No newline at end of file diff --git a/test/runtests.jl b/test/runtests.jl index 77dec82..9c69a51 100644 --- a/test/runtests.jl +++ b/test/runtests.jl @@ -7,6 +7,8 @@ using Random using DataStructures import Graphs +#Test the HIF standard support +include("hif-standard-tests.jl") h1 = Hypergraph{Float64, Int, String}(5,4) h1[1:3,1] .= 1.5 @@ -16,10 +18,14 @@ h1[4,3:4] .= 4.5 h1[5,4] = 5.5 h1[5,2] = 6.5 +#if !endswith(pwd(),"test") +# cd(@__DIR__) +#end +#h = hg_load("data/hgf/test1.hgf"; T=Int, HType=Hypergraph) @testset "SimpleHypergraphs Hypergraph " begin - h = hg_load("data/test1.hgf"; T=Int, HType=Hypergraph) + h = hg_load("data/hgf/test1.hgf"; T=Int, HType=Hypergraph) @test size(h) == (4, 4) @test nhv(h) == 4 @test nhe(h) == 4 @@ -29,55 +35,59 @@ h1[5,2] = 6.5 2 3 nothing nothing nothing nothing 5 nothing nothing nothing 6 nothing] - mktemp("data") do path, _ - println(path) - hg_save(path, h) - - loaded_hg = replace(read(path, String), r"\n*$" => "") - - @test loaded_hg == - reduce(replace, - ["\r\n"=>"\n", - r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments - r"\n*$"=>""], #remove final \n* - init=read("data/test1.hgf", String)) #no comments - - @test loaded_hg == - reduce(replace, - ["\r\n"=>"\n", - r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments - r"\n*$"=>""], #remove final \n* - init=read("data/test_singlelinecomment.hgf", String)) #single line comment - - @test loaded_hg == - reduce(replace, - ["\r\n"=>"\n", - r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments - r"\n*$"=>""], #remove final \n* - init=read("data/test_multiplelinescomment.hgf", String)) #multiple lines comment - - for v=1:nhv(h) - set_vertex_meta!(h1, v, v) - end + path1, _ = mktemp("data") + path2, _ = mktemp("data") + + hg_save(path1, h) + + loaded_hg = replace(read(path1, String), r"\n*$" => "") + + @test loaded_hg == + reduce(replace, + ["\r\n"=>"\n", + r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments + r"\n*$"=>""], #remove final \n* + init=read("data/hgf/test1.hgf", String)) #no comments + + @test loaded_hg == + reduce(replace, + ["\r\n"=>"\n", + r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments + r"\n*$"=>""], #remove final \n* + init=read("data/hgf/test_singlelinecomment.hgf", String)) #single line comment + + @test loaded_hg == + reduce(replace, + ["\r\n"=>"\n", + r"^\"\"\"(?s).*\"\"\"\n"=>"", #remove initial comments + r"\n*$"=>""], #remove final \n* + init=read("data/hgf/test_multiplelinescomment.hgf", String)) #multiple lines comment + + for v=1:nhv(h) + set_vertex_meta!(h1, v, v) + end - for he=1:nhe(h) - set_hyperedge_meta!(h1, string(he), he) - end + for he=1:nhe(h) + set_hyperedge_meta!(h1, string(he), he) + end - hg_save(path, h1; format=JSON_Format()) - loaded_hg = hg_load(path; format=JSON_Format(), HType=Hypergraph, T=Float64, V=Int, E=String) + hg_save(path1, h1; format=JSON_Format(), pretty=false) + hg_save(path2, h1; format=JSON_Format(), pretty=true) - @test h1 == loaded_hg - @test h1.v_meta == loaded_hg.v_meta - @test h1.he_meta == loaded_hg.he_meta + loaded_hg = hg_load(path1; format=JSON_Format(), HType=Hypergraph, T=Float64, V=Int, E=String) + loaded_hg2 = hg_load(path2; format=JSON_Format(), HType=Hypergraph, T=Float64, V=Int, E=String) - @test get_vertex_meta(h1, 1) == get_vertex_meta(loaded_hg, 1) - @test get_hyperedge_meta(h1, 2) == get_hyperedge_meta(loaded_hg, 2) - end + @test h1 == loaded_hg == loaded_hg2 + @test h1.v_meta == loaded_hg.v_meta == loaded_hg2.v_meta + @test h1.he_meta == loaded_hg.he_meta == loaded_hg2.he_meta + + @test get_vertex_meta(h1, 1) == get_vertex_meta(loaded_hg, 1) == get_vertex_meta(loaded_hg2, 1) + @test get_hyperedge_meta(h1, 2) == get_hyperedge_meta(loaded_hg, 2) == get_hyperedge_meta(loaded_hg2, 2) - @test_throws ArgumentError hg_load("data/test_malformedcomment.hgf"; T=Int) - @test_throws ArgumentError hg_load("data/test_argumenterror.hgf"; T=Int) + + @test_throws ArgumentError hg_load("data/hgf/test_malformedcomment.hgf"; T=Int) + @test_throws ArgumentError hg_load("data/hgf/test_argumenterror.hgf"; T=Int) h2 = Hypergraph{Float64}(0,0) @test h2 == Hypergraph{Float64,Nothing}(0,0) @@ -631,3 +641,4 @@ end; @test Graphs.diameter(h, SnodeDistanceDijkstra(1,1,2)) == typemax(Int) @test Graphs.diameter(h, SedgeDistanceDijkstra(1,1,1)) == 1 end; + diff --git a/tutorials/hif-standard/HIF-SimpleHypergraphs-demo.ipynb b/tutorials/hif-standard/HIF-SimpleHypergraphs-demo.ipynb new file mode 100644 index 0000000..5053a27 --- /dev/null +++ b/tutorials/hif-standard/HIF-SimpleHypergraphs-demo.ipynb @@ -0,0 +1,225 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 2, + "id": "aedb3468-cb90-4838-b79f-ba96b65991f0", + "metadata": {}, + "outputs": [], + "source": [ + "using SimpleHypergraphs" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "cc7cc7d8-9ba1-4bf4-8633-1d9f6e67b1df", + "metadata": {}, + "outputs": [], + "source": [ + "using Makie\n", + "using GLMakie\n", + "using NetworkLayout\n", + "using Random\n", + "using Colors\n", + "using ColorSchemes\n", + "using Graphs\n", + "using GraphMakie" + ] + }, + { + "cell_type": "code", + "execution_count": 10, + "id": "3592b678-84b2-46e5-b1d2-c06b1bde06ec", + "metadata": {}, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mNodes in the source file were not sorted or not consistent - their order will change\n", + "\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ SimpleHypergraphs /Users/Shared/AAABIBLIOTEKA/NCN2/SimpleHypergraphs.jl/src/io_hif.jl:95\u001b[39m\n", + "\u001b[33m\u001b[1m┌ \u001b[22m\u001b[39m\u001b[33m\u001b[1mWarning: \u001b[22m\u001b[39mEdges in the source file were not sorted or not consistent - their order will change\n", + "\u001b[33m\u001b[1m└ \u001b[22m\u001b[39m\u001b[90m@ SimpleHypergraphs /Users/Shared/AAABIBLIOTEKA/NCN2/SimpleHypergraphs.jl/src/io_hif.jl:99\u001b[39m\n" + ] + }, + { + "data": { + "text/plain": [ + "1960×533 Hypergraph{Bool, Union{Nothing, Dict{Symbol, Any}}, Union{Nothing, Dict{Symbol, Any}}, Dict{Int64, Bool}}:\n", + " 1 nothing nothing nothing … nothing nothing nothing\n", + " nothing 1 nothing nothing nothing nothing nothing\n", + " nothing 1 nothing nothing nothing nothing nothing\n", + " nothing 1 nothing nothing nothing nothing nothing\n", + " nothing nothing 1 1 nothing nothing nothing\n", + " nothing nothing 1 1 … nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing … nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " ⋮ ⋱ ⋮ \n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing … nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing … nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing nothing nothing nothing\n", + " nothing nothing nothing nothing 1 nothing nothing\n", + " nothing nothing nothing nothing nothing nothing 1" + ] + }, + "execution_count": 10, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "file_path = \"data/publications.hif.json\" # Your specified file path\n", + "# note that SimpleHypergraphs.jl uses 1-based numbering of vertices and hyperedges.\n", + "# Hence renumbering is required when outher ids are used in the input json\n", + "# We can however keep the original id in the vertex/edge metadata using the `add_original_id_to_meta` parameter\n", + "hg = hg_load(file_path; format=HIF_Format(), T=Bool, add_original_id_to_meta=:origid)" + ] + }, + { + "cell_type": "code", + "execution_count": 16, + "id": "42b0c1ef-1400-4180-98b3-828923813498", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "Dict{Symbol, Any} with 6 entries:\n", + " :funding_agencies => Any[]\n", + " :origid => \"Asymptotically sharp bounds for cancellative and union-…\n", + " :abstract => \"An \\$r\\$-graph is called \\$t\\$-cancellative if for arbi…\n", + " :date => \"2024-11-12\"\n", + " :tags => Any[\"Combinatorics\"]\n", + " :source => \"Arxiv\"" + ] + }, + "execution_count": 16, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "# it can be seen that the title was used as the hyperedge identifier. Similary, author name was used as the vertex identifier\n", + "hg.he_meta[end]" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "id": "3c88c705-71ec-4fb4-884e-bb536fc3f013", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "We found 16 communities \n" + ] + } + ], + "source": [ + "cmpts = get_connected_components(hg)\n", + "n, id = findmax(length, cmpts)\n", + "to_select = Set(cmpts[id])\n", + "for v in nhv(hg):-1:1\n", + " if !(v in to_select)\n", + " remove_vertex!(hg, v)\n", + " end\n", + "end\n", + "SimpleHypergraphs.prune_hypergraph(hg)\n", + "\n", + "cnm = CFModularityCNMLike(5000)\n", + "Random.seed!(1234)\n", + "cnm_comms = findcommunities(hg, cnm)\n", + "\n", + "println(\"We found $(length(cnm_comms.bp)) communities \")\n", + "\n", + "t = Graphs.Graph(get_twosection_adjacency_mx(hg;replace_weights=1))\n", + "\n", + "my_colors = vcat(ColorSchemes.rainbow[range(1, stop=length(ColorSchemes.rainbow), step=3)], ColorSchemes.rainbow[2]);\n", + "function get_color(i, comms, colors)\n", + " for j in 1:length(comms)\n", + " if length(comms[j]) > 1 && i in comms[j]\n", + " return \"#\"*hex(colors[j % length(colors) + 1])\n", + " end\n", + " end\n", + " return \"#000000\"\n", + "end;\n", + "\n", + "degrees = Graphs.degree.(Ref(t), Graphs.vertices(t));\n", + "dsize = 7 .+ 12 .* degrees./maximum(degrees);\n" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "id": "3e173ec8-f027-49c3-a160-138993a544fc", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/html": [ + "" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "fig = Figure(size = (600, 600))\n", + "ax = Axis(fig[1, 1]; aspect = DataAspect())\n", + "hidedecorations!(ax)\n", + "hidespines!(ax)\n", + "graphplot!(t;\n", + " node_size = dsize,\n", + " node_color = get_color.(1:Graphs.nv(t), Ref(cnm_comms.bp), Ref(reverse(my_colors))),\n", + " edge_width = 0.5, \n", + " edge_color = RGBA(0.5, 0.5, 0.5, 0.4) \n", + ")\n", + "\n", + "fig\n" + ] + }, + { + "cell_type": "markdown", + "id": "5aedd67c-8434-4a67-a198-c51c4f5161ec", + "metadata": {}, + "source": [ + "#### Akcnowledgments\n", + "Development of HIF-standard integration along with this tutorial was funded by the National\n", + "Science Centre (NCN), Poland (grant number: 2021/41/B/HS4/03349)." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Julia 1.12", + "language": "julia", + "name": "julia-1.12" + }, + "language_info": { + "file_extension": ".jl", + "mimetype": "application/julia", + "name": "julia", + "version": "1.12.3" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/tutorials/hif-standard/data/publications.hif.json b/tutorials/hif-standard/data/publications.hif.json new file mode 100644 index 0000000..357fb8d --- /dev/null +++ b/tutorials/hif-standard/data/publications.hif.json @@ -0,0 +1 @@ +{"edges": [{"edge": "Hypergraph: A Unified and Uniform Definition with Application to Chemical Hypergraph and More", "weight": 1, "attrs": {"tags": ["Machine Learning", "Quantitative Methods"], "abstract": "The conventional definition of hypergraph has two major issues: (1) there is not a standard definition of directed hypergraph and (2) there is not a formal definition of nested hypergraph. To resolve these issues, we propose a new definition of hypergraph that unifies the concepts of undirected, directed and nested hypergraphs, and that is uniform in using hyperedge as a single construct for representing high-order correlations among things, i.e., nodes and hyperedges. Specifically, we define a hyperedge to be a simple hyperedge, a nesting hyperedge, or a directed hyperedge. With this new definition, a hypergraph is nested if it has nesting hyperedge(s), and is directed if it has directed hyperedge(s). Otherwise, a hypergraph is a simple hypergraph. The uniformity and power of this new definition, with visualization, should facilitate the use of hypergraph for representing (hierarchical) high-order correlations in general and chemical systems in particular. Graph has been widely used as a mathematical structure for machine learning on molecular structures and 3D molecular geometries. However, graph has a major limitation: it can represent only pairwise correlations between nodes. Hypergraph extends graph with high-order correlations among nodes. This extension is significant or essential for machine learning on chemical systems. For molecules, this is significant as it allows the direct, explicit representation of multicenter bonds and molecular substructures. For chemical reactions, this is essential since most chemical reactions involve multiple participants. We propose the use of chemical hypergraph, a multilevel hypergraph with simple, nesting and directed hyperedges, as a single mathematical structure for representing chemical systems. We apply the new definition of hypergraph to chemical hypergraph and, as simplified versions, molecular hypergraph and chemical reaction hypergraph.", "source": "Arxiv", "date": "2024-05-14", "funding_agencies": []}}, {"edge": "Hypergraph C*-algebras", "weight": 1, "attrs": {"tags": ["Operator Algebras"], "abstract": "We give a definition of hypergraph C*-algebras. These generalize the well-known graph C*-algebras as well as ultragraph C*-algebras. In contrast to those objects, hypergraph C*-algebras are not always nuclear. We provide a number of non-nuclear examples, we prove a Gauge-Invariant Uniqueness Theorem for a subclass of hypergraph C*-algebras and we study moves on hypergraphs which generalize the moves in the theory of graph C*-algebras.", "source": "Arxiv", "date": "2024-05-15", "funding_agencies": []}}, {"edge": "New matrices for spectral hypergraph theory, I", "weight": 1, "attrs": {"tags": ["Combinatorics", "Graphs and matrices", "Hypergraphs", "Eigenvalues, singular values, and eigenvectors"], "abstract": "We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to use the spectrum of the unified matrix of a hypergraph as a tool to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of the unified matrix.", "source": "Arxiv", "date": "2024-11-11", "funding_agencies": []}}, {"edge": "New matrices for spectral hypergraph theory, II", "weight": 1, "attrs": {"tags": ["Combinatorics", "Spectral Theory", "Graphs and matrices", "Hypergraphs", "Eigenvalues, singular values, and eigenvectors"], "abstract": "The properties of a hypergraph explored through the spectrum of its unified matrix was made by the authors in [26]. In this paper, we introduce three different hypergraph matrices: unified Laplacian matrix, unified signless Laplacian matrix, and unified normalized Laplacian matrix, all defined using the unified matrix. We show that these three matrices of a hypergraph are respectively identical to the Laplacian matrix, signless Laplacian matrix, and normalized Laplacian matrix of the associated graph. This allows us to use the spectra of these hypergraph matrices as a means to connect the structural properties of the hypergraph with those of the associated graph. Additionally, we introduce certain hypergraph structures and invariants during this process, and relate them to the eigenvalues of these three matrices.", "source": "Arxiv", "date": "2024-11-12", "funding_agencies": []}}, {"edge": "Hyperedge Interaction-aware Hypergraph Neural Network", "weight": 1, "attrs": {"tags": ["Machine Learning", "Social and Information Networks"], "abstract": "Hypergraphs provide an effective modeling approach for modeling high-order relationships in many real-world datasets. To capture such complex relationships, several hypergraph neural networks have been proposed for learning hypergraph structure, which propagate information from nodes to hyperedges and then from hyperedges back to nodes. However, most existing methods focus on information propagation between hyperedges and nodes, neglecting the interactions among hyperedges themselves. In this paper, we propose HeIHNN, a hyperedge interaction-aware hypergraph neural network, which captures the interactions among hyperedges during the convolution process and introduce a novel mechanism to enhance information flow between hyperedges and nodes. Specifically, HeIHNN integrates the interactions between hyperedges into the hypergraph convolution by constructing a three-stage information propagation process. After propagating information from nodes to hyperedges, we introduce a hyperedge-level convolution to update the hyperedge embeddings. Finally, the embeddings that capture rich information from the interaction among hyperedges will be utilized to update the node embeddings. Additionally, we introduce a hyperedge outlier removal mechanism in the information propagation stages between nodes and hyperedges, which dynamically adjusts the hypergraph structure using the learned embeddings, effectively removing outliers. Extensive experiments conducted on real-world datasets show the competitive performance of HeIHNN compared with state-of-the-art methods.", "source": "Arxiv", "date": "2024-01-28", "funding_agencies": []}}, {"edge": "Nuclearity of Hypergraph C*-Algebras", "weight": 1, "attrs": {"tags": ["Operator Algebras"], "abstract": "We partially characterize nuclearity for the recently introduced class of hypergraph C*-algebras using a tailor-made hypergraph minor relation. The latter is generated by certain operations on hypergraphs which resemble the moves on directed graphs used by Eilers, Restorff, Ruiz and S{\\o}rensen to classify unital graph C*-algebras. In particular, we obtain a new proof of the fact that every graph C*-algebra associated to a finite graph is nuclear.", "source": "Arxiv", "date": "2024-05-16", "funding_agencies": []}}, {"edge": "Hypergraph Connectivity Augmentation in Strongly Polynomial Time", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms", "Discrete Mathematics"], "abstract": "We consider hypergraph network design problems where the goal is to construct a hypergraph that satisfies certain connectivity requirements. For graph network design problems where the goal is to construct a graph that satisfies certain connectivity requirements, the number of edges in every feasible solution is at most quadratic in the number of vertices. In contrast, for hypergraph network design problems, we might have feasible solutions in which the number of hyperedges is exponential in the number of vertices. This presents an additional technical challenge in hypergraph network design problems compared to graph network design problems: in order to solve the problem in polynomial time, we first need to show that there exists a feasible solution in which the number of hyperedges is polynomial in the input size. The central theme of this work is to show that certain hypergraph network design problems admit solutions in which the number of hyperedges is polynomial in the number of vertices and moreover, can be solved in strongly polynomial time. Our work improves on the previous fastest pseudo-polynomial run-time for these problems. In addition, we develop strongly polynomial time algorithms that return near-uniform hypergraphs as solutions (i.e., every pair of hyperedges differ in size by at most one). As applications of our results, we derive the first strongly polynomial time algorithms for (i) degree-specified hypergraph connectivity augmentation using hyperedges, (ii) degree-specified hypergraph node-to-area connectivity augmentation using hyperedges, and (iii) degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.", "source": "Arxiv", "date": "2024-02-16", "funding_agencies": []}}, {"edge": "SPHINX: Structural Prediction using Hypergraph Inference Network", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "The importance of higher-order relations is widely recognized in a large number of real-world systems. However, annotating them is a tedious and sometimes impossible task. Consequently, current approaches for data modelling either ignore the higher-order interactions altogether or simplify them into pairwise connections. In order to facilitate higher-order processing, even when a hypergraph structure is not available, we introduce Structural Prediction using Hypergraph Inference Network (SPHINX), a model that learns to infer a latent hypergraph structure in an unsupervised way, solely from the final node-level signal. The model consists of a soft, differentiable clustering method used to sequentially predict, for each hyperedge, the probability distribution over the nodes and a sampling algorithm that converts them into an explicit hypergraph structure. We show that the recent advancement in k-subset sampling represents a suitable tool for producing discrete hypergraph structures, addressing some of the training instabilities exhibited by prior works. The resulting model can generate the higher-order structure necessary for any modern hypergraph neural network, facilitating the capture of higher-order interaction in domains where annotating them is difficult. Through extensive ablation studies and experiments conducted on two challenging datasets for trajectory prediction, we demonstrate that our model is capable of inferring suitable latent hypergraphs, that are interpretable and enhance the final performance.", "source": "Arxiv", "date": "2024-10-04", "funding_agencies": []}}, {"edge": "When Joints Meet Extremal Graph Theory: Hypergraph Joints", "weight": 1, "attrs": {"tags": ["Combinatorics", "Classical Analysis and ODEs", "Arrangements of points, flats, hyperplanes", "Probabilistic methods", "Extremal set theory", "Extremal problems", "Inequalities for sums, series and integrals"], "abstract": "The Kruskal--Katona theorem determines the maximum number of $d$-cliques in an $n$-edge $(d-1)$-uniform hypergraph. A generalization of the theorem was proposed by Bollob\\'as and Eccles, called the partial shadow problem. The problem asks to determine the maximum number of $r$-sets of vertices that contain at least $d$ edges in an $n$-edge $(d-1)$-uniform hypergraph. In our previous work, we obtained an asymptotically tight upper bound via its connection to the joints problem, a problem in incidence geometry. In a different direction, Friedgut and Kahn generalized the Kruskal--Katona theorem by determining the maximum number of copies of any fixed hypergraph in an $n$-edge hypergraph, up to a multiplicative factor. In this paper, using the connection to the joints problem again, we generalize our previous work to show an analogous partial shadow phenomenon for any hypergraph, generalizing Friedgut and Kahn's result. The key idea is to encode the graph-theoretic problem with new kinds of joints that we call hypergraph joints. Our main theorem is a generalization of the joints theorem that upper bounds the number of hypergraph joints, which the partial shadow phenomenon immediately follows from. In addition, with an appropriate notion of multiplicities, our theorem also generalizes a generalization of H\\`older's inequality considered by Finner.", "source": "Arxiv", "date": "2024-10-08", "funding_agencies": []}}, {"edge": "Hypergraph Extensions of Spectral Tur\\'an Theorem", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal problems", "Graphs and matrices", "Hypergraphs"], "abstract": "The spectral Tur\\'an theorem states that the $k$-partite Tur\\'an graph is the unique graph attaining the maximum adjacency spectral radius among all graphs of order $n$ containing no the complete graph $K_{k+1}$ as a subgraph. This result is known to be stronger than the classical Tur\\'an theorem. In this paper, we consider hypergraph extensions of spectral Tur\\'an theorem. For $k\\geq r\\geq 2$, let $H_{k+1}^{(r)}$ be the $r$-uniform hypergraph obtained from $K_{k+1}$ by enlarging each edge with a new set of $(r-2)$ vertices. Let $F_{k+1}^{(r)}$ be the $r$-uniform hypergraph with edges: $\\{1,2,\\ldots,r\\} =: [r]$ and $E_{ij} \\cup\\{i,j\\}$ over all pairs $\\{i,j\\}\\in \\binom{[k+1]}{2}\\setminus\\binom{[r]}{2}$, where $E_{ij}$ are pairwise disjoint $(r-2)$-sets disjoint from $[k+1]$. Generalizing the Tur\\'an theorem to hypergraphs, Pikhurko [J. Combin. Theory Ser. B, 103 (2013) 220--225] and Mubayi and Pikhurko [J. Combin. Theory Ser. B, 97 (2007) 669--678] respectively determined the exact Tur\\'an number of $H_{k+1}^{(r)}$ and $F_{k+1}^{(r)}$, and characterized the corresponding extremal hypergraphs. Our main results show that $T_r(n,k)$, the complete $k$-partite $r$-uniform hypergraph on $n$ vertices where no two parts differ by more than one in size, is the unique hypergraph having the maximum $p$-spectral radius among all $n$-vertex $H_{k+1}^{(r)}$-free (resp. $F_{k+1}^{(r)}$-free) $r$-uniform hypergraphs for sufficiently large $n$. These findings are obtained by establishing $p$-spectral version of the stability theorems. Our results offer $p$-spectral analogues of the results by Mubayi and Pikhurko, and connect both hypergraph Tur\\'an theorem and hypergraph spectral Tur\\'an theorem in a unified form via the $p$-spectral radius.", "source": "Arxiv", "date": "2024-08-06", "funding_agencies": []}}, {"edge": "Hypergraph Unreliability in Quasi-Polynomial Time", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "The hypergraph unreliability problem asks for the probability that a hypergraph gets disconnected when every hyperedge fails independently with a given probability. For graphs, the unreliability problem has been studied over many decades, and multiple fully polynomial-time approximation schemes are known starting with the work of Karger (STOC 1995). In contrast, prior to this work, no non-trivial result was known for hypergraphs (of arbitrary rank). In this paper, we give quasi-polynomial time approximation schemes for the hypergraph unreliability problem. For any fixed $\\varepsilon \\in (0, 1)$, we first give a $(1+\\varepsilon)$-approximation algorithm that runs in $m^{O(\\log n)}$ time on an $m$-hyperedge, $n$-vertex hypergraph. Then, we improve the running time to $m\\cdot n^{O(\\log^2 n)}$ with an additional exponentially small additive term in the approximation.", "source": "Arxiv", "date": "2024-03-27", "funding_agencies": []}}, {"edge": "Localized Version of Hypergraph Erdos-Gallai Theorem", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Extremal problems"], "abstract": "This paper focuses on extensions of the classic Erd\\H{o}s-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge $e$ of an $n$-vertex uniform hypergraph $\\mathcal{H}$ is defined to a special function with respect to the number of edges of the longest Berge path containing $e$. We prove that the summation of the weighted function of all edges is at most $n$ for an $n$-vertex uniform hypergraph $\\mathcal{H}$ and characterize all extremal hypergraphs that attain the value, which strengthens and extends the hypergraph version of the classic Erd\\H{o}s-Gallai Theorem.", "source": "Arxiv", "date": "2024-03-31", "funding_agencies": []}}, {"edge": "DPHGNN: A Dual Perspective Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": ["Machine Learning", "Social and Information Networks"], "abstract": "Message passing on hypergraphs has been a standard framework for learning higher-order correlations between hypernodes. Recently-proposed hypergraph neural networks (HGNNs) can be categorized into spatial and spectral methods based on their design choices. In this work, we analyze the impact of change in hypergraph topology on the suboptimal performance of HGNNs and propose DPHGNN, a novel dual-perspective HGNN that introduces equivariant operator learning to capture lower-order semantics by inducing topology-aware spatial and spectral inductive biases. DPHGNN employs a unified framework to dynamically fuse lower-order explicit feature representations from the underlying graph into the super-imposed hypergraph structure. We benchmark DPHGNN over eight benchmark hypergraph datasets for the semi-supervised hypernode classification task and obtain superior performance compared to seven state-of-the-art baselines. We also provide a theoretical framework and a synthetic hypergraph isomorphism test to express the power of spatial HGNNs and quantify the expressivity of DPHGNN beyond the Generalized Weisfeiler Leman (1-GWL) test. Finally, DPHGNN was deployed by our partner e-commerce company for the Return-to-Origin (RTO) prediction task, which shows ~7% higher macro F1-Score than the best baseline.", "source": "Arxiv", "date": "2024-05-26", "funding_agencies": []}}, {"edge": "HyperMagNet: A Magnetic Laplacian based Hypergraph Neural Network", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "In data science, hypergraphs are natural models for data exhibiting multi-way relations, whereas graphs only capture pairwise. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via symmetrized matrix representations, potentially losing important information. We propose an alternative approach to hypergraph neural networks in which the hypergraph is represented as a non-reversible Markov chain. We use this Markov chain to construct a complex Hermitian Laplacian matrix - the magnetic Laplacian - which serves as the input to our proposed hypergraph neural network. We study HyperMagNet for the task of node classification, and demonstrate its effectiveness over graph-reduction based hypergraph neural networks.", "source": "Arxiv", "date": "2024-02-14", "funding_agencies": []}}, {"edge": "Optimal control problem of evolution equation governed by hypergraph Laplacian", "weight": 1, "attrs": {"tags": ["Optimization and Control", "Problems involving ordinary differential equations", "Hypergraphs", "Evolution inclusions", "Optimal control problems involving ordinary differential equations", "Set-valued and variational analysis"], "abstract": "In this paper, we consider an optimal control problem of an ordinary differential inclusion governed by the hypergraph Laplacian, which is defined as a subdifferential of a convex function and then is a set-valued operator. We can assure the existence of optimal control for a suitable cost function by using methods of a priori estimates established in the previous studies. However, due to the multivaluedness of the hypergraph Laplacian, it seems to be difficult to derive the necessary optimality condition for this problem. To cope with this difficulty, we introduce an approximation operator based on the approximation method of the hypergraph, so-called ``clique expansion.'' We first consider the optimality condition of the approximation problem with the clique expansion of the hypergraph Laplacian and next discuss the convergence to the original problem. In appendix, we state some basic properties of the clique expansion of the hypergraph Laplacian for future works.", "source": "Arxiv", "date": "2024-08-31", "funding_agencies": []}}, {"edge": "Autoregressive Adaptive Hypergraph Transformer for Skeleton-based Activity Recognition", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Extracting multiscale contextual information and higher-order correlations among skeleton sequences using Graph Convolutional Networks (GCNs) alone is inadequate for effective action classification. Hypergraph convolution addresses the above issues but cannot harness the long-range dependencies. Transformer proves to be effective in capturing these dependencies and making complex contextual features accessible. We propose an Autoregressive Adaptive HyperGraph Transformer (AutoregAd-HGformer) model for in-phase (autoregressive and discrete) and out-phase (adaptive) hypergraph generation. The vector quantized in-phase hypergraph equipped with powerful autoregressive learned priors produces a more robust and informative representation suitable for hyperedge formation. The out-phase hypergraph generator provides a model-agnostic hyperedge learning technique to align the attributes with input skeleton embedding. The hybrid (supervised and unsupervised) learning in AutoregAd-HGformer explores the action-dependent feature along spatial, temporal, and channel dimensions. The extensive experimental results and ablation study indicate the superiority of our model over state-of-the-art hypergraph architectures on NTU RGB+D, NTU RGB+D 120, and NW-UCLA datasets.", "source": "Arxiv", "date": "2024-11-08", "funding_agencies": []}}, {"edge": "Towards an optimal hypergraph container lemma", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "The hypergraph container lemma is a powerful tool in probabilistic combinatorics that has found many applications since it was first proved a decade ago. Roughly speaking, it asserts that the family of independent sets of every uniform hypergraph can be covered by a small number of almost-independent sets, called containers. In this article, we formulate and prove two new versions of the lemma that display the following three attractive features. First, they both admit short and simple proofs that have surprising connections to other well-studied topics in probabilistic combinatorics. Second, they use alternative notions of almost-independence in order to describe the containers. Third, they yield improved dependence of the number of containers on the uniformity of the hypergraph, hitting a natural barrier for second-moment-type approaches.", "source": "Arxiv", "date": "2024-08-13", "funding_agencies": []}}, {"edge": "Hypergraph Laplacian Eigenmaps and Face Recognition Problems", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition", "Machine Learning"], "abstract": "Face recognition is a very important topic in data science and biometric security research areas. It has multiple applications in military, finance, and retail, to name a few. In this paper, the novel hypergraph Laplacian Eigenmaps will be proposed and combine with the k nearest-neighbor method and/or with the kernel ridge regression method to solve the face recognition problem. Experimental results illustrate that the accuracy of the combination of the novel hypergraph Laplacian Eigenmaps and one specific classification system is similar to the accuracy of the combination of the old symmetric normalized hypergraph Laplacian Eigenmaps method and one specific classification system.", "source": "Arxiv", "date": "2024-05-26", "funding_agencies": []}}, {"edge": "HyCubE: Efficient Knowledge Hypergraph 3D Circular Convolutional Embedding", "weight": 1, "attrs": {"tags": ["Artificial Intelligence"], "abstract": "Knowledge hypergraph embedding models are usually computationally expensive due to the inherent complex semantic information. However, existing works mainly focus on improving the effectiveness of knowledge hypergraph embedding, making the model architecture more complex and redundant. It is desirable and challenging for knowledge hypergraph embedding to reach a trade-off between model effectiveness and efficiency. In this paper, we propose an end-to-end efficient knowledge hypergraph embedding model, HyCubE, which designs a novel 3D circular convolutional neural network and the alternate mask stack strategy to enhance the interaction and extraction of feature information comprehensively. Furthermore, our proposed model achieves a better trade-off between effectiveness and efficiency by adaptively adjusting the 3D circular convolutional layer structure to handle n-ary knowledge tuples of different arities with fewer parameters. In addition, we use a knowledge hypergraph 1-N multilinear scoring way to accelerate the model training efficiency further. Finally, extensive experimental results on all datasets demonstrate that our proposed model consistently outperforms state-of-the-art baselines, with an average improvement of 8.22% and a maximum improvement of 33.82% across all metrics. Meanwhile, HyCubE is 6.12x faster, GPU memory usage is 52.67% lower, and the number of parameters is reduced by 85.21% compared with the average metric of the latest state-of-the-art baselines.", "source": "Arxiv", "date": "2024-02-14", "funding_agencies": []}}, {"edge": "From Graphs to Hypergraphs: Hypergraph Projection and its Remediation", "weight": 1, "attrs": {"tags": ["Machine Learning", "Information Retrieval", "Social and Information Networks"], "abstract": "We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically involves an underlying projection process that maps the original hypergraph onto a graph, and is common in graph-based analysis. While hypergraph projection can potentially lead to loss of higher-order relations, there exists very limited studies on the consequences of doing so, as well as its remediation. This work fills this gap by doing two things: (1) we develop analysis based on graph and set theory, showing two ubiquitous patterns of hyperedges that are root to structural information loss in all hypergraph projections; we also quantify the combinatorial impossibility of recovering the lost higher-order structures if no extra help is provided; (2) we still seek to recover the lost higher-order structures in hypergraph projection, and in light of (1)'s findings we propose to relax the problem into a learning-based setting. Under this setting, we develop a learning-based hypergraph reconstruction method based on an important statistic of hyperedge distributions that we find. Our reconstruction method is evaluated on 8 real-world datasets under different settings, and exhibits consistently good performance. We also demonstrate benefits of the reconstructed hypergraphs via use cases of protein rankings and link predictions.", "source": "Arxiv", "date": "2024-01-16", "funding_agencies": []}}, {"edge": "Spectral decomposition of hypergraph automorphism compatible matrices", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Graphs and matrices", "Degree sequences", "Enumeration of graphs and maps"], "abstract": "This study explores the relationship between hypergraph automorphisms and the spectral properties of matrices associated with hypergraphs. For an automorphism $f$, an \\( f \\)-compatible matrices capture aspects of the symmetry, represented by \\( f \\), within the hypergraph. First, we explore rotation, a specific kind of automorphism and find that the spectrum of any matrix compatible with a rotation can be decomposed into the spectra of smaller matrices associated with that rotation. We show that the spectrum of any \\(f\\)-compatible matrix can be decomposed into the spectra of smaller matrices associated with the component rotations comprising \\( f \\). Further, we study a hypergraph symmetry termed unit-automorphism, which induces bijections on the hyperedges, though not necessarily on the vertex set. We show that unit automorphisms also lead to the spectral decomposition of compatible matrices.", "source": "Arxiv", "date": "2024-05-02", "funding_agencies": []}}, {"edge": "Expressive Higher-Order Link Prediction through Hypergraph Symmetry Breaking", "weight": 1, "attrs": {"tags": ["Machine Learning", "Machine Learning"], "abstract": "A hypergraph consists of a set of nodes along with a collection of subsets of the nodes called hyperedges. Higher-order link prediction is the task of predicting the existence of a missing hyperedge in a hypergraph. A hyperedge representation learned for higher order link prediction is fully expressive when it does not lose distinguishing power up to an isomorphism. Many existing hypergraph representation learners, are bounded in expressive power by the Generalized Weisfeiler Lehman-1 (GWL-1) algorithm, a generalization of the Weisfeiler Lehman-1 algorithm. However, GWL-1 has limited expressive power. In fact, induced subhypergraphs with identical GWL-1 valued nodes are indistinguishable. Furthermore, message passing on hypergraphs can already be computationally expensive, especially on GPU memory. To address these limitations, we devise a preprocessing algorithm that can identify certain regular subhypergraphs exhibiting symmetry. Our preprocessing algorithm runs once with complexity the size of the input hypergraph. During training, we randomly replace subhypergraphs identified by the algorithm with covering hyperedges to break symmetry. We show that our method improves the expressivity of GWL-1. Our extensive experiments also demonstrate the effectiveness of our approach for higher-order link prediction on both graph and hypergraph datasets with negligible change in computation.", "source": "Arxiv", "date": "2024-02-17", "funding_agencies": []}}, {"edge": "Federated Hypergraph Learning with Hyperedge Completion", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Hypergraph neural networks enhance conventional graph neural networks by capturing high-order relationships among nodes, which proves vital in data-rich environments where interactions are not merely pairwise. As data complexity and interconnectivity grow, it is common for graph-structured data to be split and stored in a distributed manner, underscoring the necessity of federated learning on subgraphs. In this work, we propose FedHGN, a novel algorithm for federated hypergraph learning. Our algorithm utilizes subgraphs of a hypergraph stored on distributed devices to train local HGNN models in a federated manner:by collaboratively developing an effective global HGNN model through sharing model parameters while preserving client privacy. Additionally, considering that hyperedges may span multiple clients, a pre-training step is employed before the training process in which cross-client hyperedge feature gathering is performed at the central server. In this way, the missing cross-client information can be supplemented from the central server during the node feature aggregation phase. Experimental results on seven real-world datasets confirm the effectiveness of our approach and demonstrate its performance advantages over traditional federated graph learning methods.", "source": "Arxiv", "date": "2024-08-09", "funding_agencies": []}}, {"edge": "Generalized Gradient Descent is a Hypergraph Functor", "weight": 1, "attrs": {"tags": ["Category Theory", "Machine Learning"], "abstract": "Cartesian reverse derivative categories (CRDCs) provide an axiomatic generalization of the reverse derivative, which allows generalized analogues of classic optimization algorithms such as gradient descent to be applied to a broad class of problems. In this paper, we show that generalized gradient descent with respect to a given CRDC induces a hypergraph functor from a hypergraph category of optimization problems to a hypergraph category of dynamical systems. The domain of this functor consists of objective functions that are 1) general in the sense that they are defined with respect to an arbitrary CRDC, and 2) open in that they are decorated spans that can be composed with other such objective functions via variable sharing. The codomain is specified analogously as a category of general and open dynamical systems for the underlying CRDC. We describe how the hypergraph functor induces a distributed optimization algorithm for arbitrary composite problems specified in the domain. To illustrate the kinds of problems our framework can model, we show that parameter sharing models in multitask learning, a prevalent machine learning paradigm, yield a composite optimization problem for a given choice of CRDC. We then apply the gradient descent functor to this composite problem and describe the resulting distributed gradient descent algorithm for training parameter sharing models.", "source": "Arxiv", "date": "2024-03-28", "funding_agencies": []}}, {"edge": "Hypergraph-enhanced Dual Semi-supervised Graph Classification", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Information Retrieval", "Social and Information Networks"], "abstract": "In this paper, we study semi-supervised graph classification, which aims at accurately predicting the categories of graphs in scenarios with limited labeled graphs and abundant unlabeled graphs. Despite the promising capability of graph neural networks (GNNs), they typically require a large number of costly labeled graphs, while a wealth of unlabeled graphs fail to be effectively utilized. Moreover, GNNs are inherently limited to encoding local neighborhood information using message-passing mechanisms, thus lacking the ability to model higher-order dependencies among nodes. To tackle these challenges, we propose a Hypergraph-Enhanced DuAL framework named HEAL for semi-supervised graph classification, which captures graph semantics from the perspective of the hypergraph and the line graph, respectively. Specifically, to better explore the higher-order relationships among nodes, we design a hypergraph structure learning to adaptively learn complex node dependencies beyond pairwise relations. Meanwhile, based on the learned hypergraph, we introduce a line graph to capture the interaction between hyperedges, thereby better mining the underlying semantic structures. Finally, we develop a relational consistency learning to facilitate knowledge transfer between the two branches and provide better mutual guidance. Extensive experiments on real-world graph datasets verify the effectiveness of the proposed method against existing state-of-the-art methods.", "source": "Arxiv", "date": "2024-05-07", "funding_agencies": []}}, {"edge": "Reconstructing short-lived particles using hypergraph representation learning", "weight": 1, "attrs": {"tags": ["High Energy Physics - Phenomenology"], "abstract": "In collider experiments, the kinematic reconstruction of heavy, short-lived particles is vital for precision tests of the Standard Model and in searches for physics beyond it. Performing kinematic reconstruction in collider events with many final-state jets, such as the all-hadronic decay of top-antitop quark pairs, is challenging. We present HyPER: Hypergraph for Particle Event Reconstruction, a novel architecture based on graph neural networks that uses hypergraph representation learning to build more powerful and efficient representations of collider events. HyPER is used to reconstruct parent particles from sets of final-state objects. Trained and tested on simulation, the HyPER model is shown to perform favorably when compared to existing state-of-the-art reconstruction techniques, while demonstrating superior parameter efficiency. The novel hypergraph approach allows the method to be applied to particle reconstruction in a multitude of different physics processes.", "source": "Arxiv", "date": "2024-02-15", "funding_agencies": []}}, {"edge": "FedHCDR: Federated Cross-Domain Recommendation with Hypergraph Signal Decoupling", "weight": 1, "attrs": {"tags": ["Machine Learning", "Information Retrieval", "Social and Information Networks"], "abstract": "In recent years, Cross-Domain Recommendation (CDR) has drawn significant attention, which utilizes user data from multiple domains to enhance the recommendation performance. However, current CDR methods require sharing user data across domains, thereby violating the General Data Protection Regulation (GDPR). Consequently, numerous approaches have been proposed for Federated Cross-Domain Recommendation (FedCDR). Nevertheless, the data heterogeneity across different domains inevitably influences the overall performance of federated learning. In this study, we propose FedHCDR, a novel Federated Cross-Domain Recommendation framework with Hypergraph signal decoupling. Specifically, to address the data heterogeneity across domains, we introduce an approach called hypergraph signal decoupling (HSD) to decouple the user features into domain-exclusive and domain-shared features. The approach employs high-pass and low-pass hypergraph filters to decouple domain-exclusive and domain-shared user representations, which are trained by the local-global bi-directional transfer algorithm. In addition, a hypergraph contrastive learning (HCL) module is devised to enhance the learning of domain-shared user relationship information by perturbing the user hypergraph. Extensive experiments conducted on three real-world scenarios demonstrate that FedHCDR outperforms existing baselines significantly.", "source": "Arxiv", "date": "2024-03-04", "funding_agencies": []}}, {"edge": "CURSOR: Scalable Mixed-Order Hypergraph Matching with CUR Decomposition", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "To achieve greater accuracy, hypergraph matching algorithms require exponential increases in computational resources. Recent kd-tree-based approximate nearest neighbor (ANN) methods, despite the sparsity of their compatibility tensor, still require exhaustive calculations for large-scale graph matching. This work utilizes CUR tensor decomposition and introduces a novel cascaded second and third-order hypergraph matching framework (CURSOR) for efficient hypergraph matching. A CUR-based second-order graph matching algorithm is used to provide a rough match, and then the core of CURSOR, a fiber-CUR-based tensor generation method, directly calculates entries of the compatibility tensor by leveraging the initial second-order match result. This significantly decreases the time complexity and tensor density. A probability relaxation labeling (PRL)-based matching algorithm, especially suitable for sparse tensors, is developed. Experiment results on large-scale synthetic datasets and widely-adopted benchmark sets demonstrate the superiority of CURSOR over existing methods. The tensor generation method in CURSOR can be integrated seamlessly into existing hypergraph matching methods to improve their performance and lower their computational costs.", "source": "Arxiv", "date": "2024-02-26", "funding_agencies": []}}, {"edge": "Creating Subgraphs in Semi-Random Hypergraph Games", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "The semi-random hypergraph process is a natural generalisation of the semi-random graph process, which can be thought of as a one player game. For fixed $r < s$, starting with an empty hypergraph on $n$ vertices, in each round a set of $r$ vertices $U$ is presented to the player independently and uniformly at random. The player then selects a set of $s-r$ vertices $V$ and adds the hyperedge $U \\cup V$ to the $s$-uniform hypergraph. For a fixed (monotone) increasing graph property, the player's objective is to force the graph to satisfy this property with high probability in as few rounds as possible. We focus on the case where the player's objective is to construct a subgraph isomorphic to an arbitrary, fixed hypergraph $H$. In the case $r=1$ the threshold for the number of rounds required was already known in terms of the degeneracy of $H$. In the case $2 \\le r < s$, we give upper and lower bounds on this threshold for general $H$, and find further improved upper bounds for cliques in particular. We identify cases where the upper and lower bounds match. We also demonstrate that the lower bounds are not always tight by finding exact thresholds for various paths and cycles.", "source": "Arxiv", "date": "2024-09-28", "funding_agencies": []}}, {"edge": "Applications of Sparse Hypergraph Colorings", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Many problems in extremal combinatorics can be reduced to determining the independence number of a specific auxiliary hypergraph. We present two such problems, one from discrete geometry and one from hypergraph Tur\\'an theory. Using results on hypergraph colorings by Cooper-Mubayi and Li-Postle, we demonstrate that for those two problems the trivial lower bound on the independence number can be improved upon: Erd\\H{o}s, Graham, Ruzsa and Taylor asked to determine the largest size, denoted by $g(n)$, of a subset $P$ of the grid $[n]^2$ such that every pair of points in $P$ span a different slope. Improving on a lower bound by Zhang from 1993, we show that $$g(n)=\\Omega \\left( \\frac{n^{2/3} (\\log \\log n)^{1/3} }{ \\log^{1/3}n} \\right).$$ Let $H^r_3$ denote an $r$-graph with $r+1$ vertices and $3$ edges. Recently, Sidorenko proved the following lower bounds for the Tur\\'an density of this $r$-graph: $\\pi(H^r_3)\\geq r^{-2}$ for every $r$, and $\\pi(H^r_3)\\geq (1.7215 - o(1)) r^{-2}$. We present an improved asymptotic bound: $\\pi(H^r_3)=\\Omega\\left(r^{-2} \\log^{1/2} r \\right).$", "source": "Arxiv", "date": "2024-06-03", "funding_agencies": []}}, {"edge": "Hypergraph $p$-Laplacian regularization on point clouds for data interpolation", "weight": 1, "attrs": {"tags": ["Numerical Analysis", "Machine Learning", "Numerical Analysis", "Analysis of PDEs", "Problems involving randomness", "Variational methods for second-order, elliptic equations", "Stability and convergence of numerical methods"], "abstract": "As a generalization of graphs, hypergraphs are widely used to model higher-order relations in data. This paper explores the benefit of the hypergraph structure for the interpolation of point cloud data that contain no explicit structural information. We define the $\\varepsilon_n$-ball hypergraph and the $k_n$-nearest neighbor hypergraph on a point cloud and study the $p$-Laplacian regularization on the hypergraphs. We prove the variational consistency between the hypergraph $p$-Laplacian regularization and the continuum $p$-Laplacian regularization in a semisupervised setting when the number of points $n$ goes to infinity while the number of labeled points remains fixed. A key improvement compared to the graph case is that the results rely on weaker assumptions on the upper bound of $\\varepsilon_n$ and $k_n$. To solve the convex but non-differentiable large-scale optimization problem, we utilize the stochastic primal-dual hybrid gradient algorithm. Numerical experiments on data interpolation verify that the hypergraph $p$-Laplacian regularization outperforms the graph $p$-Laplacian regularization in preventing the development of spikes at the labeled points.", "source": "Arxiv", "date": "2024-05-02", "funding_agencies": []}}, {"edge": "Conflict-free Hypergraph Matchings and Coverings", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Recent work showing the existence of conflict-free almost-perfect hypergraph matchings has found many applications. We show that, assuming certain simple degree and codegree conditions on the hypergraph $ \\mathcal{H} $ and the conflicts to be avoided, a conflict-free almost-perfect matching can be extended to one covering all of the vertices in a particular subset of $ V(\\mathcal{H}) $, by using an additional set of edges; in particular, we ensure that our matching avoids all of a further set of conflicts, which may consist of both old and new edges. This setup is useful for various applications, and our main theorem provides a black box which encapsulates many long and tedious calculations, massively simplifying the proofs of results in generalised Ramsey theory.", "source": "Arxiv", "date": "2024-07-25", "funding_agencies": []}}, {"edge": "A Survey on Hypergraph Mining: Patterns, Tools, and Generators", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Databases", "Physics and Society"], "abstract": "Hypergraphs are a natural and powerful choice for modeling group interactions in the real world, which are often referred to as higher-order networks. For example, when modeling collaboration networks, where collaborations can involve not just two but three or more people, employing hypergraphs allows us to explore beyond pairwise (dyadic) patterns and capture groupwise (polyadic) patterns. The mathematical complexity of hypergraphs offers both opportunities and challenges for learning and mining on hypergraphs, and hypergraph mining, which seeks to enhance our understanding of underlying systems through hypergraph modeling, gained increasing attention in research. Researchers have discovered various structural patterns in real-world hypergraphs, leading to the development of mining tools. Moreover, they have designed generators with the aim of reproducing and thereby shedding light on these patterns. In this survey, we provide a comprehensive overview of the current landscape of hypergraph mining, covering patterns, tools, and generators. We provide comprehensive taxonomies for them, and we also provide in-depth discussions to provide insights into future research on hypergraph mining.", "source": "Arxiv", "date": "2024-01-16", "funding_agencies": []}}, {"edge": "Hyper-YOLO: When Visual Object Detection Meets Hypergraph Computation", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "We introduce Hyper-YOLO, a new object detection method that integrates hypergraph computations to capture the complex high-order correlations among visual features. Traditional YOLO models, while powerful, have limitations in their neck designs that restrict the integration of cross-level features and the exploitation of high-order feature interrelationships. To address these challenges, we propose the Hypergraph Computation Empowered Semantic Collecting and Scattering (HGC-SCS) framework, which transposes visual feature maps into a semantic space and constructs a hypergraph for high-order message propagation. This enables the model to acquire both semantic and structural information, advancing beyond conventional feature-focused learning. Hyper-YOLO incorporates the proposed Mixed Aggregation Network (MANet) in its backbone for enhanced feature extraction and introduces the Hypergraph-Based Cross-Level and Cross-Position Representation Network (HyperC2Net) in its neck. HyperC2Net operates across five scales and breaks free from traditional grid structures, allowing for sophisticated high-order interactions across levels and positions. This synergy of components positions Hyper-YOLO as a state-of-the-art architecture in various scale models, as evidenced by its superior performance on the COCO dataset. Specifically, Hyper-YOLO-N significantly outperforms the advanced YOLOv8-N and YOLOv9-T with 12\\% $\\text{AP}^{val}$ and 9\\% $\\text{AP}^{val}$ improvements. The source codes are at ttps://github.com/iMoonLab/Hyper-YOLO.", "source": "Arxiv", "date": "2024-08-08", "funding_agencies": []}}, {"edge": "On off-diagonal hypergraph Ramsey numbers", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \\ge 2^{\\Omega_H(n \\log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.", "source": "Arxiv", "date": "2024-04-02", "funding_agencies": []}}, {"edge": "HyperSMOTE: A Hypergraph-based Oversampling Approach for Imbalanced Node Classifications", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Hypergraphs are increasingly utilized in both unimodal and multimodal data scenarios due to their superior ability to model and extract higher-order relationships among nodes, compared to traditional graphs. However, current hypergraph models are encountering challenges related to imbalanced data, as this imbalance can lead to biases in the model towards the more prevalent classes. While the existing techniques, such as GraphSMOTE, have improved classification accuracy for minority samples in graph data, they still fall short when addressing the unique structure of hypergraphs. Inspired by SMOTE concept, we propose HyperSMOTE as a solution to alleviate the class imbalance issue in hypergraph learning. This method involves a two-step process: initially synthesizing minority class nodes, followed by the nodes integration into the original hypergraph. We synthesize new nodes based on samples from minority classes and their neighbors. At the same time, in order to solve the problem on integrating the new node into the hypergraph, we train a decoder based on the original hypergraph incidence matrix to adaptively associate the augmented node to hyperedges. We conduct extensive evaluation on multiple single-modality datasets, such as Cora, Cora-CA and Citeseer, as well as multimodal conversation dataset MELD to verify the effectiveness of HyperSMOTE, showing an average performance gain of 3.38% and 2.97% on accuracy, respectively.", "source": "Arxiv", "date": "2024-09-09", "funding_agencies": []}}, {"edge": "Exploring Quantum Contextuality with the Quantum Moebius-Escher-Penrose hypergraph", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "This study presents the quantum Moebius-Escher-Penrose hypergraph, drawing inspiration from paradoxical constructs such as the Moeobius strip and Penrose's `impossible objects'. The hypergraph is constructed using faithful orthogonal representations in Hilbert space, thereby embedding the graph within a quantum framework. Additionally, a quasi-classical realization is achieved through two-valued states and partition logic, leading to an embedding within Boolean algebra. This dual representation delineates the distinctions between classical and quantum embeddings, with a particular focus on contextuality, highlighted by violations of exclusivity and completeness, quantified through classical and quantum probabilities. The study also examines violations of Boole's conditions of possible experience using correlation polytopes, underscoring the inherent contextuality of the hypergraph. These results offer deeper insights into quantum contextuality and its intricate relationship with classical logic structures.", "source": "Arxiv", "date": "2024-09-16", "funding_agencies": []}}, {"edge": "Heterogeneous Hypergraph Embedding for Recommendation Systems", "weight": 1, "attrs": {"tags": ["Information Retrieval", "Artificial Intelligence", "Machine Learning", "Social and Information Networks", "Machine Learning"], "abstract": "Recent advancements in recommender systems have focused on integrating knowledge graphs (KGs) to leverage their auxiliary information. The core idea of KG-enhanced recommenders is to incorporate rich semantic information for more accurate recommendations. However, two main challenges persist: i) Neglecting complex higher-order interactions in the KG-based user-item network, potentially leading to sub-optimal recommendations, and ii) Dealing with the heterogeneous modalities of input sources, such as user-item bipartite graphs and KGs, which may introduce noise and inaccuracies. To address these issues, we present a novel Knowledge-enhanced Heterogeneous Hypergraph Recommender System (KHGRec). KHGRec captures group-wise characteristics of both the interaction network and the KG, modeling complex connections in the KG. Using a collaborative knowledge heterogeneous hypergraph (CKHG), it employs two hypergraph encoders to model group-wise interdependencies and ensure explainability. Additionally, it fuses signals from the input graphs with cross-view self-supervised learning and attention mechanisms. Extensive experiments on four real-world datasets show our model's superiority over various state-of-the-art baselines, with an average 5.18\\% relative improvement. Additional tests on noise resilience, missing data, and cold-start problems demonstrate the robustness of our KHGRec framework. Our model and evaluation datasets are publicly available at \\url{https://github.com/viethungvu1998/KHGRec}.", "source": "Arxiv", "date": "2024-07-04", "funding_agencies": []}}, {"edge": "HyperBERT: Mixing Hypergraph-Aware Layers with Language Models for Node Classification on Text-Attributed Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Computation and Language", "Machine Learning"], "abstract": "Hypergraphs are characterized by complex topological structure, representing higher-order interactions among multiple entities through hyperedges. Lately, hypergraph-based deep learning methods to learn informative data representations for the problem of node classification on text-attributed hypergraphs have garnered increasing research attention. However, existing methods struggle to simultaneously capture the full extent of hypergraph structural information and the rich linguistic attributes inherent in the nodes attributes, which largely hampers their effectiveness and generalizability. To overcome these challenges, we explore ways to further augment a pretrained BERT model with specialized hypergraph-aware layers for the task of node classification. Such layers introduce higher-order structural inductive bias into the language model, thus improving the model's capacity to harness both higher-order context information from the hypergraph structure and semantic information present in text. In this paper, we propose a new architecture, HyperBERT, a mixed text-hypergraph model which simultaneously models hypergraph relational structure while maintaining the high-quality text encoding capabilities of a pre-trained BERT. Notably, HyperBERT presents results that achieve a new state-of-the-art on five challenging text-attributed hypergraph node classification benchmarks.", "source": "Arxiv", "date": "2024-02-11", "funding_agencies": []}}, {"edge": "Short proof of the hypergraph container theorem", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We present a short and simple proof of the celebrated hypergraph container theorem of Balogh--Morris--Samotij and Saxton--Thomason. On a high level, our argument utilises the idea of iteratively taking vertices of largest degree from an independent set and constructing a hypergraph of lower uniformity which preserves independent sets and inherits edge distribution. The original algorithms for constructing containers also remove in each step vertices of high degree which are not in the independent set. Our modified algorithm postpones this until the end, which surprisingly results in a significantly simplified analysis.", "source": "Arxiv", "date": "2024-08-15", "funding_agencies": []}}, {"edge": "A Hypergraph Approach to Distributed Broadcast", "weight": 1, "attrs": {"tags": ["Information Theory", "Multiagent Systems", "Systems and Control", "Systems and Control", "Information Theory"], "abstract": "This paper explores the distributed broadcast problem within the context of network communications, a critical challenge in decentralized information dissemination. We put forth a novel hypergraph-based approach to address this issue, focusing on minimizing the number of broadcasts to ensure comprehensive data sharing among all network users. The key contributions of this work include the establishment of a general lower bound for the problem using the min-cut capacity of hypergraphs, and a distributed broadcast for quasi-trees (DBQT) algorithm tailored for the unique structure of quasi-trees, which is proven to be optimal. This paper advances both network communication strategies and hypergraph theory, with implications for a wide range of real-world applications, from vehicular and sensor networks to distributed storage systems.", "source": "Arxiv", "date": "2024-04-25", "funding_agencies": []}}, {"edge": "HYGENE: A Diffusion-based Hypergraph Generation Method", "weight": 1, "attrs": {"tags": ["Machine Learning", "Discrete Mathematics"], "abstract": "Hypergraphs are powerful mathematical structures that can model complex, high-order relationships in various domains, including social networks, bioinformatics, and recommender systems. However, generating realistic and diverse hypergraphs remains challenging due to their inherent complexity and lack of effective generative models. In this paper, we introduce a diffusion-based Hypergraph Generation (HYGENE) method that addresses these challenges through a progressive local expansion approach. HYGENE works on the bipartite representation of hypergraphs, starting with a single pair of connected nodes and iteratively expanding it to form the target hypergraph. At each step, nodes and hyperedges are added in a localized manner using a denoising diffusion process, which allows for the construction of the global structure before refining local details. Our experiments demonstrated the effectiveness of HYGENE, proving its ability to closely mimic a variety of properties in hypergraphs. To the best of our knowledge, this is the first attempt to employ deep learning models for hypergraph generation, and our work aims to lay the groundwork for future research in this area.", "source": "Arxiv", "date": "2024-08-29", "funding_agencies": []}}, {"edge": "Adaptive Hypergraph Network for Trust Prediction", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Artificial Intelligence"], "abstract": "Trust plays an essential role in an individual's decision-making. Traditional trust prediction models rely on pairwise correlations to infer potential relationships between users. However, in the real world, interactions between users are usually complicated rather than pairwise only. Hypergraphs offer a flexible approach to modeling these complex high-order correlations (not just pairwise connections), since hypergraphs can leverage hyperedeges to link more than two nodes. However, most hypergraph-based methods are generic and cannot be well applied to the trust prediction task. In this paper, we propose an Adaptive Hypergraph Network for Trust Prediction (AHNTP), a novel approach that improves trust prediction accuracy by using higher-order correlations. AHNTP utilizes Motif-based PageRank to capture high-order social influence information. In addition, it constructs hypergroups from both node-level and structure-level attributes to incorporate complex correlation information. Furthermore, AHNTP leverages adaptive hypergraph Graph Convolutional Network (GCN) layers and multilayer perceptrons (MLPs) to generate comprehensive user embeddings, facilitating trust relationship prediction. To enhance model generalization and robustness, we introduce a novel supervised contrastive learning loss for optimization. Extensive experiments demonstrate the superiority of our model over the state-of-the-art approaches in terms of trust prediction accuracy. The source code of this work can be accessed via https://github.com/Sherry-XU1995/AHNTP.", "source": "Arxiv", "date": "2024-02-07", "funding_agencies": []}}, {"edge": "SHyPar: A Spectral Coarsening Approach to Hypergraph Partitioning", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Machine Learning"], "abstract": "State-of-the-art hypergraph partitioners utilize a multilevel paradigm to construct progressively coarser hypergraphs across multiple layers, guiding cut refinements at each level of the hierarchy. Traditionally, these partitioners employ heuristic methods for coarsening and do not consider the structural features of hypergraphs. In this work, we introduce a multilevel spectral framework, SHyPar, for partitioning large-scale hypergraphs by leveraging hyperedge effective resistances and flow-based community detection techniques. Inspired by the latest theoretical spectral clustering frameworks, such as HyperEF and HyperSF, SHyPar aims to decompose large hypergraphs into multiple subgraphs with few inter-partition hyperedges (cut size). A key component of SHyPar is a flow-based local clustering scheme for hypergraph coarsening, which incorporates a max-flow-based algorithm to produce clusters with substantially improved conductance. Additionally, SHyPar utilizes an effective resistance-based rating function for merging nodes that are strongly connected (coupled). Compared with existing state-of-the-art hypergraph partitioning methods, our extensive experimental results on real-world VLSI designs demonstrate that SHyPar can more effectively partition hypergraphs, achieving state-of-the-art solution quality.", "source": "Arxiv", "date": "2024-10-08", "funding_agencies": []}}, {"edge": "LightHGNN: Distilling Hypergraph Neural Networks into MLPs for $100\\times$ Faster Inference", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Hypergraph Neural Networks (HGNNs) have recently attracted much attention and exhibited satisfactory performance due to their superiority in high-order correlation modeling. However, it is noticed that the high-order modeling capability of hypergraph also brings increased computation complexity, which hinders its practical industrial deployment. In practice, we find that one key barrier to the efficient deployment of HGNNs is the high-order structural dependencies during inference. In this paper, we propose to bridge the gap between the HGNNs and inference-efficient Multi-Layer Perceptron (MLPs) to eliminate the hypergraph dependency of HGNNs and thus reduce computational complexity as well as improve inference speed. Specifically, we introduce LightHGNN and LightHGNN$^+$ for fast inference with low complexity. LightHGNN directly distills the knowledge from teacher HGNNs to student MLPs via soft labels, and LightHGNN$^+$ further explicitly injects reliable high-order correlations into the student MLPs to achieve topology-aware distillation and resistance to over-smoothing. Experiments on eight hypergraph datasets demonstrate that even without hypergraph dependency, the proposed LightHGNNs can still achieve competitive or even better performance than HGNNs and outperform vanilla MLPs by $16.3$ on average. Extensive experiments on three graph datasets further show the average best performance of our LightHGNNs compared with all other methods. Experiments on synthetic hypergraphs with 5.5w vertices indicate LightHGNNs can run $100\\times$ faster than HGNNs, showcasing their ability for latency-sensitive deployments.", "source": "Arxiv", "date": "2024-02-06", "funding_agencies": []}}, {"edge": "Instruction-based Hypergraph Pretraining", "weight": 1, "attrs": {"tags": ["Information Retrieval"], "abstract": "Pretraining has been widely explored to augment the adaptability of graph learning models to transfer knowledge from large datasets to a downstream task, such as link prediction or classification. However, the gap between training objectives and the discrepancy between data distributions in pretraining and downstream tasks hinders the transfer of the pretrained knowledge. Inspired by instruction-based prompts widely used in pretrained language models, we introduce instructions into graph pretraining. In this paper, we propose a novel pretraining framework named Instruction-based Hypergraph Pretraining. To overcome the discrepancy between pretraining and downstream tasks, text-based instructions are applied to provide explicit guidance on specific tasks for representation learning. Compared to learnable prompts, whose effectiveness depends on the quality and the diversity of training data, text-based instructions intrinsically encapsulate task information and support the model to generalize beyond the structure seen during pretraining. To capture high-order relations with task information in a context-aware manner, a novel prompting hypergraph convolution layer is devised to integrate instructions into information propagation in hypergraphs. Extensive experiments conducted on three public datasets verify the superiority of IHP in various scenarios.", "source": "Arxiv", "date": "2024-03-27", "funding_agencies": []}}, {"edge": "Against Filter Bubbles: Diversified Music Recommendation via Weighted Hypergraph Embedding Learning", "weight": 1, "attrs": {"tags": ["Information Retrieval", "Machine Learning"], "abstract": "Recommender systems serve a dual purpose for users: sifting out inappropriate or mismatched information while accurately identifying items that align with their preferences. Numerous recommendation algorithms are designed to provide users with a personalized array of information tailored to their preferences. Nevertheless, excessive personalization can confine users within a `filter bubble`. Consequently, achieving the right balance between accuracy and diversity in recommendations is a pressing concern. To address this challenge, exemplified by music recommendation, we introduce the Diversified Weighted Hypergraph music Recommendation algorithm (DWHRec). In the DWHRec algorithm, the initial connections between users and listened tracks are represented by a weighted hypergraph. Simultaneously, associations between artists, albums and tags with tracks are also appended to the hypergraph. To explore users' latent preferences, a hypergraph-based random walk embedding method is applied to the constructed hypergraph. In our investigation, accuracy is gauged by the alignment between the user and the track, whereas the array of recommended track types measures diversity. We rigorously compared DWHRec against seven state-of-the-art recommendation algorithms using two real-world music datasets. The experimental results validate DWHRec as a solution that adeptly harmonizes accuracy and diversity, delivering a more enriched musical experience. Beyond music recommendation, DWHRec can be extended to cater to other scenarios with similar data structures.", "source": "Arxiv", "date": "2024-02-25", "funding_agencies": []}}, {"edge": "Near-optimal Size Linear Sketches for Hypergraph Cut Sparsifiers", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "A $(1 \\pm \\epsilon)$-sparsifier of a hypergraph $G(V,E)$ is a (weighted) subgraph that preserves the value of every cut to within a $(1 \\pm \\epsilon)$-factor. It is known that every hypergraph with $n$ vertices admits a $(1 \\pm \\epsilon)$-sparsifier with $\\tilde{O}(n/\\epsilon^2)$ hyperedges. In this work, we explore the task of building such a sparsifier by using only linear measurements (a \\emph{linear sketch}) over the hyperedges of $G$, and provide nearly-matching upper and lower bounds for this task. Specifically, we show that there is a randomized linear sketch of size $\\widetilde{O}(n r \\log(m) / \\epsilon^2)$ bits which with high probability contains sufficient information to recover a $(1 \\pm \\epsilon)$ cut-sparsifier with $\\tilde{O}(n/\\epsilon^2)$ hyperedges for any hypergraph with at most $m$ edges each of which has arity bounded by $r$. This immediately gives a dynamic streaming algorithm for hypergraph cut sparsification with an identical space complexity, improving on the previous best known bound of $\\widetilde{O}(n r^2 \\log^4(m) / \\epsilon^2)$ bits of space (Guha, McGregor, and Tench, PODS 2015). We complement our algorithmic result above with a nearly-matching lower bound. We show that for every $\\epsilon \\in (0,1)$, one needs $\\Omega(nr \\log(m/n) / \\log(n))$ bits to construct a $(1 \\pm \\epsilon)$-sparsifier via linear sketching, thus showing that our linear sketch achieves an optimal dependence on both $r$ and $\\log(m)$.", "source": "Arxiv", "date": "2024-07-04", "funding_agencies": []}}, {"edge": "Hypergraph-Aided Task-Resource Matching for Maximizing Value of Task Completion in Collaborative IoT Systems", "weight": 1, "attrs": {"tags": ["Systems and Control", "Systems and Control"], "abstract": "With the growing scale and intrinsic heterogeneity of Internet of Things (IoT) systems, distributed device collaboration becomes essential for effective task completion by dynamically utilizing limited communication and computing resources. However, the separated design and situation-agnostic operation of computing, communication and application layers create a fundamental challenge for rapid task-resource matching, which further deteriorate the overall task completion effectiveness. To overcome this challenge, we utilize hypergraph as a new tool to vertically unify computing, communication, and task aspects of IoT systems for an effective matching by accurately capturing the relationships between tasks and communication and computing resources. Specifically, a state-of-the-art task-resource matching hypergraph (TRM-hypergraph) model is proposed in this paper, which is used to effectively transform the process of allocating complex heterogeneous resources to convoluted tasks into a hypergraph matching problem. Taking into account computational complexity and storage, a game-theoretic hypergraph matching algorithm is proposed via considering the hypergraph matching problem as a non-cooperative multi-player clustering game. Numerical results demonstrate that the proposed TRM-hypergraph model achieves superior performance in matching of tasks and resources compared with comparison algorithms.", "source": "Arxiv", "date": "2024-05-30", "funding_agencies": []}}, {"edge": "Loose Hamilton paths in the 3-uniform cube hypergraph", "weight": 1, "attrs": {"tags": ["Combinatorics", "Paths and cycles"], "abstract": "It is well-known that the $d$-dimensional hypercube contains a Hamilton cycle for $d\\ge 2$. In this paper we address the analogous problem in the $3$-uniform cube hypergraph, a $3$-uniform analogue of the hypercube: for simple parity reasons, the $3$-uniform cube hypergraph can never admit a loose Hamilton cycle in any dimension, so we do the next best thing and consider loose Hamilton paths, and determine for which dimensions these exist.", "source": "Arxiv", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Explaining Hypergraph Neural Networks: From Local Explanations to Global Concepts", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Hypergraph neural networks are a class of powerful models that leverage the message passing paradigm to learn over hypergraphs, a generalization of graphs well-suited to describing relational data with higher-order interactions. However, such models are not naturally interpretable, and their explainability has received very limited attention. We introduce SHypX, the first model-agnostic post-hoc explainer for hypergraph neural networks that provides both local and global explanations. At the instance-level, it performs input attribution by discretely sampling explanation subhypergraphs optimized to be faithful and concise. At the model-level, it produces global explanation subhypergraphs using unsupervised concept extraction. Extensive experiments across four real-world and four novel, synthetic hypergraph datasets demonstrate that our method finds high-quality explanations which can target a user-specified balance between faithfulness and concision, improving over baselines by 25 percent points in fidelity on average.", "source": "Arxiv", "date": "2024-10-10", "funding_agencies": []}}, {"edge": "Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Graphs and matrices", "Exact enumeration problems, generating functions", "Multilinear algebra, tensor products"], "abstract": "It is a classical result due to Jacobi in algebraic combinatorics that the generating function of closed walks at a vertex $u$ in a graph $G$ is determined by the rational function \\[ \\frac{\\phi_{G-u}(t)}{\\phi_G(t)} \\] where $\\phi_G(t)$ is the characteristic polynomial of $G$. In this paper, we show that the corresponding rational function for a hypergraph is also a generating function for some combinatorial objects in the hypergraph. We make use of the Heaps of Pieces framework, developed by Viennot, demonstrating its use on graphs, digraphs, and multigraphs before using it on hypergraphs. In the case of a graph $G$, the pieces are cycles and the concurrence relation is sharing a vertex. The pyramids with maximal piece containing a vertex $u \\in V(G)$ are in one-to-one correspondence with closed walks at $u$. In the case of a hypergraph $\\mathcal{H}$, connected `infragraphs` can be defined as the set of pieces, with the same concurrence relation: sharing a vertex. Our main results are established by analyzing multivariate resultants of polynomial systems associated to adjacency hypermatrices.", "source": "Arxiv", "date": "2024-11-05", "funding_agencies": []}}, {"edge": "Hypergraph-based Motion Generation with Multi-modal Interaction Relational Reasoning", "weight": 1, "attrs": {"tags": ["Robotics", "Artificial Intelligence", "Machine Learning", "Multiagent Systems"], "abstract": "The intricate nature of real-world driving environments, characterized by dynamic and diverse interactions among multiple vehicles and their possible future states, presents considerable challenges in accurately predicting the motion states of vehicles and handling the uncertainty inherent in the predictions. Addressing these challenges requires comprehensive modeling and reasoning to capture the implicit relations among vehicles and the corresponding diverse behaviors. This research introduces an integrated framework for autonomous vehicles (AVs) motion prediction to address these complexities, utilizing a novel Relational Hypergraph Interaction-informed Neural mOtion generator (RHINO). RHINO leverages hypergraph-based relational reasoning by integrating a multi-scale hypergraph neural network to model group-wise interactions among multiple vehicles and their multi-modal driving behaviors, thereby enhancing motion prediction accuracy and reliability. Experimental validation using real-world datasets demonstrates the superior performance of this framework in improving predictive accuracy and fostering socially aware automated driving in dynamic traffic scenarios.", "source": "Arxiv", "date": "2024-09-17", "funding_agencies": []}}, {"edge": "Structure-Aware Simplification for Hypergraph Visualization", "weight": 1, "attrs": {"tags": ["Graphics"], "abstract": "Hypergraphs provide a natural way to represent polyadic relationships in network data. For large hypergraphs, it is often difficult to visually detect structures within the data. Recently, a scalable polygon-based visualization approach was developed allowing hypergraphs with thousands of hyperedges to be simplified and examined at different levels of detail. However, this approach is not guaranteed to eliminate all of the visual clutter caused by unavoidable overlaps. Furthermore, meaningful structures can be lost at simplified scales, making their interpretation unreliable. In this paper, we define hypergraph structures using the bipartite graph representation, allowing us to decompose the hypergraph into a union of structures including topological blocks, bridges, and branches, and to identify exactly where unavoidable overlaps must occur. We also introduce a set of topology preserving and topology altering atomic operations, enabling the preservation of important structures while reducing unavoidable overlaps to improve visual clarity and interpretability in simplified scales. We demonstrate our approach in several real-world applications.", "source": "Arxiv", "date": "2024-07-28", "funding_agencies": []}}, {"edge": "Variational Multi-Modal Hypergraph Attention Network for Multi-Modal Relation Extraction", "weight": 1, "attrs": {"tags": ["Computation and Language"], "abstract": "Multi-modal relation extraction (MMRE) is a challenging task that aims to identify relations between entities in text leveraging image information. Existing methods are limited by their neglect of the multiple entity pairs in one sentence sharing very similar contextual information (ie, the same text and image), resulting in increased difficulty in the MMRE task. To address this limitation, we propose the Variational Multi-Modal Hypergraph Attention Network (VM-HAN) for multi-modal relation extraction. Specifically, we first construct a multi-modal hypergraph for each sentence with the corresponding image, to establish different high-order intra-/inter-modal correlations for different entity pairs in each sentence. We further design the Variational Hypergraph Attention Networks (V-HAN) to obtain representational diversity among different entity pairs using Gaussian distribution and learn a better hypergraph structure via variational attention. VM-HAN achieves state-of-the-art performance on the multi-modal relation extraction task, outperforming existing methods in terms of accuracy and efficiency.", "source": "Arxiv", "date": "2024-04-18", "funding_agencies": []}}, {"edge": "The Linear $q$-Hypergraph Process", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We analyze a random greedy process to construct $q$-uniform linear hypergraphs using the differential equation method. We show for $q=o(\\sqrt{\\log n})$, that this process yields a hypergraph with $\\frac{n(n-1)}{q(q-1)}(1-o(1))$ edges. We also give some bounds for maximal linear hypergraphs.", "source": "Arxiv", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Temporal Knowledge Graph Reasoning with Dynamic Hypergraph Embedding", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph associahedra and compactifications of moduli spaces of points", "weight": 1, "attrs": {"tags": ["Algebraic Geometry", "Combinatorics", "Families, moduli (algebraic)", "Algebraic moduli problems, moduli of vector bundles", "$n$-dimensional polytopes", "Toric varieties, Newton polyhedra", "Hypergraphs", "Special polytopes (linear programming, centrally symmetric, etc.)"], "abstract": "We prove that every Hassett compactification of the moduli space of weighted stable rational curves that admits both a reduction map from the Losev-Manin compactification and a reduction map to projective space is a toric variety, whose corresponding polytope is a hypergraph associahedron (also known as a nestohedron). In addition, we present an analogous result for the moduli space of labeled weighted points in affine space up to translation and scaling. These results are interconnected, and we make their relationship explicit through the concept of ``inflation` of a hypergraph associahedron.", "source": "Arxiv", "date": "2024-09-13", "funding_agencies": []}}, {"edge": "Random Tur\\'an Problems for Hypergraph Expansions", "weight": 1, "attrs": {"tags": ["Combinatorics", "Probabilistic methods", "Extremal problems", "Hypergraphs", "Random graphs"], "abstract": "The random Tur\\'an number $\\mathrm{ex}(G_{n,p}^r,F)$ is the maximum number of edges in an $F$-free subgraph of the random $r$-uniform hypergraph $G_{n,p}^r$. We prove general results which (informally) shows that if $F$ is an $r_0$-graph, then upper bounds for $\\mathrm{ex}(G_{n,p}^{r_0},F)$ can be lifted into upper bounds for $\\mathrm{ex}(G_{n,p}^{r},F^{(r)})$ where $F^{(r)}$ is the $r$-uniform expansion of $F$, i.e.\\ the hypergraph obtained from $F$ by inserting $r-r_0$ distinct vertices into each edge of $F$. These results unify and generalize most known upper bounds for random Tur\\'an numbers of degenerate hypergraphs of uniformity at least 3, and also provide new tight bounds for the random Tur\\'an numbers of expansions of theta graphs and complete bipartite graphs.", "source": "Arxiv", "date": "2024-08-06", "funding_agencies": []}}, {"edge": "A spatial hypergraph model where epidemic spread demonstrates clear higher-order effects", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Physics and Society"], "abstract": "We demonstrate a spatial hypergraph model that allows us to vary the amount of higher-order structure in the generated hypergraph. Specifically, we can vary from a model that is a pure pairwise graph into a model that is almost a pure hypergraph. We use this spatial hypergraph model to study higher-order effects in epidemic spread. We use a susceptible-infected-recovered-susceptible (SIRS) epidemic model designed to mimic the spread of an airborne pathogen. We study three types of airborne effects that emulate airborne dilution effects. For the scenario of linear dilution, which roughly correspond to constant ventilation per person as required in many building codes, we see essentially no impact from introducing small hyperedges up to size 15 whereas we do see effects when the hyperedge set is dominated by large hyperedges. Specifically, we track the mean infections after the SIRS epidemic has run for awhile so it is in a `steady state` and find the mean is higher in the large hyperedge regime wheras it is unchanged from pairwise to small hyperedge regime.", "source": "Arxiv", "date": "2024-10-16", "funding_agencies": []}}, {"edge": "The connection between the chromatic numbers of a hypergraph and its $1$-intersection graph", "weight": 1, "attrs": {"tags": ["Combinatorics", "Coloring of graphs and hypergraphs"], "abstract": "A well known problem from an excellent book of Lov\\'asz states that any hypergraph with the property that no pair of hyperedges intersect in exactly one vertex can be properly 2-colored. Motivated by this as well as recent works of Keszegh and of Gy\\'arf\\'as et al we study the $1$-intersection graph of a hypergraph. The $1$-intersection graph encodes those pairs of hyperedges in a hypergraph that intersect in exactly one vertex. We prove for $k\\in\\{2,4\\}$ that all hypergraphs whose $1$-intersection graph is $k$-partite can be properly $k$-colored.", "source": "Arxiv", "date": "2024-06-17", "funding_agencies": []}}, {"edge": "Hyper-3DG: Text-to-3D Gaussian Generation via Hypergraph", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Text-to-3D generation represents an exciting field that has seen rapid advancements, facilitating the transformation of textual descriptions into detailed 3D models. However, current progress often neglects the intricate high-order correlation of geometry and texture within 3D objects, leading to challenges such as over-smoothness, over-saturation and the Janus problem. In this work, we propose a method named ``3D Gaussian Generation via Hypergraph (Hyper-3DG)'', designed to capture the sophisticated high-order correlations present within 3D objects. Our framework is anchored by a well-established mainflow and an essential module, named ``Geometry and Texture Hypergraph Refiner (HGRefiner)''. This module not only refines the representation of 3D Gaussians but also accelerates the update process of these 3D Gaussians by conducting the Patch-3DGS Hypergraph Learning on both explicit attributes and latent visual features. Our framework allows for the production of finely generated 3D objects within a cohesive optimization, effectively circumventing degradation. Extensive experimentation has shown that our proposed method significantly enhances the quality of 3D generation while incurring no additional computational overhead for the underlying framework. (Project code: https://github.com/yjhboy/Hyper3DG)", "source": "Arxiv", "date": "2024-03-14", "funding_agencies": []}}, {"edge": "Quantum teleportation of a qutrit state via a hypergraph state in a noisy environment", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "The concept of quantum teleportation is fundamental in the theory of quantum communication. Developing models of quantum teleportation in different physical scenario is a modern trend of research in this direction. This work is at the interface of hypergraph theory and quantum teleportation. We propose a new teleportation protocol for qutrit states in a noisy environment. This protocol utilizes a shared quantum hypergraph state between parties as quantum channels, to carry quantum information. During the preparation of the shared state different quantum noise may act on it. In this article, we consider six types of noises which are generalized for qutrits using Wyel operators. They are the qutrit-flip noise, qutrit-phase-flip noise, depolarizing noise, Markovian and non-Markovian amplitude damping channel, Markovian and non-Markovian dephasing channel, and non-Markovian depolarization noise. There are only five qutrit hypergraph states which satisfies our requirements to be used as a channel. We consider all of them in this work. For different hypergraph states and different noises we work out the analytical expressions of quantum teleportation fidelity.", "source": "Arxiv", "date": "2024-09-09", "funding_agencies": []}}, {"edge": "Bilevel Hypergraph Networks for Multi-Modal Alzheimer's Diagnosis", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Early detection of Alzheimer's disease's precursor stages is imperative for significantly enhancing patient outcomes and quality of life. This challenge is tackled through a semi-supervised multi-modal diagnosis framework. In particular, we introduce a new hypergraph framework that enables higher-order relations between multi-modal data, while utilising minimal labels. We first introduce a bilevel hypergraph optimisation framework that jointly learns a graph augmentation policy and a semi-supervised classifier. This dual learning strategy is hypothesised to enhance the robustness and generalisation capabilities of the model by fostering new pathways for information propagation. Secondly, we introduce a novel strategy for generating pseudo-labels more effectively via a gradient-driven flow. Our experimental results demonstrate the superior performance of our framework over current techniques in diagnosing Alzheimer's disease.", "source": "Arxiv", "date": "2024-03-19", "funding_agencies": []}}, {"edge": "Hypergraph product code with 0.2 constant coding rate and high code capacity noise threshold", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "The low coding rate of quantum stabilizer codes results in formidable physical qubit overhead when realizing quantum error correcting in engineering. In this letter, we propose a new class of hypergraph-product code called TGRE-hypergraph-product code. This code has constant coding rate 0.2, which is the highest constant coding rate of quantum stabilizer codes to our best knowledge. We perform simulations to test the error correcting capability TGRE-hypergraph-product code and find their code capacity noise threshold in depolarizing noise channel is around 0.096.", "source": "Arxiv", "date": "2024-02-14", "funding_agencies": []}}, {"edge": "MSHyper: Multi-Scale Hypergraph Transformer for Long-Range Time Series Forecasting", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Demystifying interactions between temporal patterns of different scales is fundamental to precise long-range time series forecasting. However, previous works lack the ability to model high-order interactions. To promote more comprehensive pattern interaction modeling for long-range time series forecasting, we propose a Multi-Scale Hypergraph Transformer (MSHyper) framework. Specifically, a multi-scale hypergraph is introduced to provide foundations for modeling high-order pattern interactions. Then by treating hyperedges as nodes, we also build a hyperedge graph to enhance hypergraph modeling. In addition, a tri-stage message passing mechanism is introduced to aggregate pattern information and learn the interaction strength between temporal patterns of different scales. Extensive experiments on five real-world datasets demonstrate that MSHyper achieves state-of-the-art performance, reducing prediction errors by an average of 8.73% and 7.15% over the best baseline in MSE and MAE, respectively.", "source": "Arxiv", "date": "2024-01-17", "funding_agencies": []}}, {"edge": "NeuralQP: A General Hypergraph-based Optimization Framework for Large-scale QCQPs", "weight": 1, "attrs": {"tags": ["Optimization and Control", "Machine Learning"], "abstract": "Machine Learning (ML) optimization frameworks have gained attention for their ability to accelerate the optimization of large-scale Quadratically Constrained Quadratic Programs (QCQPs) by learning shared problem structures. However, existing ML frameworks often rely heavily on strong problem assumptions and large-scale solvers. This paper introduces NeuralQP, a general hypergraph-based framework for large-scale QCQPs. NeuralQP features two main components: Hypergraph-based Neural Prediction, which generates embeddings and predicted solutions for QCQPs without problem assumptions, and Parallel Neighborhood Optimization, which employs a McCormick relaxation-based repair strategy to identify and correct illegal variables, iteratively improving the solution with a small-scale solver. We further prove that our framework UniEGNN with our hypergraph representation is equivalent to the Interior-Point Method (IPM) for quadratic programming. Experiments on two benchmark problems and large-scale real-world instances from QPLIB demonstrate that NeuralQP outperforms state-of-the-art solvers (e.g., Gurobi and SCIP) in both solution quality and time efficiency, further validating the efficiency of ML optimization frameworks for QCQPs.", "source": "Arxiv", "date": "2024-09-28", "funding_agencies": []}}, {"edge": "Inferring gene regulatory networks by hypergraph variational autoencoder", "weight": 1, "attrs": {"tags": ["bioinformatics", "1", "bioinformatics"], "abstract": "In constructing Gene Regulatory Networks (GRNs), it is crucial to consider cellular heterogeneity and differential gene regulatory modules. However, traditional methods have predominantly focused on cellular heterogeneity, approaching the subject from a relatively narrow scope. We present HyperG-VAE, a Bayesian deep generative model that utilizes a hypergraph to model single-cell RNA sequencing (scRNA-seq) data. HyperG-VAE employs a cell encoder with a Structural Equation Model to address cellular heterogeneity and build GRNs, alongside a gene encoder using hypergraph self-attention to identify gene modules. Encoders are synergistically optimized by a decoder, enabling HyperG-VAE to excel in GRN inference, single-cell clustering, and data visualization, evidenced by benchmarks. Additionally, HyperG-VAE effectively reveals gene regulation patterns and shows robustness in varied downstream analyses, demonstrated using B cell development data in bone marrow. The interplay of encoders by the overlapping genes between predicted GRNs and gene modules is further validated by gene set enrichment analysis, underscoring that the gene encoder boosts the GRN inference. HyperG-VAE proves efficient in scRNA-seq data analysis and GRN inference.", "source": "Biorxiv", "date": "2024-04-02", "funding_agencies": []}}, {"edge": "Assigning Entities to Teams as a Hypergraph Discovery Problem", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Spectral Theory", "Physics and Society"], "abstract": "We propose a team assignment algorithm based on a hypergraph approach focusing on resilience and diffusion optimization. Specifically, our method is based on optimizing the algebraic connectivity of the Laplacian matrix of an edge-dependent vertex-weighted hypergraph. We used constrained simulated annealing, where we constrained the effort agents can exert to perform a task and the minimum effort a task requires to be completed. We evaluated our methods in terms of the number of unsuccessful patches to drive our solution into the feasible region and the cost of patching. We showed that our formulation provides more robust solutions than the original data and the greedy approach. We hope that our methods motivate further research in applying hypergraphs to similar problems in different research areas and in exploring variations of our methods.", "source": "Arxiv", "date": "2024-03-06", "funding_agencies": []}}, {"edge": "Relational Learning in Pre-Trained Models: A Theory from Hypergraph Recovery Perspective", "weight": 1, "attrs": {"tags": ["Artificial Intelligence", "Machine Learning"], "abstract": "Foundation Models (FMs) have demonstrated remarkable insights into the relational dynamics of the world, leading to the crucial question: how do these models acquire an understanding of world hybrid relations? Traditional statistical learning, particularly for prediction problems, may overlook the rich and inherently structured information from the data, especially regarding the relationships between objects. We introduce a mathematical model that formalizes relational learning as hypergraph recovery to study pre-training of FMs. In our framework, the world is represented as a hypergraph, with data abstracted as random samples from hyperedges. We theoretically examine the feasibility of a Pre-Trained Model (PTM) to recover this hypergraph and analyze the data efficiency in a minimax near-optimal style. By integrating rich graph theories into the realm of PTMs, our mathematical framework offers powerful tools for an in-depth understanding of pre-training from a unique perspective and can be used under various scenarios. As an example, we extend the framework to entity alignment in multimodal learning.", "source": "Arxiv", "date": "2024-06-17", "funding_agencies": []}}, {"edge": "Beyond hypergraph acyclicity: limits of tractability for pseudo-Boolean optimization", "weight": 1, "attrs": {"tags": ["Optimization and Control", "Discrete Mathematics"], "abstract": "In this paper, we study the problem of minimizing a polynomial function with literals over all binary points, often referred to as pseudo-Boolean optimization. We investigate the fundamental limits of computation for this problem by providing new necessary conditions and sufficient conditions for tractability. On one hand, we obtain the first intractability results for pseudo-Boolean optimization problems on signed hypergraphs with bounded rank, in terms of the treewidth of the intersection graph. On the other hand, we introduce the nest-set gap, a new hypergraph-theoretic notion that enables us to move beyond hypergraph acyclicity, and obtain a polynomial-size extended formulation for the pseudo-Boolean polytope of a class of signed hypergraphs whose underlying hypergraphs contain beta-cycles.", "source": "Arxiv", "date": "2024-10-30", "funding_agencies": []}}, {"edge": "New bounds of two hypergraph Ramsey problems", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We focus on two hypergraph Ramsey problems. First, we consider the Erd\\H{o}s-Hajnal function $r_k(k+1,t;n)$. In 1972, Erd\\H{o}s and Hajnal conjectured that the tower growth rate of $r_k(k+1,t;n)$ is $t-1$ for each $2\\le t\\le k$. To finish this conjecture, it remains to show that the tower growth rate of $r_4(5,4;n)$ is three. We prove a superexponential lower bound for $r_4(5,4;n)$, which improves the previous best lower bound $r_4(5,4;n)\\geq 2^{\\Omega(n^2)}$ from Mubayi and Suk (\\emph{J. Eur. Math. Soc., 2020}). Second, we prove an upper bound for the hypergraph Erd\\H{o}s-Rogers function $f^{(k)}_{k+1,k+2}(N)$ that is an iterated $(k-3)$-fold logarithm in $N$ for each $k\\geq 5$. This improves the previous upper bound that is an iterated $(k-13)$-fold logarithm in $N$ for $k\\ge14$ due to Mubayi and Suk (\\emph{J. London Math. Soc., 2018}), in which they conjectured that $f^{(k)}_{k+1,k+2}(N)$ is an iterated $(k-2)$-fold logarithm in $N$ for each $k\\ge3$.", "source": "Arxiv", "date": "2024-10-29", "funding_agencies": []}}, {"edge": "Covering a supermodular-like function in a mixed hypergraph", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences, submitted], which characterizes a mixed hypergraph $\\mathcal{F}=(V, \\mathcal{E} \\cup \\mathcal{A})$ having an orientation $\\overrightarrow{\\mathcal{E}}$ of $\\mathcal{E}$ such that $e_{\\overrightarrow{\\mathcal{E}} \\cup \\mathcal{A}} (\\mathcal{P}) \\geq \\sum_{X \\in \\mathcal{P}}h(X) -b(\\cup \\mathcal{P})$ for every subpartition $\\mathcal{P}$ of $V$, where $h$ is an integer-valued, intersecting supermodular function on $V$ and $b$ a submodular function on $V$. As a corollary, another conjecture in the same paper is confirmed, which characterizes a mixed hypergraph having a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex $v$ belongs to exactly $k$ of them and is the root of at least $f(v)$ and at most $g(v)$ of them.", "source": "Arxiv", "date": "2024-02-08", "funding_agencies": []}}, {"edge": "Hypergraph-based multi-scale spatio-temporal graph convolution network for Time-Series anomaly detection", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Multivariate time series anomaly detection technology plays an important role in many fields including aerospace, water treatment, cloud service providers, etc. Excellent anomaly detection models can greatly improve work efficiency and avoid major economic losses. However, with the development of technology, the increasing size and complexity of data, and the lack of labels for relevant abnormal data, it is becoming increasingly challenging to perform effective and accurate anomaly detection in high-dimensional and complex data sets. In this paper, we propose a hypergraph based spatiotemporal graph convolutional neural network model STGCN_Hyper, which explicitly captures high-order, multi-hop correlations between multiple variables through a hypergraph based dynamic graph structure learning module. On this basis, we further use the hypergraph based spatiotemporal graph convolutional network to utilize the learned hypergraph structure to effectively propagate and aggregate one-hop and multi-hop related node information in the convolutional network, thereby obtaining rich spatial information. Furthermore, through the multi-scale TCN dilated convolution module, the STGCN_hyper model can also capture the dependencies of features at different scales in the temporal dimension. An unsupervised anomaly detector based on PCA and GMM is also integrated into the STGCN_hyper model. Through the anomaly score of the detector, the model can detect the anomalies in an unsupervised way. Experimental results on multiple time series datasets show that our model can flexibly learn the multi-scale time series features in the data and the dependencies between features, and outperforms most existing baseline models in terms of precision, recall, F1-score on anomaly detection tasks. Our code is available on: https://git.ecdf.ed.ac.uk/msc-23-24/s2044819", "source": "Arxiv", "date": "2024-10-29", "funding_agencies": []}}, {"edge": "When LLM Meets Hypergraph: A Sociological Analysis on Personality via Online Social Networks", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Information Retrieval"], "abstract": "Individual personalities significantly influence our perceptions, decisions, and social interactions, which is particularly crucial for gaining insights into human behavior patterns in online social network analysis. Many psychological studies have observed that personalities are strongly reflected in their social behaviors and social environments. In light of these problems, this paper proposes a sociological analysis framework for one's personality in an environment-based view instead of individual-level data mining. Specifically, to comprehensively understand an individual's behavior from low-quality records, we leverage the powerful associative ability of LLMs by designing an effective prompt. In this way, LLMs can integrate various scattered information with their external knowledge to generate higher-quality profiles, which can significantly improve the personality analysis performance. To explore the interactive mechanism behind the users and their online environments, we design an effective hypergraph neural network where the hypergraph nodes are users and the hyperedges in the hypergraph are social environments. We offer a useful dataset with user profile data, personality traits, and several detected environments from the real-world social platform. To the best of our knowledge, this is the first network-based dataset containing both hypergraph structure and social information, which could push forward future research in this area further. By employing the framework on this dataset, we can effectively capture the nuances of individual personalities and their online behaviors, leading to a deeper understanding of human interactions in the digital world.", "source": "Arxiv", "date": "2024-07-03", "funding_agencies": []}}, {"edge": "Hypergraph Learning based Recommender System for Anomaly Detection, Control and Optimization", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Anomaly detection is fundamental yet, challenging problem with practical applications in industry. The current approaches neglect the higher-order dependencies within the networks of interconnected sensors in the high-dimensional time series(multisensor data) for anomaly detection. To this end, we present a self-adapting anomaly detection framework for joint learning of (a) discrete hypergraph structure and (b) modeling the temporal trends and spatial relations among the interdependent sensors using the hierarchical encoder-decoder architecture to overcome the challenges. The hypergraph representation learning-based framework exploits the relational inductive biases in the hypergraph-structured data to learn the pointwise single-step-ahead forecasts through the self-supervised autoregressive task and predicts the anomalies based on the forecast error. Furthermore, our framework incentivizes learning the anomaly-diagnosis ontology through a differentiable approach. It derives the anomaly information propagation-based computational hypergraphs for root cause analysis and provides recommendations through an offline, optimal predictive control policy to remedy an anomaly. We conduct extensive experiments to evaluate the proposed method on the benchmark datasets for fair and rigorous comparison with the popular baselines. The proposed method outperforms the baseline models and achieves SOTA performance. We report the ablation studies to support the efficacy of the framework.", "source": "Arxiv", "date": "2024-08-21", "funding_agencies": []}}, {"edge": "Hyper-SAMARL: Hypergraph-based Coordinated Task Allocation and Socially-aware Navigation for Multi-Robot Systems", "weight": 1, "attrs": {"tags": ["Robotics", "Multiagent Systems"], "abstract": "A team of multiple robots seamlessly and safely working in human-filled public environments requires adaptive task allocation and socially-aware navigation that account for dynamic human behavior. Current approaches struggle with highly dynamic pedestrian movement and the need for flexible task allocation. We propose Hyper-SAMARL, a hypergraph-based system for multi-robot task allocation and socially-aware navigation, leveraging multi-agent reinforcement learning (MARL). Hyper-SAMARL models the environmental dynamics between robots, humans, and points of interest (POIs) using a hypergraph, enabling adaptive task assignment and socially-compliant navigation through a hypergraph diffusion mechanism. Our framework, trained with MARL, effectively captures interactions between robots and humans, adapting tasks based on real-time changes in human activity. Experimental results demonstrate that Hyper-SAMARL outperforms baseline models in terms of social navigation, task completion efficiency, and adaptability in various simulated scenarios.", "source": "Arxiv", "date": "2024-09-17", "funding_agencies": []}}, {"edge": "Hypergraph Self-supervised Learning with Sampling-efficient Signals", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Self-supervised learning (SSL) provides a promising alternative for representation learning on hypergraphs without costly labels. However, existing hypergraph SSL models are mostly based on contrastive methods with the instance-level discrimination strategy, suffering from two significant limitations: (1) They select negative samples arbitrarily, which is unreliable in deciding similar and dissimilar pairs, causing training bias. (2) They often require a large number of negative samples, resulting in expensive computational costs. To address the above issues, we propose SE-HSSL, a hypergraph SSL framework with three sampling-efficient self-supervised signals. Specifically, we introduce two sampling-free objectives leveraging the canonical correlation analysis as the node-level and group-level self-supervised signals. Additionally, we develop a novel hierarchical membership-level contrast objective motivated by the cascading overlap relationship in hypergraphs, which can further reduce membership sampling bias and improve the efficiency of sample utilization. Through comprehensive experiments on 7 real-world hypergraphs, we demonstrate the superiority of our approach over the state-of-the-art method in terms of both effectiveness and efficiency.", "source": "Arxiv", "date": "2024-04-17", "funding_agencies": []}}, {"edge": "Hypergraph based Understanding for Document Semantic Entity Recognition", "weight": 1, "attrs": {"tags": ["Artificial Intelligence"], "abstract": "Semantic entity recognition is an important task in the field of visually-rich document understanding. It distinguishes the semantic types of text by analyzing the position relationship between text nodes and the relation between text content. The existing document understanding models mainly focus on entity categories while ignoring the extraction of entity boundaries. We build a novel hypergraph attention document semantic entity recognition framework, HGA, which uses hypergraph attention to focus on entity boundaries and entity categories at the same time. It can conduct a more detailed analysis of the document text representation analyzed by the upstream model and achieves a better performance of semantic information. We apply this method on the basis of GraphLayoutLM to construct a new semantic entity recognition model HGALayoutLM. Our experiment results on FUNSD, CORD, XFUND and SROIE show that our method can effectively improve the performance of semantic entity recognition tasks based on the original model. The results of HGALayoutLM on FUNSD and XFUND reach the new state-of-the-art results.", "source": "Arxiv", "date": "2024-07-09", "funding_agencies": []}}, {"edge": "Towards Multi-agent Policy-based Directed Hypergraph Learning for Traffic Signal Control", "weight": 1, "attrs": {"tags": ["Multiagent Systems"], "abstract": "Deep reinforcement learning (DRL) methods that incorporate graph neural networks (GNNs) have been extensively studied for intelligent traffic signal control, which aims to coordinate traffic signals effectively across multiple intersections. Despite this progress, the standard graph learning used in these methods still struggles to capture higher-order correlations in real-world traffic flow. In this paper, we propose a multi-agent proximal policy optimization framework DHG-PPO, which incorporates PPO and directed hypergraph module to extract the spatio-temporal attributes of the road networks. DHG-PPO enables multiple agents to ingeniously interact through the dynamical construction of hypergraph. The effectiveness of DHG-PPO is validated in terms of average travel time and throughput against state-of-the-art baselines through extensive experiments.", "source": "Arxiv", "date": "2024-09-08", "funding_agencies": []}}, {"edge": "Bootstrap Percolation on the Binomial Random $k$-uniform Hypergraph", "weight": 1, "attrs": {"tags": ["Probability"], "abstract": "We investigate the behaviour of $r$-neighbourhood bootstrap percolation on the binomial $k$-uniform random hypergraph $H_k(n,p)$ for given integers $k\\geq 2$ and $r\\geq 2$. In $r$-neighbourhood bootstrap percolation, infection spreads through the hypergraph, starting from a set of initially infected vertices, and in each subsequent step of the process every vertex with at least $r$ infected neighbours becomes infected. For our analysis the set of initially infected vertices is chosen uniformly at random from all sets of given size. In the regime $n^{-1}\\ll n^{k-2}p \\ll n^{-1/r}$ we establish a threshold such that if the number of initially infected vertices remains below the threshold, then with high probability only a few additional vertices become infected, while if the number of initially infected vertices exceeds the threshold then with high probability almost every vertex becomes infected. In fact we show that the probability of failure decreases exponentially.", "source": "Arxiv", "date": "2024-03-19", "funding_agencies": []}}, {"edge": "Hypergraph adjusted plus-minus", "weight": 1, "attrs": {"tags": ["Methodology", "Applications"], "abstract": "In team sports, traditional ranking statistics do not allow for the simultaneous evaluation of both individuals and combinations of players. Metrics for individual player rankings often fail to consider the interaction effects between groups of players, while methods for assessing full lineups cannot be used to identify the value of lower-order combinations of players (pairs, trios, etc.). Given that player and lineup rankings are inherently dependent on each other, these limitations may affect the accuracy of performance evaluations. To address this, we propose a novel adjusted box score plus-minus (APM) approach that allows for the simultaneous ranking of individual players, lower-order combinations of players, and full lineups. The method adjusts for the complete dependency structure and is motivated by the connection between APM and the hypergraph representation of a team. We discuss the similarities of our approach to other advanced metrics, demonstrate it using NBA data from 2012-2022, and suggest potential directions for future work.", "source": "Arxiv", "date": "2024-03-29", "funding_agencies": []}}, {"edge": "A hypergraph bipartite Tur\\'an problem with odd uniformity", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "In this paper, we investigate the hypergraph Tur\\'an number $ex(n,K^{(r)}_{s,t})$. Here, $K^{(r)}_{s,t}$ denotes the $r$-uniform hypergraph with vertex set $\\left(\\cup_{i\\in [t]}X_i\\right)\\cup Y$ and edge set $\\{X_i\\cup \\{y\\}: i\\in [t], y\\in Y\\}$, where $X_1,X_2,\\cdots,X_t$ are $t$ pairwise disjoint sets of size $r-1$ and $Y$ is a set of size $s$ disjoint from each $X_i$. This study was initially explored by Erd\\H{o}s and has since received substantial attention in research. Recent advancements by Brada\\v{c}, Gishboliner, Janzer and Sudakov have greatly contributed to a better understanding of this problem. They proved that $ex(n,K_{s,t}^{(r)})=O_{s,t}(n^{r-\\frac{1}{s-1}})$ holds for any $r\\geq 3$ and $s,t\\geq 2$. They also provided constructions illustrating the tightness of this bound if $r\\geq 4$ is {\\it even} and $t\\gg s\\geq 2$. Furthermore, they proved that $ex(n,K_{s,t}^{(3)})=O_{s,t}(n^{3-\\frac{1}{s-1}-\\varepsilon_s})$ holds for $s\\geq 3$ and some $\\epsilon_s>0$. Addressing this intriguing discrepancy between the behavior of this number for $r=3$ and the even cases, Brada\\v{c} et al. post a question of whether \\begin{equation*} \\mbox{$ex(n,K_{s,t}^{(r)})= O_{r,s,t}(n^{r-\\frac{1}{s-1}- \\varepsilon})$ holds for odd $r\\geq 5$ and any $s\\geq 3$.} \\end{equation*} In this paper, we provide an affirmative answer to this question, utilizing novel techniques to identify regular and dense substructures. This result highlights a rare instance in hypergraph Tur\\'an problems where the solution depends on the parity of the uniformity.", "source": "Arxiv", "date": "2024-03-07", "funding_agencies": []}}, {"edge": "Multi-Channel Hypergraph Contrastive Learning for Matrix Completion", "weight": 1, "attrs": {"tags": ["Machine Learning", "Information Retrieval"], "abstract": "Rating is a typical user explicit feedback that visually reflects how much a user likes a related item. The (rating) matrix completion is essentially a rating prediction process, which is also a significant problem in recommender systems. Recently, graph neural networks (GNNs) have been widely used in matrix completion, which captures users' preferences over items by formulating a rating matrix as a bipartite graph. However, existing methods are susceptible due to data sparsity and long-tail distribution in real-world scenarios. Moreover, the messaging mechanism of GNNs makes it difficult to capture high-order correlations and constraints between nodes, which are essentially useful in recommendation tasks. To tackle these challenges, we propose a Multi-Channel Hypergraph Contrastive Learning framework for matrix completion, named MHCL. Specifically, MHCL adaptively learns hypergraph structures to capture high-order correlations between nodes and jointly captures local and global collaborative relationships through attention-based cross-view aggregation. Additionally, to consider the magnitude and order information of ratings, we treat different rating subgraphs as different channels, encourage alignment between adjacent ratings, and further achieve the mutual enhancement between different ratings through multi-channel cross-rating contrastive learning. Extensive experiments on five public datasets demonstrate that the proposed method significantly outperforms the current state-of-the-art approaches.", "source": "Arxiv", "date": "2024-11-02", "funding_agencies": []}}, {"edge": "Co-Representation Neural Hypergraph Diffusion for Edge-Dependent Node Classification", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Hypergraphs are widely employed to represent complex higher-order relations in real-world applications. Most hypergraph learning research focuses on node-level or edge-level tasks. A practically relevant but more challenging task, edge-dependent node classification (ENC), is only recently proposed. In ENC, a node can have different labels across different hyperedges, which requires the modeling of node-edge pairs instead of single nodes or hyperedges. Existing solutions for this task are based on message passing and model interactions in within-edge and within-node structures as multi-input single-output functions. This brings three limitations: (1) non-adaptive representation size, (2) non-adaptive messages, and (3) insufficient direct interactions among nodes or edges. To tackle these limitations, we propose CoNHD, a new ENC solution that models both within-edge and within-node interactions as multi-input multi-output functions. Specifically, we represent these interactions as a hypergraph diffusion process on node-edge co-representations. We further develop a neural implementation for this diffusion process, which can adapt to a specific ENC dataset. Extensive experiments demonstrate the effectiveness and efficiency of the proposed CoNHD method.", "source": "Arxiv", "date": "2024-05-23", "funding_agencies": []}}, {"edge": "HyperGALE: ASD Classification via Hypergraph Gated Attention with Learnable Hyperedges", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Computer Vision and Pattern Recognition", "Neural and Evolutionary Computing"], "abstract": "Autism Spectrum Disorder (ASD) is a neurodevelopmental condition characterized by varied social cognitive challenges and repetitive behavioral patterns. Identifying reliable brain imaging-based biomarkers for ASD has been a persistent challenge due to the spectrum's diverse symptomatology. Existing baselines in the field have made significant strides in this direction, yet there remains room for improvement in both performance and interpretability. We propose \\emph{HyperGALE}, which builds upon the hypergraph by incorporating learned hyperedges and gated attention mechanisms. This approach has led to substantial improvements in the model's ability to interpret complex brain graph data, offering deeper insights into ASD biomarker characterization. Evaluated on the extensive ABIDE II dataset, \\emph{HyperGALE} not only improves interpretability but also demonstrates statistically significant enhancements in key performance metrics compared to both previous baselines and the foundational hypergraph model. The advancement \\emph{HyperGALE} brings to ASD research highlights the potential of sophisticated graph-based techniques in neurodevelopmental studies. The source code and implementation instructions are available at GitHub:https://github.com/mehular0ra/HyperGALE.", "source": "Arxiv", "date": "2024-03-21", "funding_agencies": []}}, {"edge": "Gene Expression Prediction from Histology Images via Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": ["bioinformatics", "1", "bioinformatics"], "abstract": "Spatial transcriptomics reveals the spatial distribution of genes in complex tissues, providing crucial insights into biological processes, disease mechanisms, and drug development. The prediction of gene expression based on cost-effective histology images is a promising yet challenging field of research. Existing methods for gene prediction from histology images exhibit two major limitations. First, they ignore the intricate relationship between cell morphological information and gene expression. Second, these methods do not fully utilize the different latent stages of features extracted from the images. To address these limitations, we propose a novel hypergraph neural network model, HGGEP, to predict gene expressions from histology images. HGGEP includes a gradient enhancement module to enhance the models perception of cell morphological information. A lightweight backbone network extracts multiple latent stage features from the image, followed by attention mechanisms to refine the representation of features at each latent stage and capture their relations with nearby features. To explore higher-order associations among multiple latent stage features, we stack them and feed into the hypergraph to establish associations among features at different scales. Experimental results on multiple datasets from disease samples including cancers and tumor disease, demonstrate the superior performance of our HGGEP model than existing methods.Key PointsWe develop a novel histology image-based gene prediction model named HGGEP, which demonstrates high accuracy and robust performance.To reveal the intricate relationship between cell morphology and gene expression in images, we propose a gradient enhancement module, which effectively improves the models capability in perceiving cell morphology in images.HGGEP includes a hypergraph module that efficiently models higher-order associations among features across multiple latent stages, resulting in significant performance improvement.", "source": "Biorxiv", "date": "2024-08-07", "funding_agencies": []}}, {"edge": "An application of node and edge nonlinear hypergraph centrality to a protein complex hypernetwork", "weight": 1, "attrs": {"tags": ["Quantitative Methods", "Combinatorics"], "abstract": "The use of graph centrality measures applied to biological networks, such as protein interaction networks, underpins much research into identifying key players within biological processes. This approach however is restricted to dyadic interactions and it is well-known that in many instances interactions are polyadic. In this study we illustrate the merit of using hypergraph centrality applied to a hypernetwork as an alternative. Specifically, we review and propose an extension to a recently introduced node and edge nonlinear hypergraph centrality model which provides mutually dependent node and edge centralities. A Saccharomyces Cerevisiae protein complex hypernetwork is used as an example application with nodes representing proteins and hyperedges representing protein complexes. The resulting rankings of the nodes and edges are considered to see if they provide insight into the essentiality of the proteins and complexes. We find that certain variations of the model predict essentiality more accurately and that the degree-based variation illustrates that the centrality-lethality rule extends to a hypergraph setting. In particular, through exploitation of the models flexibility, we identify small sets of proteins densely populated with essential proteins. One of the key advantages of applying this model to a protein complex hypernetwork is that it also provides a classification method for protein complexes, unlike previous approaches which are only concerned with classifying proteins.", "source": "Arxiv", "date": "2024-06-03", "funding_agencies": []}}, {"edge": "Multi-Scale Heterogeneity-Aware Hypergraph Representation for Histopathology Whole Slide Images", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Survival prediction is a complex ordinal regression task that aims to predict the survival coefficient ranking among a cohort of patients, typically achieved by analyzing patients' whole slide images. Existing deep learning approaches mainly adopt multiple instance learning or graph neural networks under weak supervision. Most of them are unable to uncover the diverse interactions between different types of biological entities(\\textit{e.g.}, cell cluster and tissue block) across multiple scales, while such interactions are crucial for patient survival prediction. In light of this, we propose a novel multi-scale heterogeneity-aware hypergraph representation framework. Specifically, our framework first constructs a multi-scale heterogeneity-aware hypergraph and assigns each node with its biological entity type. It then mines diverse interactions between nodes on the graph structure to obtain a global representation. Experimental results demonstrate that our method outperforms state-of-the-art approaches on three benchmark datasets. Code is publicly available at \\href{https://github.com/Hanminghao/H2GT}{https://github.com/Hanminghao/H2GT}.", "source": "Arxiv", "date": "2024-04-30", "funding_agencies": []}}, {"edge": "Unveiling Tissue Structure and Tumor Microenvironment from Spatially Resolved Transcriptomics by Hypergraph Learning", "weight": 1, "attrs": {"tags": ["bioinformatics", "1", "bioinformatics"], "abstract": "Spatially resolved transcriptomics (SRT) technologies acquire gene expressions and spatial information simultaneously, reshaping the perspectives of life sciences. Identifying spatial patterns is essential for exploring organ development and tumor microenvironment. Nevertheless, emerging SRT technologies have also introduced diverse spatial resolutions, posing challenges in characterizing spatial domains with finer resolutions. Here we propose a hypergraph-based method, termed HyperSTAR to precisely recognize spatial domains across varying spatial resolutions by utilizing higher-order relationships among spatially adjacent tissue programs. Specifically, a gene expression-guided hyperedge decomposition module is incorporated to refine the structure of the hypergraph to precisely delineate the boundaries of spatial domains. A hypergraph attention convolutional neural network is designed to adaptively learn the significance of each hyperedge. With the power of capturing intricate higher-order relationships within spatially neighboring multi-spots/cells, HyperSTAR demonstrates superior performance across different technologies with various resolutions compared to existing advanced graph neural network models in multiple tasks including uncovering tissue sub-structure, inferring spatiotemporal patterns, and denoising spatially resolved gene expressions. It successfully reveals spatial heterogeneity in breast cancer section and its findings are further validated through functional and survival analyses of independent clinical data. Notably, HyperSTAR performs well with diverse spatial omics data types and seamlessly extends to large-scale datasets.", "source": "Biorxiv", "date": "2024-05-16", "funding_agencies": []}}, {"edge": "Hypergraph saturation for the bow tie", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal problems", "Hypergraphs", "Factorization, matching, covering and packing"], "abstract": "Erd\\H{o}s and S\\'os initiated the study of the maximum size of a $k$-uniform set system, for $k \\geq 4$, with no singleton intersections $50$ years ago. In this work, we investigate the dual problem: finding the minimum size of a $k$-uniform hypergraph with no singleton intersections, such that adding any missing hyperedge forces a singleton intersection. These problems, known as saturation and semi-saturation, are typically challenging. Our focus is on an elementary-to-state case in the line of work by Erd\\H{o}s, F\\`uredi and Tuza. We establish tight linear bounds for $k=4$, marking one of the first non-obvious cases with such a bound.", "source": "Arxiv", "date": "2024-08-29", "funding_agencies": []}}, {"edge": "Hypergraph rewriting and Causal structure of $\\lambda-$calculus", "weight": 1, "attrs": {"tags": ["Discrete Mathematics", "Logic in Computer Science"], "abstract": "In this paper, we first study hypergraph rewriting in categorical terms in an attempt to define the notion of events and develop foundations of causality in graph rewriting. We introduce novel concepts within the framework of double-pushout rewriting in adhesive categories. Secondly, we will study the notion of events in $\\lambda-$calculus, wherein we construct an algorithm to determine causal relations between events following the evaluation of a $\\lambda-$expression satisfying certain conditions. Lastly, we attempt to extend this definition to arbitrary $\\lambda-$expressions.", "source": "Arxiv", "date": "2024-09-02", "funding_agencies": []}}, {"edge": "Proportion-Based Hypergraph Burning", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Graph burning is a discrete process that models the spread of influence through a network using a fire as a proxy for the type of influence being spread. This process was recently extended to hypergraphs. We introduce a variant of hypergraph burning that uses an alternative propagation rule for how the fire spreads - if some fixed proportion of vertices are on fire in a hyperedge, then in the next round the entire hyperedge catches fire. This new variant has more potential for applications than the original model, and it is similarly viable for obtaining deep theoretical results. We obtain bounds which apply to general hypergraphs, and introduce the concept of the burning distribution, which describes how the model changes as the proportion ranges over (0,1). We also obtain computational results which suggest there is a strong correlation between the automorphism group order and the lazy burning number of a balanced incomplete block design.", "source": "Arxiv", "date": "2024-08-12", "funding_agencies": []}}, {"edge": "New Graph and Hypergraph Container Lemmas with Applications in Property Testing", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "The graph and hypergraph container methods are powerful tools with a wide range of applications across combinatorics. Recently, Blais and Seth (FOCS 2023) showed that the graph container method is particularly well-suited for the analysis of the natural canonical tester for two fundamental graph properties: having a large independent set and $k$-colorability. In this work, we show that the connection between the container method and property testing extends further along two different directions. First, we show that the container method can be used to analyze the canonical tester for many other properties of graphs and hypergraphs. We introduce a new hypergraph container lemma and use it to give an upper bound of $\\widetilde{O}(kq^3/\\epsilon)$ on the sample complexity of $\\epsilon$-testing satisfiability, where $q$ is the number of variables per constraint and $k$ is the size of the alphabet. This is the first upper bound for the problem that is polynomial in all of $k$, $q$ and $1/\\epsilon$. As a corollary, we get new upper bounds on the sample complexity of the canonical testers for hypergraph colorability and for every semi-homogeneous graph partition property. Second, we show that the container method can also be used to study the query complexity of (non-canonical) graph property testers. This result is obtained by introducing a new container lemma for the class of all independent set stars, a strict superset of the class of all independent sets. We use this container lemma to give a new upper bound of $\\widetilde{O}(\\rho^5/\\epsilon^{7/2})$ on the query complexity of $\\epsilon$-testing the $\\rho$-independent set property. This establishes for the first time the non-optimality of the canonical tester for a non-homogeneous graph partition property.", "source": "Arxiv", "date": "2024-03-27", "funding_agencies": []}}, {"edge": "Hypergraph-based Multi-View Action Recognition using Event Cameras", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition", "Artificial Intelligence", "Machine Learning"], "abstract": "Action recognition from video data forms a cornerstone with wide-ranging applications. Single-view action recognition faces limitations due to its reliance on a single viewpoint. In contrast, multi-view approaches capture complementary information from various viewpoints for improved accuracy. Recently, event cameras have emerged as innovative bio-inspired sensors, leading to advancements in event-based action recognition. However, existing works predominantly focus on single-view scenarios, leaving a gap in multi-view event data exploitation, particularly in challenges like information deficit and semantic misalignment. To bridge this gap, we introduce HyperMV, a multi-view event-based action recognition framework. HyperMV converts discrete event data into frame-like representations and extracts view-related features using a shared convolutional network. By treating segments as vertices and constructing hyperedges using rule-based and KNN-based strategies, a multi-view hypergraph neural network that captures relationships across viewpoint and temporal features is established. The vertex attention hypergraph propagation is also introduced for enhanced feature fusion. To prompt research in this area, we present the largest multi-view event-based action dataset $\\text{THU}^{\\text{MV-EACT}}\\text{-50}$, comprising 50 actions from 6 viewpoints, which surpasses existing datasets by over tenfold. Experimental results show that HyperMV significantly outperforms baselines in both cross-subject and cross-view scenarios, and also exceeds the state-of-the-arts in frame-based multi-view action recognition.", "source": "Arxiv", "date": "2024-03-28", "funding_agencies": []}}, {"edge": "On the energy barrier of hypergraph product codes", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "A macroscopic energy barrier is a necessary condition for self-correcting quantum memory. In this paper, we prove tight bounds on the energy barrier applicable to any quantum code obtained from the hypergraph product of two classical codes. If the underlying classical codes are low-density parity-check codes (LDPC), the energy barrier of the quantum code is shown to be the minimum energy barrier of the underlying classical codes (and their transposes) up to an additive $O(1)$ constant.", "source": "Arxiv", "date": "2024-07-29", "funding_agencies": []}}, {"edge": "Ada-MSHyper: Adaptive Multi-Scale Hypergraph Transformer for Time Series Forecasting", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Although transformer-based methods have achieved great success in multi-scale temporal pattern interaction modeling, two key challenges limit their further development: (1) Individual time points contain less semantic information, and leveraging attention to model pair-wise interactions may cause the information utilization bottleneck. (2) Multiple inherent temporal variations (e.g., rising, falling, and fluctuating) entangled in temporal patterns. To this end, we propose Adaptive Multi-Scale Hypergraph Transformer (Ada-MSHyper) for time series forecasting. Specifically, an adaptive hypergraph learning module is designed to provide foundations for modeling group-wise interactions, then a multi-scale interaction module is introduced to promote more comprehensive pattern interactions at different scales. In addition, a node and hyperedge constraint mechanism is introduced to cluster nodes with similar semantic information and differentiate the temporal variations within each scales. Extensive experiments on 11 real-world datasets demonstrate that Ada-MSHyper achieves state-of-the-art performance, reducing prediction errors by an average of 4.56%, 10.38%, and 4.97% in MSE for long-range, short-range, and ultra-long-range time series forecasting, respectively. Code is available at https://github.com/shangzongjiang/Ada-MSHyper.", "source": "Arxiv", "date": "2024-10-31", "funding_agencies": []}}, {"edge": "Hyperedge Representations with Hypergraph Wavelets: Applications to Spatial Transcriptomics", "weight": 1, "attrs": {"tags": ["Machine Learning", "Machine Learning", "Signal Processing", "Quantitative Methods"], "abstract": "In many data-driven applications, higher-order relationships among multiple objects are essential in capturing complex interactions. Hypergraphs, which generalize graphs by allowing edges to connect any number of nodes, provide a flexible and powerful framework for modeling such higher-order relationships. In this work, we introduce hypergraph diffusion wavelets and describe their favorable spectral and spatial properties. We demonstrate their utility for biomedical discovery in spatially resolved transcriptomics by applying the method to represent disease-relevant cellular niches for Alzheimer's disease.", "source": "Arxiv", "date": "2024-09-14", "funding_agencies": []}}, {"edge": "Hypergraph Change Point Detection using Adapted Cardinality-Based Gadgets: Applications in Dynamic Legal Structures", "weight": 1, "attrs": {"tags": ["Social and Information Networks"], "abstract": "Hypergraphs provide a robust framework for modeling complex systems with higher-order interactions. However, analyzing them in dynamic settings presents significant computational challenges. To address this, we introduce a novel method that adapts the cardinality-based gadget to convert hypergraphs into strongly connected weighted directed graphs, complemented by a symmetrized combinatorial Laplacian. We demonstrate that the harmonic mean of the conductance and edge expansion of the original hypergraph can be upper-bounded by the conductance of the transformed directed graph, effectively preserving crucial cut information. Additionally, we analyze how the resulting Laplacian relates to that derived from the star expansion. Our approach was validated through change point detection experiments on both synthetic and real datasets, showing superior performance over clique and star expansions in maintaining spectral information in dynamic settings. Finally, we applied our method to analyze a dynamic legal hypergraph constructed from extensive United States court opinion data.", "source": "Arxiv", "date": "2024-09-12", "funding_agencies": []}}, {"edge": "Multimodal Fusion via Hypergraph Autoencoder and Contrastive Learning for Emotion Recognition in Conversation", "weight": 1, "attrs": {"tags": ["Multimedia"], "abstract": "Multimodal emotion recognition in conversation (MERC) seeks to identify the speakers' emotions expressed in each utterance, offering significant potential across diverse fields. The challenge of MERC lies in balancing speaker modeling and context modeling, encompassing both long-distance and short-distance contexts, as well as addressing the complexity of multimodal information fusion. Recent research adopts graph-based methods to model intricate conversational relationships effectively. Nevertheless, the majority of these methods utilize a fixed fully connected structure to link all utterances, relying on convolution to interpret complex context. This approach can inherently heighten the redundancy in contextual messages and excessive graph network smoothing, particularly in the context of long-distance conversations. To address this issue, we propose a framework that dynamically adjusts hypergraph connections by variational hypergraph autoencoder (VHGAE), and employs contrastive learning to mitigate uncertainty factors during the reconstruction process. Experimental results demonstrate the effectiveness of our proposal against the state-of-the-art methods on IEMOCAP and MELD datasets. We release the code to support the reproducibility of this work at https://github.com/yzjred/-HAUCL.", "source": "Arxiv", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Dynamic Hypergraph-Enhanced Prediction of Sequential Medical Visits", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "This study introduces a pioneering Dynamic Hypergraph Networks (DHCE) model designed to predict future medical diagnoses from electronic health records with enhanced accuracy. The DHCE model innovates by identifying and differentiating acute and chronic diseases within a patient's visit history, constructing dynamic hypergraphs that capture the complex, high-order interactions between diseases. It surpasses traditional recurrent neural networks and graph neural networks by effectively integrating clinical event data, reflected through medical language model-assisted encoding, into a robust patient representation. Through extensive experiments on two benchmark datasets, MIMIC-III and MIMIC-IV, the DHCE model exhibits superior performance, significantly outpacing established baseline models in the precision of sequential diagnosis prediction.", "source": "Arxiv", "date": "2024-08-08", "funding_agencies": []}}, {"edge": "Accurate and Fast Pixel Retrieval with Spatial and Uncertainty Aware Hypergraph Diffusion", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "This paper presents a novel method designed to enhance the efficiency and accuracy of both image retrieval and pixel retrieval. Traditional diffusion methods struggle to propagate spatial information effectively in conventional graphs due to their reliance on scalar edge weights. To overcome this limitation, we introduce a hypergraph-based framework, uniquely capable of efficiently propagating spatial information using local features during query time, thereby accurately retrieving and localizing objects within a database. Additionally, we innovatively utilize the structural information of the image graph through a technique we term `community selection`. This approach allows for the assessment of the initial search result's uncertainty and facilitates an optimal balance between accuracy and speed. This is particularly crucial in real-world applications where such trade-offs are often necessary. Our experimental results, conducted on the (P)ROxford and (P)RParis datasets, demonstrate the significant superiority of our method over existing diffusion techniques. We achieve state-of-the-art (SOTA) accuracy in both image-level and pixel-level retrieval, while also maintaining impressive processing speed. This dual achievement underscores the effectiveness of our hypergraph-based framework and community selection technique, marking a notable advancement in the field of content-based image retrieval.", "source": "Arxiv", "date": "2024-06-17", "funding_agencies": []}}, {"edge": "Quantum Networks: from Multipartite Entanglement to Hypergraph Immersion", "weight": 1, "attrs": {"tags": ["Quantum Physics", "Discrete Mathematics", "Social and Information Networks", "Computational Physics", "Physics and Society"], "abstract": "Multipartite entanglement, a higher-order interaction unique to quantum information, offers various advantages over bipartite entanglement in quantum network (QN) applications. Establishing multipartite entanglement across remote parties in QN requires entanglement routing, which irreversibly transforms the QN topology at the cost of existing entanglement links. Here, we address the question of whether a QN can be topologically transformed into another via entanglement routing. Our key result is an exact mapping from multipartite entanglement routing to Nash-Williams's graph immersion problem, extended to hypergraphs. This generalized hypergraph immersion problem introduces a partial order between QN topologies, permitting certain topological transformations while precluding others, offering discerning insights into the design and manipulation of higher-order network topologies in QNs.", "source": "Arxiv", "date": "2024-06-19", "funding_agencies": []}}, {"edge": "Anomaly Resilient Temporal QoS Prediction using Hypergraph Convoluted Transformer Network", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Quality-of-Service (QoS) prediction is a critical task in the service lifecycle, enabling precise and adaptive service recommendations by anticipating performance variations over time in response to evolving network uncertainties and user preferences. However, contemporary QoS prediction methods frequently encounter data sparsity and cold-start issues, which hinder accurate QoS predictions and limit the ability to capture diverse user preferences. Additionally, these methods often assume QoS data reliability, neglecting potential credibility issues such as outliers and the presence of greysheep users and services with atypical invocation patterns. Furthermore, traditional approaches fail to leverage diverse features, including domain-specific knowledge and complex higher-order patterns, essential for accurate QoS predictions. In this paper, we introduce a real-time, trust-aware framework for temporal QoS prediction to address the aforementioned challenges, featuring an end-to-end deep architecture called the Hypergraph Convoluted Transformer Network (HCTN). HCTN combines a hypergraph structure with graph convolution over hyper-edges to effectively address high-sparsity issues by capturing complex, high-order correlations. Complementing this, the transformer network utilizes multi-head attention along with parallel 1D convolutional layers and fully connected dense blocks to capture both fine-grained and coarse-grained dynamic patterns. Additionally, our approach includes a sparsity-resilient solution for detecting greysheep users and services, incorporating their unique characteristics to improve prediction accuracy. Trained with a robust loss function resistant to outliers, HCTN demonstrated state-of-the-art performance on the large-scale WSDREAM-2 datasets for response time and throughput.", "source": "Arxiv", "date": "2024-10-23", "funding_agencies": []}}, {"edge": "Borromean Hypergraph Formation in Dense Random Rectangles", "weight": 1, "attrs": {"tags": ["Soft Condensed Matter", "Statistical Mechanics", "General Topology"], "abstract": "We develop a minimal system to study the stochastic formation of Borromean links within topologically entangled networks without requiring the use of knot invariants. Borromean linkages may form in entangled solutions of open polymer chains or in Olympic gel systems such as kinetoplast DNA, but it is challenging to investigate this due to the difficulty of computing three-body link invariants. Here, we investigate randomly oriented rectangles densely packed within a volume, and evaluate them for Hopf linking and Borromean link formation. We show that dense packings of rectangles can form Borromean triplets and larger clusters, and that in high enough density the combination of Hopf and Borromean linking can create a percolating hypergraph through the network. We present data for the percolation threshold of Borromean hypergraphs, and discuss implications for the existence of Borromean connectivity within kinetoplast DNA.", "source": "Arxiv", "date": "2024-05-31", "funding_agencies": []}}, {"edge": "Text-Video Retrieval via Variational Multi-Modal Hypergraph Networks", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition", "Computation and Language"], "abstract": "Text-video retrieval is a challenging task that aims to identify relevant videos given textual queries. Compared to conventional textual retrieval, the main obstacle for text-video retrieval is the semantic gap between the textual nature of queries and the visual richness of video content. Previous works primarily focus on aligning the query and the video by finely aggregating word-frame matching signals. Inspired by the human cognitive process of modularly judging the relevance between text and video, the judgment needs high-order matching signal due to the consecutive and complex nature of video contents. In this paper, we propose chunk-level text-video matching, where the query chunks are extracted to describe a specific retrieval unit, and the video chunks are segmented into distinct clips from videos. We formulate the chunk-level matching as n-ary correlations modeling between words of the query and frames of the video and introduce a multi-modal hypergraph for n-ary correlation modeling. By representing textual units and video frames as nodes and using hyperedges to depict their relationships, a multi-modal hypergraph is constructed. In this way, the query and the video can be aligned in a high-order semantic space. In addition, to enhance the model's generalization ability, the extracted features are fed into a variational inference component for computation, obtaining the variational representation under the Gaussian distribution. The incorporation of hypergraphs and variational inference allows our model to capture complex, n-ary interactions among textual and visual contents. Experimental results demonstrate that our proposed method achieves state-of-the-art performance on the text-video retrieval task.", "source": "Arxiv", "date": "2024-01-06", "funding_agencies": []}}, {"edge": "Hyper-STTN: Social Group-aware Spatial-Temporal Transformer Network for Human Trajectory Prediction with Hypergraph Reasoning", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition", "Machine Learning"], "abstract": "Predicting crowded intents and trajectories is crucial in varouls real-world applications, including service robots and autonomous vehicles. Understanding environmental dynamics is challenging, not only due to the complexities of modeling pair-wise spatial and temporal interactions but also the diverse influence of group-wise interactions. To decode the comprehensive pair-wise and group-wise interactions in crowded scenarios, we introduce Hyper-STTN, a Hypergraph-based Spatial-Temporal Transformer Network for crowd trajectory prediction. In Hyper-STTN, crowded group-wise correlations are constructed using a set of multi-scale hypergraphs with varying group sizes, captured through random-walk robability-based hypergraph spectral convolution. Additionally, a spatial-temporal transformer is adapted to capture pedestrians' pair-wise latent interactions in spatial-temporal dimensions. These heterogeneous group-wise and pair-wise are then fused and aligned though a multimodal transformer network. Hyper-STTN outperformes other state-of-the-art baselines and ablation models on 5 real-world pedestrian motion datasets.", "source": "Arxiv", "date": "2024-01-11", "funding_agencies": []}}, {"edge": "VLSI Hypergraph Partitioning with Deep Learning", "weight": 1, "attrs": {"tags": ["Hardware Architecture", "Artificial Intelligence", "Distributed, Parallel, and Cluster Computing", "Machine Learning"], "abstract": "Partitioning is a known problem in computer science and is critical in chip design workflows, as advancements in this area can significantly influence design quality and efficiency. Deep Learning (DL) techniques, particularly those involving Graph Neural Networks (GNNs), have demonstrated strong performance in various node, edge, and graph prediction tasks using both inductive and transductive learning methods. A notable area of recent interest within GNNs are pooling layers and their application to graph partitioning. While these methods have yielded promising results across social, computational, and other random graphs, their effectiveness has not yet been explored in the context of VLSI hypergraph netlists. In this study, we introduce a new set of synthetic partitioning benchmarks that emulate real-world netlist characteristics and possess a known upper bound for solution cut quality. We distinguish these benchmarks with the prior work and evaluate existing state-of-the-art partitioning algorithms alongside GNN-based approaches, highlighting their respective advantages and disadvantages.", "source": "Arxiv", "date": "2024-09-02", "funding_agencies": []}}, {"edge": "Almost Tight Bounds for Online Hypergraph Matching", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "In the online hypergraph matching problem, hyperedges of size $k$ over a common ground set arrive online in adversarial order. The goal is to obtain a maximum matching (disjoint set of hyperedges). A na\\`ive greedy algorithm for this problem achieves a competitive ratio of $\\frac{1}{k}$. We show that no (randomized) online algorithm has competitive ratio better than $\\frac{2+o(1)}{k}$. If edges are allowed to be assigned fractionally, we give a deterministic online algorithm with competitive ratio $\\frac{1-o(1)}{\\ln(k)}$ and show that no online algorithm can have competitive ratio strictly better than $\\frac{1+o(1)}{\\ln(k)}$. Lastly, we give a $\\frac{1-o(1)}{\\ln(k)}$ competitive algorithm for the fractional edge-weighted version of the problem under a free disposal assumption.", "source": "Arxiv", "date": "2024-02-13", "funding_agencies": []}}, {"edge": "Basket-Enhanced Heterogenous Hypergraph for Price-Sensitive Next Basket Recommendation", "weight": 1, "attrs": {"tags": ["Information Retrieval"], "abstract": "Next Basket Recommendation (NBR) is a new type of recommender system that predicts combinations of items users are likely to purchase together. Existing NBR models often overlook a crucial factor, which is price, and do not fully capture item-basket-user interactions. To address these limitations, we propose a novel method called Basket-augmented Dynamic Heterogeneous Hypergraph (BDHH). BDHH utilizes a heterogeneous multi-relational graph to capture the intricate relationships among item features, with price as a critical factor. Moreover, our approach includes a basket-guided dynamic augmentation network that could dynamically enhances item-basket-user interactions. Experiments on real-world datasets demonstrate that BDHH significantly improves recommendation accuracy, providing a more comprehensive understanding of user behavior.", "source": "Arxiv", "date": "2024-09-18", "funding_agencies": []}}, {"edge": "Multiplexed Quantum Communication with Surface and Hypergraph Product Codes", "weight": 1, "attrs": {"tags": ["Quantum Physics", "Networking and Internet Architecture", "CODING AND INFORMATION THEORY", "COMPUTER-COMMUNICATION NETWORKS", "DISCRETE MATHEMATICS"], "abstract": "Connecting multiple processors via quantum interconnect technologies could help to overcome issues of scalability in single-processor quantum computers. Transmission via these interconnects can be performed more efficiently using quantum multiplexing, where information is encoded in high-dimensional photonic degrees of freedom. We explore the effects of multiplexing on logical error rates in surface codes and hypergraph product codes. We show that, although multiplexing makes loss errors more damaging, assigning qubits to photons in an intelligent manner can minimize these effects, and the ability to encode higher-distance codes in a smaller number of photons can result in overall lower logical error rates. This multiplexing technique can also be adapted to quantum communication and multimode quantum memory with high-dimensional qudit systems.", "source": "Arxiv", "date": "2024-06-13", "funding_agencies": []}}, {"edge": "High-order Neighborhoods Know More: HyperGraph Learning Meets Source-free Unsupervised Domain Adaptation", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Source-free Unsupervised Domain Adaptation (SFDA) aims to classify target samples by only accessing a pre-trained source model and unlabelled target samples. Since no source data is available, transferring the knowledge from the source domain to the target domain is challenging. Existing methods normally exploit the pair-wise relation among target samples and attempt to discover their correlations by clustering these samples based on semantic features. The drawback of these methods includes: 1) the pair-wise relation is limited to exposing the underlying correlations of two more samples, hindering the exploration of the structural information embedded in the target domain; 2) the clustering process only relies on the semantic feature, while overlooking the critical effect of domain shift, i.e., the distribution differences between the source and target domains. To address these issues, we propose a new SFDA method that exploits the high-order neighborhood relation and explicitly takes the domain shift effect into account. Specifically, we formulate the SFDA as a Hypergraph learning problem and construct hyperedges to explore the local group and context information among multiple samples. Moreover, we integrate a self-loop strategy into the constructed hypergraph to elegantly introduce the domain uncertainty of each sample. By clustering these samples based on hyperedges, both the semantic feature and domain shift effects are considered. We then describe an adaptive relation-based objective to tune the model with soft attention levels for all samples. Extensive experiments are conducted on Office-31, Office-Home, VisDA, and PointDA-10 datasets. The results demonstrate the superiority of our method over state-of-the-art counterparts.", "source": "Arxiv", "date": "2024-05-11", "funding_agencies": []}}, {"edge": "HINTs: Sensemaking on large collections of documents with Hypergraph visualization and INTelligent agents", "weight": 1, "attrs": {"tags": ["Human-Computer Interaction"], "abstract": "Sensemaking on a large collection of documents (corpus) is a challenging task often found in fields such as market research, legal studies, intelligence analysis, political science, computational linguistics, etc. Previous works approach this problem either from a topic- or entity-based perspective, but they lack interpretability and trust due to poor model alignment. In this paper, we present HINTs, a visual analytics approach that combines topic- and entity-based techniques seamlessly and integrates Large Language Models (LLMs) as both a general NLP task solver and an intelligent agent. By leveraging the extraction capability of LLMs in the data preparation stage, we model the corpus as a hypergraph that matches the user's mental model when making sense of the corpus. The constructed hypergraph is hierarchically organized with an agglomerative clustering algorithm by combining semantic and connectivity similarity. The system further integrates an LLM-based intelligent chatbot agent in the interface to facilitate sensemaking. To demonstrate the generalizability and effectiveness of the HINTs system, we present two case studies on different domains and a comparative user study. We report our insights on the behavior patterns and challenges when intelligent agents are used to facilitate sensemaking. We find that while intelligent agents can address many challenges in sensemaking, the visual hints that visualizations provide are necessary to address the new problems brought by intelligent agents. We discuss limitations and future work for combining interactive visualization and LLMs more profoundly to better support corpus analysis.", "source": "Arxiv", "date": "2024-03-05", "funding_agencies": []}}, {"edge": "On contention resolution for the hypergraph matching, knapsack, and $k$-column sparse packing problems", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms", "Optimization and Control"], "abstract": "The contention resolution framework is a versatile rounding technique used as a part of the relaxation and rounding approach for solving constrained submodular function maximization problems. We apply this framework to the hypergraph matching, knapsack, and $k$-column sparse packing problems. In the hypergraph matching setting, we adapt the technique of Guruganesh, Lee (2018) to non-constructively prove that the correlation gap is at least $\\frac{1-e^{-k}}{k}$ and provide a monotone $\\left(b,\\frac{1-e^{-bk}}{bk}\\right)$-balanced contention resolution scheme, generalizing the results of Bruggmann, Zenklusen (2019). For the knapsack problem, we prove that the correlation gap of instances where exactly $k$ copies of each item fit into the knapsack is at least $\\frac{1-e^{-2}}{2}$ and provide several monotone contention resolution schemes: a $\\frac{1-e^{-2}}{2}$-balanced scheme for instances where all item sizes are strictly bigger than $\\frac{1}{2}$, a $\\frac{4}{9}$-balanced scheme for instances where all item sizes are at most $\\frac{1}{2}$, and a $0.279$-balanced scheme for instances with arbitrary item sizes. For $k$-column sparse packing integer programs, we slightly modify the $\\left(2k+o\\left(k\\right)\\right)$-approximation algorithm for $k$-CS-PIP based on the strengthened LP relaxation presented in Brubach et al. (2019) to obtain a $\\frac{1}{4k+o\\left(k\\right)}$-balanced contention resolution scheme and hence a $\\left(4k+o\\left(k\\right)\\right)$-approximation algorithm for $k$-CS-PIP based on the natural LP relaxation.", "source": "Arxiv", "date": "2024-03-24", "funding_agencies": []}}, {"edge": "Community detection in the hypergraph stochastic block model and reconstruction on hypertrees", "weight": 1, "attrs": {"tags": ["Probability", "Information Theory", "Information Theory"], "abstract": "We study the weak recovery problem on the $r$-uniform hypergraph stochastic block model ($r$-HSBM) with two balanced communities. In this model, $n$ vertices are randomly divided into two communities, and size-$r$ hyperedges are added randomly depending on whether all vertices in the hyperedge are in the same community. The goal of weak recovery is to recover a non-trivial fraction of the communities given the hypergraph. Pal and Zhu (2021); Stephan and Zhu (2022) established that weak recovery is always possible above a natural threshold called the Kesten-Stigum (KS) threshold. For assortative models (i.e., monochromatic hyperedges are preferred), Gu and Polyanskiy (2023) proved that the KS threshold is tight if $r\\le 4$ or the expected degree $d$ is small. For other cases, the tightness of the KS threshold remained open. In this paper we determine the tightness of the KS threshold for a wide range of parameters. We prove that for $r\\le 6$ and $d$ large enough, the KS threshold is tight. This shows that there is no information-computation gap in this regime and partially confirms a conjecture of Angelini et al. (2015). On the other hand, we show that for $r\\ge 5$, there exist parameters for which the KS threshold is not tight. In particular, for $r\\ge 7$, the KS threshold is not tight if the model is disassortative (i.e., polychromatic hyperedges are preferred) or $d$ is large enough. This provides more evidence supporting the existence of an information-computation gap in these cases. Furthermore, we establish asymptotic bounds on the weak recovery threshold for fixed $r$ and large $d$. We also obtain a number of results regarding the broadcasting on hypertrees (BOHT) model, including the asymptotics of the reconstruction threshold for $r\\ge 7$ and impossibility of robust reconstruction at criticality.", "source": "Arxiv", "date": "2024-02-09", "funding_agencies": []}}, {"edge": "Ada-HGNN: Adaptive Sampling for Scalable Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Hypergraphs serve as an effective model for depicting complex connections in various real-world scenarios, from social to biological networks. The development of Hypergraph Neural Networks (HGNNs) has emerged as a valuable method to manage the intricate associations in data, though scalability is a notable challenge due to memory limitations. In this study, we introduce a new adaptive sampling strategy specifically designed for hypergraphs, which tackles their unique complexities in an efficient manner. We also present a Random Hyperedge Augmentation (RHA) technique and an additional Multilayer Perceptron (MLP) module to improve the robustness and generalization capabilities of our approach. Thorough experiments with real-world datasets have proven the effectiveness of our method, markedly reducing computational and memory demands while maintaining performance levels akin to conventional HGNNs and other baseline models. This research paves the way for improving both the scalability and efficacy of HGNNs in extensive applications. We will also make our codebase publicly accessible.", "source": "Arxiv", "date": "2024-05-22", "funding_agencies": []}}, {"edge": "Single-shot preparation of hypergraph product codes via dimension jump", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "Quantum error correction is a fundamental primitive of fault-tolerant quantum computing. But in order for error correction to proceed, one must first prepare the codespace of the underlying error-correcting code. A popular method for encoding quantum low-density parity-check codes is transversal initialization, where one begins in a product state and measures a set of stabilizer generators. In the presence of measurement errors however, this procedure is generically not fault-tolerant, and so one typically needs to repeat the measurements many times, resulting in a deep initialization circuit. We present a protocol that prepares the codespace of constant-rate hypergraph product codes in constant depth with $O(\\sqrt{n})$ spatial overhead, and we show that the protocol is robust even in the presence of measurement errors. Our construction is inspired by dimension-jumping in topological codes and leverages two properties that arise from the homological product of codes. We provide some improvements to lower the spatial overhead and discuss applications to fault-tolerant architectures.", "source": "Arxiv", "date": "2024-10-07", "funding_agencies": []}}, {"edge": "Mapping Cancer Stem Cell Markers Distribution:A Hypergraph Analysis Across Organs", "weight": 1, "attrs": {"tags": ["Biological Physics", "Quantitative Methods", "Hypergraphs", "Applications to biology and medical sciences", "Biophysics", "Applications of discrete Markov processes (social mobility, learning theory, industrial processes, etc.)", "General", "Graph Theory", "Applications"], "abstract": "This study presents an interdisciplinary approach to analyse the distribution of cancer stem cell markers (CSCMs) across various cancer-affected organs using hypergraphs. Cancer stem cells (CSCs) play a crucial role in cancer initiation, progression, and metastasis. By employing hypergraphs, we model the relationships between CSCM locations and cancerous organs, providing a comprehensive representation of these interactions. Initially, we utilised an unweighted incidence matrix and its Markov transition matrices to gain a dynamic perspective on CSCM distributions. This method allows us to observe how these markers spread and influence cancer progression in a dynamical context. By calculating mutual information for each node and hyperedge, our analysis uncovers complex interaction patterns between CSCMs and organs, highlighting the critical roles of certain markers in cancer progression and metastasis. Our approach offers a detailed representation of cancer stem cell networks, enhancing our understanding of the mechanisms driving cancer heterogeneity and metastasis. By integrating hypergraph theory with cancer biology, this study provides valuable insights for developing targeted cancer therapies.", "source": "Arxiv", "date": "2024-07-27", "funding_agencies": []}}, {"edge": "Principal eigenvectors and principal ratios in hypergraph Tur\\'an problems", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Graphs and matrices"], "abstract": "For a general class of hypergraph Tur\\'an problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Tur\\'an tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu, Cioab\\u{a}--Gregory, and Zhang. The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues, and provide some preliminary evidence for our conjecture.", "source": "Arxiv", "date": "2024-01-18", "funding_agencies": []}}, {"edge": "Code Reviewer Recommendation Based on a Hypergraph with Multiplex Relationships", "weight": 1, "attrs": {"tags": ["Software Engineering"], "abstract": "Code review is an essential component of software development, playing a vital role in ensuring a comprehensive check of code changes. However, the continuous influx of pull requests and the limited pool of available reviewer candidates pose a significant challenge to the review process, making the task of assigning suitable reviewers to each review request increasingly difficult. To tackle this issue, we present MIRRec, a novel code reviewer recommendation method that leverages a hypergraph with multiplex relationships. MIRRec encodes high-order correlations that go beyond traditional pairwise connections using degree-free hyperedges among pull requests and developers. This way, it can capture high-order implicit connectivity and identify potential reviewers. To validate the effectiveness of MIRRec, we conducted experiments using a dataset comprising 48,374 pull requests from ten popular open-source software projects hosted on GitHub. The experiment results demonstrate that MIRRec, especially without PR-Review Commenters relationship, outperforms existing stateof-the-art code reviewer recommendation methods in terms of ACC and MRR, highlighting its significance in improving the code review process.", "source": "Arxiv", "date": "2024-01-19", "funding_agencies": []}}, {"edge": "Vision HgNN: An Electron-Micrograph is Worth Hypergraph of Hypernodes", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Material characterization using electron micrographs is a crucial but challenging task with applications in various fields, such as semiconductors, quantum materials, batteries, etc. The challenges in categorizing electron micrographs include but are not limited to the complexity of patterns, high level of detail, and imbalanced data distribution(long-tail distribution). Existing methods have difficulty in modeling the complex relational structure in electron micrographs, hindering their ability to effectively capture the complex relationships between different spatial regions of micrographs. We propose a hypergraph neural network(HgNN) backbone architecture, a conceptually alternative approach, to better model the complex relationships in electron micrographs and improve material characterization accuracy. By utilizing cost-effective GPU hardware, our proposed framework outperforms popular baselines. The results of the ablation studies demonstrate that the proposed framework is effective in achieving state-of-the-art performance on benchmark datasets and efficient in terms of computational and memory requirements for handling large-scale electron micrograph-based datasets.", "source": "Arxiv", "date": "2024-08-21", "funding_agencies": []}}, {"edge": "From Basic to Extra Features: Hypergraph Transformer Pretrain-then-Finetuning for Balanced Clinical Predictions on EHR", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Electronic Health Records (EHRs) contain rich patient information and are crucial for clinical research and practice. In recent years, deep learning models have been applied to EHRs, but they often rely on massive features, which may not be readily available for all patients. We propose HTP-Star, which leverages hypergraph structures with a pretrain-then-finetune framework for modeling EHR data, enabling seamless integration of additional features. Additionally, we design two techniques, namely (1) Smoothness-inducing Regularization and (2) Group-balanced Reweighting, to enhance the model's robustness during fine-tuning. Through experiments conducted on two real EHR datasets, we demonstrate that HTP-Star consistently outperforms various baselines while striking a balance between patients with basic and extra features.", "source": "Arxiv", "date": "2024-06-09", "funding_agencies": []}}, {"edge": "SE3Set: Harnessing equivariant hypergraph neural networks for molecular representation learning", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Computational Physics"], "abstract": "In this paper, we develop SE3Set, an SE(3) equivariant hypergraph neural network architecture tailored for advanced molecular representation learning. Hypergraphs are not merely an extension of traditional graphs; they are pivotal for modeling high-order relationships, a capability that conventional equivariant graph-based methods lack due to their inherent limitations in representing intricate many-body interactions. To achieve this, we first construct hypergraphs via proposing a new fragmentation method that considers both chemical and three-dimensional spatial information of molecular system. We then design SE3Set, which incorporates equivariance into the hypergragh neural network. This ensures that the learned molecular representations are invariant to spatial transformations, thereby providing robustness essential for accurate prediction of molecular properties. SE3Set has shown performance on par with state-of-the-art (SOTA) models for small molecule datasets like QM9 and MD17. It excels on the MD22 dataset, achieving a notable improvement of approximately 20% in accuracy across all molecules, which highlights the prevalence of complex many-body interactions in larger molecules. This exceptional performance of SE3Set across diverse molecular structures underscores its transformative potential in computational chemistry, offering a route to more accurate and physically nuanced modeling.", "source": "Arxiv", "date": "2024-05-26", "funding_agencies": []}}, {"edge": "Hypergraph-Transformer (HGT) for Interactive Event Prediction in Laparoscopic and Robotic Surgery", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Understanding and anticipating intraoperative events and actions is critical for intraoperative assistance and decision-making during minimally invasive surgery. Automated prediction of events, actions, and the following consequences is addressed through various computational approaches with the objective of augmenting surgeons' perception and decision-making capabilities. We propose a predictive neural network that is capable of understanding and predicting critical interactive aspects of surgical workflow from intra-abdominal video, while flexibly leveraging surgical knowledge graphs. The approach incorporates a hypergraph-transformer (HGT) structure that encodes expert knowledge into the network design and predicts the hidden embedding of the graph. We verify our approach on established surgical datasets and applications, including the detection and prediction of action triplets, and the achievement of the Critical View of Safety (CVS). Moreover, we address specific, safety-related tasks, such as predicting the clipping of cystic duct or artery without prior achievement of the CVS. Our results demonstrate the superiority of our approach compared to unstructured alternatives.", "source": "Arxiv", "date": "2024-02-02", "funding_agencies": []}}, {"edge": "LLM-Guided Multi-View Hypergraph Learning for Human-Centric Explainable Recommendation", "weight": 1, "attrs": {"tags": ["Information Retrieval"], "abstract": "As personalized recommendation systems become vital in the age of information overload, traditional methods relying solely on historical user interactions often fail to fully capture the multifaceted nature of human interests. To enable more human-centric modeling of user preferences, this work proposes a novel explainable recommendation framework, i.e., LLMHG, synergizing the reasoning capabilities of large language models (LLMs) and the structural advantages of hypergraph neural networks. By effectively profiling and interpreting the nuances of individual user interests, our framework pioneers enhancements to recommendation systems with increased explainability. We validate that explicitly accounting for the intricacies of human preferences allows our human-centric and explainable LLMHG approach to consistently outperform conventional models across diverse real-world datasets. The proposed plug-and-play enhancement framework delivers immediate gains in recommendation performance while offering a pathway to apply advanced LLMs for better capturing the complexity of human interests across machine learning applications.", "source": "Arxiv", "date": "2024-01-16", "funding_agencies": []}}, {"edge": "Cellular sheaf Laplacians on the set of simplices of symmetric simplicial set induced by hypergraph", "weight": 1, "attrs": {"tags": ["Algebraic Topology", "Simplicial sets and complexes", "Cech types", "Sheaf cohomology"], "abstract": "We generalize cellular sheaf Laplacians on an ordered finite abstract simplicial complex to the set of simplices of a symmetric simplicial set. We construct a functor from the category of hypergraphs to the category of finite symmetric simplicial sets and define cellular sheaf Laplacians on the set of simplices of finite symmetric simplicial set induced by hypergraph. We provide formulas for cellular sheaf Laplacians and show that cellular sheaf Laplacian on an ordered finite abstract simplicial complex is exactly the ordered cellular sheaf Laplacian on the set of simplices induced by abstract simplicial complex.", "source": "Arxiv", "date": "2024-11-13", "funding_agencies": []}}, {"edge": "Hypergraph Cortical Cytoarchitectonic Parcellation with Multimodal Canine Brain Atlas", "weight": 1, "attrs": {"tags": ["neuroscience", "1", "neuroscience"], "abstract": "Brain atlases are vital tools in exploring the brain structure-function relationship. The burgeoning cross-species atlases have significantly accelerated our understanding of human brain development, evolution, function, and diseases. However, the existing coarse-grained macroscopic canine brain atlases greatly constrain their utility as an animal model for neurocognition research. Finer-grained brain atlas and partitions are crucial for decoding brain spatial heterogeneity and topology at different scales. Therefore, we conduct macroscopic and microscopic brain imaging to construct an interactive online dataset of multimodal canine brain atlas. Additionally, we develop a pioneering method for cortical cytoarchitectonic partitioning based on hypergraph learning. By integrating high-dimensional cytoarchitectonic features and spatial connections between cortical columns, the method leads to fine-grained partitioning patterns. This innovative approach aims to decode the biological heterogeneity of cortical microstructures, contributing to the structural annotation of canine atlas as well as public human brain atlases. The study not only offers valuable resources but also presents a novel zonation approach to investigate the cellular organization pattern and topology of the cortex.", "source": "Biorxiv", "date": "2024-05-09", "funding_agencies": []}}, {"edge": "Hyperedge Modeling in Hypergraph Neural Networks by using Densest Overlapping Subgraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Social and Information Networks"], "abstract": "Hypergraphs tackle the limitations of traditional graphs by introducing {\\em hyperedges}. While graph edges connect only two nodes, hyperedges connect an arbitrary number of nodes along their edges. Also, the underlying message-passing mechanisms in Hypergraph Neural Networks (HGNNs) are in the form of vertex-hyperedge-vertex, which let HGNNs capture and utilize richer and more complex structural information than traditional Graph Neural Networks (GNNs). More recently, the idea of overlapping subgraphs has emerged. These subgraphs can capture more information about subgroups of vertices without limiting one vertex belonging to just one group, allowing vertices to belong to multiple groups or subgraphs. In addition, one of the most important problems in graph clustering is to find densest overlapping subgraphs (DOS). In this paper, we propose a solution to the DOS problem via Agglomerative Greedy Enumeration (DOSAGE) algorithm as a novel approach to enhance the process of generating the densest overlapping subgraphs and, hence, a robust construction of the hypergraphs. Experiments on standard benchmarks show that the DOSAGE algorithm significantly outperforms the HGNNs and six other methods on the node classification task.", "source": "Arxiv", "date": "2024-09-16", "funding_agencies": []}}, {"edge": "HyperBrain: Anomaly Detection for Temporal Hypergraph Brain Networks", "weight": 1, "attrs": {"tags": ["Machine Learning", "Neurons and Cognition"], "abstract": "Identifying unusual brain activity is a crucial task in neuroscience research, as it aids in the early detection of brain disorders. It is common to represent brain networks as graphs, and researchers have developed various graph-based machine learning methods for analyzing them. However, the majority of existing graph learning tools for the brain face a combination of the following three key limitations. First, they focus only on pairwise correlations between regions of the brain, limiting their ability to capture synchronized activity among larger groups of regions. Second, they model the brain network as a static network, overlooking the temporal changes in the brain. Third, most are designed only for classifying brain networks as healthy or disordered, lacking the ability to identify abnormal brain activity patterns linked to biomarkers associated with disorders. To address these issues, we present HyperBrain, an unsupervised anomaly detection framework for temporal hypergraph brain networks. HyperBrain models fMRI time series data as temporal hypergraphs capturing dynamic higher-order interactions. It then uses a novel customized temporal walk (BrainWalk) and neural encodings to detect abnormal co-activations among brain regions. We evaluate the performance of HyperBrain in both synthetic and real-world settings for Autism Spectrum Disorder and Attention Deficit Hyperactivity Disorder(ADHD). HyperBrain outperforms all other baselines on detecting abnormal co-activations in brain networks. Furthermore, results obtained from HyperBrain are consistent with clinical research on these brain disorders. Our findings suggest that learning temporal and higher-order connections in the brain provides a promising approach to uncover intricate connectivity patterns in brain networks, offering improved diagnosis.", "source": "Arxiv", "date": "2024-10-02", "funding_agencies": []}}, {"edge": "Non-symmetric GHZ states; weighted hypergraph and controlled-unitary graph representations", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "Non-symmetric GHZ states ($n$-GHZ$_\\alpha$), characterized by unequal superpositions of $|00...0>$ and $|11...1>$, represent a significant yet underexplored class of multipartite entangled states with potential applications in quantum information. Despite their importance, the lack of a well-defined stabilizer formalism and corresponding graph representation has hindered their comprehensive study. In this paper, we address this gap by introducing two novel graph formalisms and stabilizers for non-symmetric GHZ states. First, we provide a weighted hypergraph representation and demonstrate that non-symmetric GHZ states are local unitary (LU) equivalent to fully connected weighted hypergraphs. Although these weighted hypergraphs are not stabilizer states, we show that they can be stabilized using local operations, and an ancilla. We further extend this framework to qudits, offering a specific form for non-symmetric qudit GHZ states and their LU equivalent weighted qudit hypergraphs. Second, we propose a graph formalism using controlled-unitary (CU) operations, showing that non-symmetric qudit GHZ states can be described using star-shaped CU graphs. Our findings enhance the understanding of non-symmetric GHZ states and their potential applications in quantum information science.", "source": "Arxiv", "date": "2024-08-05", "funding_agencies": []}}, {"edge": "FGeo-HyperGNet: Geometric Problem Solving Integrating Formal Symbolic System and Hypergraph Neural Network", "weight": 1, "attrs": {"tags": ["Artificial Intelligence"], "abstract": "Geometric problem solving has always been a long-standing challenge in the fields of automated reasoning and artificial intelligence. We built a neural-symbolic system to automatically perform human-like geometric deductive reasoning. The symbolic part is a formal system built on FormalGeo, which can automatically perform geomertic relational reasoning and algebraic calculations and organize the solving process into a solution hypertree with conditions as hypernodes and theorems as hyperedges. The neural part, called HyperGNet, is a hypergraph neural network based on the attention mechanism, including a encoder to effectively encode the structural and semantic information of the hypertree, and a solver to provide problem-solving guidance. The neural part predicts theorems according to the hypertree, and the symbolic part applies theorems and updates the hypertree, thus forming a predict-apply cycle to ultimately achieve readable and traceable automatic solving of geometric problems. Experiments demonstrate the correctness and effectiveness of this neural-symbolic architecture. We achieved a step-wised accuracy of 87.65% and an overall accuracy of 85.53% on the formalgeo7k datasets.", "source": "Arxiv", "date": "2024-02-18", "funding_agencies": []}}, {"edge": "Multi-view Hypergraph-based Contrastive Learning Model for Cold-Start Micro-video Recommendation", "weight": 1, "attrs": {"tags": ["Multimedia"], "abstract": "With the widespread use of mobile devices and the rapid growth of micro-video platforms such as TikTok and Kwai, the demand for personalized micro-video recommendation systems has significantly increased. Micro-videos typically contain diverse information, such as textual metadata, visual cues (e.g., cover images), and dynamic video content, significantly affecting user interaction and engagement patterns. However, most existing approaches often suffer from the problem of over-smoothing, which limits their ability to capture comprehensive interaction information effectively. Additionally, cold-start scenarios present ongoing challenges due to sparse interaction data and the underutilization of available interaction signals. To address these issues, we propose a Multi-view Hypergraph-based Contrastive learning model for cold-start micro-video Recommendation (MHCR). MHCR introduces a multi-view multimodal feature extraction layer to capture interaction signals from various perspectives and incorporates multi-view self-supervised learning tasks to provide additional supervisory signals. Through extensive experiments on two real-world datasets, we show that MHCR significantly outperforms existing video recommendation models and effectively mitigates cold-start challenges. Our code is available at https://anonymous.4open.science/r/MHCR-02EF.", "source": "Arxiv", "date": "2024-09-15", "funding_agencies": []}}, {"edge": "A Survey on Hypergraph Neural Networks: An In-Depth and Step-By-Step Guide", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Higher-order interactions (HOIs) are ubiquitous in real-world complex systems and applications. Investigation of deep learning for HOIs, thus, has become a valuable agenda for the data mining and machine learning communities. As networks of HOIs are expressed mathematically as hypergraphs, hypergraph neural networks (HNNs) have emerged as a powerful tool for representation learning on hypergraphs. Given the emerging trend, we present the first survey dedicated to HNNs, with an in-depth and step-by-step guide. Broadly, the present survey overviews HNN architectures, training strategies, and applications. First, we break existing HNNs down into four design components: (i) input features, (ii) input structures, (iii) message-passing schemes, and (iv) training strategies. Second, we examine how HNNs address and learn HOIs with each of their components. Third, we overview the recent applications of HNNs in recommendation, bioinformatics and medical science, time series analysis, and computer vision. Lastly, we conclude with a discussion on limitations and future directions.", "source": "Arxiv", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "A hypergraph model shows the carbon reduction potential of effective space use in housing", "weight": 1, "attrs": {"tags": ["Computational Geometry", "Graphics", "Data Analysis, Statistics and Probability"], "abstract": "Humans spend over 90% of their time in buildings which account for 40% of anthropogenic greenhouse gas (GHG) emissions, making buildings the leading cause of climate change. To incentivize more sustainable construction, building codes are used to enforce indoor comfort standards and maximum energy use. However, they currently only reward energy efficiency measures such as equipment or envelope upgrades and disregard the actual spatial configuration and usage. Using a new hypergraph model that encodes building floorplan organization and facilitates automatic geometry creation, we demonstrate that space efficiency outperforms envelope upgrades in terms of operational carbon emissions in 72%, 61% and 33% of surveyed buildings in Zurich, New York, and Singapore. Automatically generated floorplans for a case study in Zurich further increase access to daylight by up to 24%, revealing that auto-generated floorplans have the potential to improve the quality of residential spaces in terms of environmental performance and access to daylight.", "source": "Arxiv", "date": "2024-05-02", "funding_agencies": []}}, {"edge": "SMA-Hyper: Spatiotemporal Multi-View Fusion Hypergraph Learning for Traffic Accident Prediction", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Predicting traffic accidents is the key to sustainable city management, which requires effective address of the dynamic and complex spatiotemporal characteristics of cities. Current data-driven models often struggle with data sparsity and typically overlook the integration of diverse urban data sources and the high-order dependencies within them. Additionally, they frequently rely on predefined topologies or weights, limiting their adaptability in spatiotemporal predictions. To address these issues, we introduce the Spatiotemporal Multiview Adaptive HyperGraph Learning (SMA-Hyper) model, a dynamic deep learning framework designed for traffic accident prediction. Building on previous research, this innovative model incorporates dual adaptive spatiotemporal graph learning mechanisms that enable high-order cross-regional learning through hypergraphs and dynamic adaptation to evolving urban data. It also utilises contrastive learning to enhance global and local data representations in sparse datasets and employs an advance attention mechanism to fuse multiple views of accident data and urban functional features, thereby enriching the contextual understanding of risk factors. Extensive testing on the London traffic accident dataset demonstrates that the SMA-Hyper model significantly outperforms baseline models across various temporal horizons and multistep outputs, affirming the effectiveness of its multiview fusion and adaptive learning strategies. The interpretability of the results further underscores its potential to improve urban traffic management and safety by leveraging complex spatiotemporal urban data, offering a scalable framework adaptable to diverse urban environments.", "source": "Arxiv", "date": "2024-07-24", "funding_agencies": []}}, {"edge": "Hypergraph Neural Networks Reveal Spatial Domains from Single-cell Transcriptomics Data", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "The task of spatial clustering of transcriptomics data is of paramount importance. It enables the classification of tissue samples into diverse subpopulations of cells, which, in turn, facilitates the analysis of the biological functions of clusters, tissue reconstruction, and cell-cell interactions. Many approaches leverage gene expressions, spatial locations, and histological images to detect spatial domains; however, Graph Neural Networks (GNNs) as state of the art models suffer from a limitation in the assumption of pairwise connections between nodes. In the case of domain detection in spatial transcriptomics, some cells are found to be not directly related. Still, they are grouped as the same domain, which shows the incapability of GNNs for capturing implicit connections among the cells. While graph edges connect only two nodes, hyperedges connect an arbitrary number of nodes along their edges, which lets Hypergraph Neural Networks (HGNNs) capture and utilize richer and more complex structural information than traditional GNNs. We use autoencoders to address the limitation of not having the actual labels, which are well-suited for unsupervised learning. Our model has demonstrated exceptional performance, achieving the highest iLISI score of 1.843 compared to other methods. This score indicates the greatest diversity of cell types identified by our method. Furthermore, our model outperforms other methods in downstream clustering, achieving the highest ARI values of 0.51 and Leiden score of 0.60.", "source": "Arxiv", "date": "2024-10-23", "funding_agencies": []}}, {"edge": "Joint Hypergraph Rewiring and Memory-Augmented Forecasting Techniques in Digital Twin Technology", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Digital Twin technology creates virtual replicas of physical objects, processes, or systems by replicating their properties, data, and behaviors. This advanced technology offers a range of intelligent functionalities, such as modeling, simulation, and data-driven decision-making, that facilitate design optimization, performance estimation, and monitoring operations. Forecasting plays a pivotal role in Digital Twin technology, as it enables the prediction of future outcomes, supports informed decision-making, minimizes risks, driving improvements in efficiency, productivity, and cost reduction. Recently, Digital Twin technology has leveraged Graph forecasting techniques in large-scale complex sensor networks to enable accurate forecasting and simulation of diverse scenarios, fostering proactive and data-driven decision making. However, existing Graph forecasting techniques lack scalability for many real-world applications. They have limited ability to adapt to non-stationary environments, retain past knowledge, lack a mechanism to capture the higher order spatio-temporal dynamics, and estimate uncertainty in model predictions. To surmount the challenges, we introduce a hybrid architecture that enhances the hypergraph representation learning backbone by incorporating fast adaptation to new patterns and memory-based retrieval of past knowledge. This balance aims to improve the slowly-learned backbone and achieve better performance in adapting to recent changes. In addition, it models the time-varying uncertainty of multi-horizon forecasts, providing estimates of prediction uncertainty. Our forecasting architecture has been validated through ablation studies and has demonstrated promising results across multiple benchmark datasets, surpassing state-ofthe-art forecasting methods by a significant margin.", "source": "Arxiv", "date": "2024-08-22", "funding_agencies": []}}, {"edge": "Multimodal Fusion of EHR in Structures and Semantics: Integrating Clinical Records and Notes with Hypergraph and LLM", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Computation and Language"], "abstract": "Electronic Health Records (EHRs) have become increasingly popular to support clinical decision-making and healthcare in recent decades. EHRs usually contain heterogeneous information, such as structural data in tabular form and unstructured data in textual notes. Different types of information in EHRs can complement each other and provide a more complete picture of the health status of a patient. While there has been a lot of research on representation learning of structured EHR data, the fusion of different types of EHR data (multimodal fusion) is not well studied. This is mostly because of the complex medical coding systems used and the noise and redundancy present in the written notes. In this work, we propose a new framework called MINGLE, which integrates both structures and semantics in EHR effectively. Our framework uses a two-level infusion strategy to combine medical concept semantics and clinical note semantics into hypergraph neural networks, which learn the complex interactions between different types of data to generate visit representations for downstream prediction. Experiment results on two EHR datasets, the public MIMIC-III and private CRADLE, show that MINGLE can effectively improve predictive performance by 11.83% relatively, enhancing semantic integration as well as multimodal fusion for structural and textual EHR data.", "source": "Arxiv", "date": "2024-02-19", "funding_agencies": []}}, {"edge": "Enumeration of minimal transversals of hypergraphs of bounded VC-dimension", "weight": 1, "attrs": {"tags": ["Combinatorics", "Computational Complexity", "Discrete Mathematics", "Data Structures and Algorithms"], "abstract": "We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\\mathcal{H}$. An equivalent formulation of this problem known as the \\emph{transversal hypergraph} problem (or \\emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \\cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bounded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.", "source": "Arxiv", "date": "2024-06-30", "funding_agencies": []}}, {"edge": "Hypergraph Multi-modal Large Language Model: Exploiting EEG and Eye-tracking Modalities to Evaluate Heterogeneous Responses for Video Understanding", "weight": 1, "attrs": {"tags": ["Computer Vision and Pattern Recognition"], "abstract": "Understanding of video creativity and content often varies among individuals, with differences in focal points and cognitive levels across different ages, experiences, and genders. There is currently a lack of research in this area, and most existing benchmarks suffer from several drawbacks: 1) a limited number of modalities and answers with restrictive length; 2) the content and scenarios within the videos are excessively monotonous, transmitting allegories and emotions that are overly simplistic. To bridge the gap to real-world applications, we introduce a large-scale Subjective Response Indicators for Advertisement Videos dataset, namely SRI-ADV. Specifically, we collected real changes in Electroencephalographic (EEG) and eye-tracking regions from different demographics while they viewed identical video content. Utilizing this multi-modal dataset, we developed tasks and protocols to analyze and evaluate the extent of cognitive understanding of video content among different users. Along with the dataset, we designed a Hypergraph Multi-modal Large Language Model (HMLLM) to explore the associations among different demographics, video elements, EEG, and eye-tracking indicators. HMLLM could bridge semantic gaps across rich modalities and integrate information beyond different modalities to perform logical reasoning. Extensive experimental evaluations on SRI-ADV and other additional video-based generative performance benchmarks demonstrate the effectiveness of our method. The codes and dataset will be released at https://github.com/mininglamp-MLLM/HMLLM.", "source": "Arxiv", "date": "2024-07-10", "funding_agencies": []}}, {"edge": "Anti-Ramsey numbers of loose paths and cycles in uniform hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "For a fixed family of $r$-uniform hypergraphs $\\mathcal{F}$, the anti-Ramsey number of $\\mathcal{F}$, denoted by $ ar(n,r,\\mathcal{F})$, is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-uniform hypergraph on $n$ vertices with at least $c$ colors, there is a rainbow copy of some hypergraph in $\\mathcal{F}$. Here, a rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. Let $\\mathcal{P}_k$ and $\\mathcal{C}_k$ be the families of loose paths and loose cycles with $k$ edges in an $r$-uniform hypergraph, respectively. In this paper, we determine the exact values of $ ar(n,r,\\mathcal{P}_k)$ and $ ar(n,r,\\mathcal{C}_k)$ for all $k\\geq 4$ and $r\\geq 3$.", "source": "Arxiv", "date": "2024-05-07", "funding_agencies": []}}, {"edge": "Toward a comprehensive simulation framework for hypergraphs: a Python-base approach", "weight": 1, "attrs": {"tags": ["Mathematical Software"], "abstract": "Hypergraphs, or generalization of graphs such that edges can contain more than two nodes, have become increasingly prominent in understanding complex network analysis. Unlike graphs, hypergraphs have relatively few supporting platforms, and such dearth presents a barrier to more widespread adaptation of hypergraph computational toolboxes that could enable further research in several areas. Here, we introduce HyperRD, a Python package for hypergraph computation, simulation, and interoperability with other powerful Python packages in graph and hypergraph research. Then, we will introduce two models on hypergraph, the general Schelling's model and the SIR model, and simulate them with HyperRD.", "source": "Arxiv", "date": "2024-01-08", "funding_agencies": []}}, {"edge": "Extending Graph Burning to Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Graph burning is a round-based game or process that discretely models the spread of influence throughout a network. We introduce a generalization of graph burning which applies to hypergraphs, as well as a variant called ''lazy'' hypergraph burning. Interestingly, lazily burning a graph is trivial, while lazily burning a hypergraph can be quite complicated. Moreover, the lazy burning model is a useful tool for analyzing the round-based model. One of our key results is that arbitrary hypergraphs do not satisfy a bound analogous to the one in the Burning Number Conjecture for graphs. We also obtain bounds on the burning number and lazy burning number of a hypergraph in terms of its parameters, and present several open problems in the field of (lazy) hypergraph burning.", "source": "Arxiv", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Almost-Tight Bounds on Preserving Cuts in Classes of Submodular Hypergraphs", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.", "source": "Arxiv", "date": "2024-02-20", "funding_agencies": []}}, {"edge": "The complexity of recognizing $ABAB$-free hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs"], "abstract": "The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\\in A\\setminus B$ and $v_2,v_4\\in B\\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete.", "source": "Arxiv", "date": "2024-09-03", "funding_agencies": []}}, {"edge": "Community detection in hypergraphs via mutual information maximization", "weight": 1, "attrs": {"tags": ["97 MATHEMATICS AND COMPUTING", "Hypergraph Clustering", "AC05-76RL01830", "DMS 1916439", "AC05-76RL0 1830", "PNNL-SA-188428"], "abstract": "The hypergraph community detection problem seeks to identify groups of related vertices in hypergraph data. We propose an information-theoretic hypergraph community detection algorithm which compresses the observed data in terms of community labels and community-edge intersections. This algorithm can also be viewed as maximum-likelihood inference in a degree-corrected microcanonical stochastic blockmodel. We perform the compression/inference step via simulated annealing. Unlike several recent algorithms based on canonical models, our microcanonical algorithm does not require inference of statistical parameters such as vertex degrees or pairwise group connection rates. Through synthetic experiments, we find that our algorithm succeeds down to recently-conjectured thresholds for sparse random hypergraphs. We also find competitive performance in cluster recovery tasks on several hypergraph data sets.", "source": "Osti", "date": "2024-03-23", "funding_agencies": ["USDOE", "National Science Foundation (NSF)"]}}, {"edge": "Associating hypergraphs defined on loops", "weight": 1, "attrs": {"tags": ["Combinatorics", "Group Theory"], "abstract": "In this paper, we define a new hypergraph $\\mathcal{H(V,E)}$ on a loop $L$, where $\\mathcal{V}$ is the set of points of the loop $L$ and $\\mathcal{E}$ is the set of hyperedges $e=\\{x,y,z\\}$ such that $x,y$ and $z$ associate in the order they are written. We call this hypergraph as the associating hypergraph on a loop $L$. We study certain properites of associating hypergraphs on the Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.", "source": "Arxiv", "date": "2024-08-29", "funding_agencies": []}}, {"edge": "Off-diagonal Ramsey numbers for slowly growing hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Ramsey theory"], "abstract": "For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is $\\textit{slowly growing}$ if there is an ordering $e_1,e_2,\\dots,e_t$ of its edges such that $|e_i \\setminus \\bigcup_{j = 1}^{i - 1}e_j| \\leq 1$ for each $i \\in \\{2, \\ldots, t\\}$. We prove that if $k \\geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\\ge2$, \\[ r(F,n) = \\Omega\\Bigl(\\frac{n^k}{(\\log n)^{2k - 2}}\\Bigr).\\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\\mbox{polylog}(n)$, where $F_5$ is the triple system $\\{123, 124, 345\\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.", "source": "Arxiv", "date": "2024-09-02", "funding_agencies": []}}, {"edge": "On the Incidence matrices of hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Vector spaces, linear dependence, rank", "Graphs and matrices", "Degree sequences", "Enumeration of graphs and maps"], "abstract": "This study delves into the incidence matrices of hypergraphs, with a focus on two types: the edge-vertex incidence matrix and the vertex-edge incidence matrix. The edge-vertex incidence matrix is a matrix in which the rows represent hyperedges and the columns represent vertices. For a given hyperedge $e$ and vertex $u$, the $(e,u)$-th entry of the matrix is $1$ if $u$ is incident to $e$; otherwise, this entry is $0$. The vertex-edge incidence matrix is simply the transpose of the edge-vertex incidence matrix. This study examines the ranks and null spaces of these incidence matrices. It is shown that certain hypergraph structures, such as $k$-uniform cycles, units, and equal partitions of hyperedges and vertices, can influence specific vectors in the null space. In a hypergraph, a unit is a maximal collection of vertices that are incident with the same set of hyperedges. Identification of vertices within the same unit leads to a smaller hypergraph, known as unit contraction. The rank of the edge-vertex incidence matrix remains the same for both the original hypergraph and its unit contraction. Additionally, this study establishes connections between the edge-vertex incidence matrix and certain eigenvalues of the adjacency matrix of the hypergraph.", "source": "Arxiv", "date": "2024-09-24", "funding_agencies": []}}, {"edge": "When $t$-intersecting hypergraphs admit bounded $c$-strong colourings", "weight": 1, "attrs": {"tags": ["Combinatorics", "Coloring of graphs and hypergraphs"], "abstract": "The $c$-strong chromatic number of a hypergraph is the smallest number of colours needed to colour its vertices so that every edge sees at least $c$ colours or is rainbow. We show that every $t$-intersecting hypergraph has bounded $(t + 1)$-strong chromatic number, resolving a problem of Blais, Weinstein and Yoshida. In fact, we characterise when a $t$-intersecting hypergraph has large $c$-strong chromatic number for $c\\geq t+2$. Our characterisation also applies to hypergraphs which exclude sunflowers with specified parameters.", "source": "Arxiv", "date": "2024-06-19", "funding_agencies": []}}, {"edge": "HypeBoy: Generative Self-Supervised Representation Learning on Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Hypergraphs are marked by complex topology, expressing higher-order interactions among multiple nodes with hyperedges, and better capturing the topology is essential for effective representation learning. Recent advances in generative self-supervised learning (SSL) suggest that hypergraph neural networks learned from generative self supervision have the potential to effectively encode the complex hypergraph topology. Designing a generative SSL strategy for hypergraphs, however, is not straightforward. Questions remain with regard to its generative SSL task, connection to downstream tasks, and empirical properties of learned representations. In light of the promises and challenges, we propose a novel generative SSL strategy for hypergraphs. We first formulate a generative SSL task on hypergraphs, hyperedge filling, and highlight its theoretical connection to node classification. Based on the generative SSL task, we propose a hypergraph SSL method, HypeBoy. HypeBoy learns effective general-purpose hypergraph representations, outperforming 16 baseline methods across 11 benchmark datasets.", "source": "Arxiv", "date": "2024-03-31", "funding_agencies": []}}, {"edge": "Saturation in Random Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Probability"], "abstract": "Let $K^r_n$ be the complete $r$-uniform hypergraph on $n$ vertices, that is, the hypergraph whose vertex set is $[n]:=\\{1,2,...,n\\}$ and whose edge set is $\\binom{[n]}{r}$. We form $G^r(n,p)$ by retaining each edge of $K^r_n$ independently with probability $p$. An $r$-uniform hypergraph $H\\subseteq G$ is $F$-saturated if $H$ does not contain any copy of $F$, but any missing edge of $H$ in $G$ creates a copy of $F$. Furthermore, we say that $H$ is weakly $F$-saturated in $G$ if $H$ does not contain any copy of $F$, but the missing edges of $H$ in $G$ can be added back one-by-one, in some order, such that every edge creates a new copy of $F$. The smallest number of edges in an $F$-saturated hypergraph in $G$ is denoted by $sat(G,F)$, and in a weakly $F$-saturated hypergraph in $G$ by $wsat(G,F)$. In 2017, Kor\\'andi and Sudakov initiated the study of saturation in random graphs, showing that for constant $p$, with high probability $sat(G(n,p),K_s)=(1+o(1))n\\log_{\\frac{1}{1-p}}n$, and $wsat(G(n,p),K_s)=wsat(K_n,K_s)$. Generalising their results, in this paper, we solve the suturation problem for random hypergraphs for every $2\\le r < s$ and constant $p$.", "source": "Arxiv", "date": "2024-05-05", "funding_agencies": []}}, {"edge": "On the Matching Problem in Random Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal set theory"], "abstract": "We study a variant of the Erd\\H{o}s Matching Problem in random hypergraphs. Let $\\mathcal{K}_p(n,k)$ denote the Erd\\H{o}s-R\\'enyi random $k$-uniform hypergraph on $n$ vertices where each possible edge is included with probability $p$. We show that when $n\\gg k^{2}s$ and $p$ is not too small, with high probability, the maximum number of edges in a sub-hypergraph of $\\mathcal{K}_p(n,k)$ with matching number $s$ is obtained by the trivial sub-hypergraphs, i.e. the sub-hypergraph consisting of all edges containing at least one vertex in a fixed set of $s$ vertices.", "source": "Arxiv", "date": "2024-10-20", "funding_agencies": []}}, {"edge": "Mimicking Networks for Constrained Multicuts in Hypergraphs", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "In this paper, we study a \\emph{multicut-mimicking network} for a hypergraph over terminals $T$ with a parameter $c$. It is a hypergraph preserving the minimum multicut values of any set of pairs over $T$ where the value is at most $c$. This is a new variant of the multicut-mimicking network of a graph in [Wahlstr\\`om ICALP'20], which introduces a parameter $c$ and extends it to handle hypergraphs. Additionally, it is a natural extension of the \\emph{connectivity-$c$ mimicking network} introduced by [Chalermsook et al. SODA'21] and [Jiang et al. ESA'22] that is a (hyper)graph preserving the minimum cut values between two subsets of terminals where the value is at most $c$. We propose an algorithm for a hypergraph that returns a multicut-mimicking network over terminals $T$ with a parameter $c$ having $|T|c^{O(r\\log c)}$ hyperedges in $p^{1+o(1)}+|T|(c^r\\log n)^{\\tilde{O}(rc)}m$ time, where $p$ and $r$ are the total size and the rank, respectively, of the hypergraph.", "source": "Arxiv", "date": "2024-09-19", "funding_agencies": []}}, {"edge": "Berge Pancyclic hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "A Berge cycle of length $\\ell$ in a hypergraph is an alternating sequence of $\\ell$ distinct vertices and $\\ell$ distinct edges $v_1,e_1,v_2, \\ldots, v_\\ell, e_{\\ell}$ such that $\\{v_i, v_{i+1}\\} \\subseteq e_i$ for all $i$, with indices taken modulo $\\ell$. We call an $n$-vertex hypergraph pancyclic if it contains Berge cycles of every length from $3$ to $n$. We prove a sharp Dirac-type result guaranteeing pancyclicity in uniform hypergraphs: for $n \\geq 70$, $3 \\leq r \\leq \\lfloor (n-1)/2\\rfloor - 2$, if $\\cH$ is an $n$-vertex, $r$-uniform hypergraph with minimum degree at least ${\\lfloor (n-1)/2 \\rfloor \\choose r-1} + 1$, then $\\cH$ is pancyclic.", "source": "Arxiv", "date": "2024-10-29", "funding_agencies": []}}, {"edge": "Star clusters in independence complexes of hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Algebraic Topology", "Simplicial sets and complexes"], "abstract": "We study the concept of star clusters in simplicial complexes, which was introduced by Barmak in 2013, by relating it with the structure of hypergraphs that correspond to the simplicial complexes. This implies that for a hypergraph $H$ with a vertex $v$ that is not isolated and not contained in an induced Berge cycle of length $3$, there exists a hypergraph $H'$ with fewer vertices than $H$ such that the independence complex of $H$ is homotopy equivalent to the suspension of the independence complex of $H'$. As an application, we also prove that if a hypergraph has no Berge cycle of length $0$ mod $3$, then the sum of all reduced Betti numbers of its independence complex is at most $1$.", "source": "Arxiv", "date": "2024-08-26", "funding_agencies": []}}, {"edge": "Randomized algorithms to generate hypergraphs with given degree sequences", "weight": 1, "attrs": {"tags": ["Combinatorics", "Probability"], "abstract": "The question whether there exists a hypergraph whose degrees are equal to a given sequence of integers is a well-known reconstruction problem in graph theory, which is motivated by discrete tomography. In this paper we approach the problem by randomized algorithms which generate the required hypergraph with positive probability if the sequence satisfies certain constraints.", "source": "Arxiv", "date": "2024-02-07", "funding_agencies": []}}, {"edge": "Random matchings in linear hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "For a given hypergraph $H$ and a vertex $v\\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\\in V(H).$ This conjecture was proved for $k = 2$ by Kahn and Kim in $1998.$ In this paper, we disprove this conjecture for all $k \\geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $\\mathcal{P}(v_1 \\notin M) = 1 - \\frac{(1 + o_d(1))}{d^{k-2}}$ and $\\mathcal{P}(v_2 \\notin M) = \\frac{(1 + o_d(1))}{d+1}.$ The gap between $\\mathcal{P}(v_1 \\notin M)$ and $\\mathcal{P}(v_2 \\notin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil's result on matching polynomials and paths in graphs, which is of independent interest.", "source": "Arxiv", "date": "2024-06-10", "funding_agencies": []}}, {"edge": "Modularity Based Community Detection in Hypergraphs", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Machine Learning", "Model Development", "MATHEMATICAL SOFTWARE"], "abstract": "In this paper, we propose a scalable community detection algorithm using hypergraph modularity function, h-Louvain. It is an adaptation of the classical Louvain algorithm in the context of hypergraphs. We observe that a direct application of the Louvain algorithm to optimize the hypergraph modularity function often fails to find meaningful communities. We propose a solution to this issue by adjusting the initial stage of the algorithm via carefully and dynamically tuned linear combination of the graph modularity function of the corresponding two-section graph and the desired hypergraph modularity function. The process is guided by Bayesian optimization of the hyper-parameters of the proposed procedure. Various experiments on synthetic as well as real-world networks are performed showing that this process yields improved results in various regimes.", "source": "Arxiv", "date": "2024-06-25", "funding_agencies": []}}, {"edge": "Understanding High-Order Network Structure using Permissible Walks on Attributed Hypergraphs", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Combinatorics"], "abstract": "Hypergraphs have been a recent focus of study in mathematical data science as a tool to understand complex networks with high-order connections. One question of particular relevance is how to leverage information carried in hypergraph attributions when doing walk-based techniques. In this work, we focus on a new generalization of a walk in a network that recovers previous approaches and allows for a description of permissible walks in hypergraphs. Permissible walk graphs are constructed by intersecting the attributed $s$-line graph of a hypergraph with a relation respecting graph. The attribution of the hypergraph's line graph commonly carries over information from categorical and temporal attributions of the original hypergraph. To demonstrate this approach on a temporally attributed example, we apply our framework to a Reddit data set composed of hyperedges as threads and authors as nodes where post times are tracked.", "source": "Arxiv", "date": "2024-05-07", "funding_agencies": []}}, {"edge": "Encoding Reusable Multi-Robot Planning Strategies as Abstract Hypergraphs", "weight": 1, "attrs": {"tags": ["Robotics", "Artificial Intelligence", "Multiagent Systems"], "abstract": "Multi-Robot Task Planning (MR-TP) is the search for a discrete-action plan a team of robots should take to complete a task. The complexity of such problems scales exponentially with the number of robots and task complexity, making them challenging for online solution. To accelerate MR-TP over a system's lifetime, this work looks at combining two recent advances: (i) Decomposable State Space Hypergraph (DaSH), a novel hypergraph-based framework to efficiently model and solve MR-TP problems; and \\mbox{(ii) learning-by-abstraction,} a technique that enables automatic extraction of generalizable planning strategies from individual planning experiences for later reuse. Specifically, we wish to extend this strategy-learning technique, originally designed for single-robot planning, to benefit multi-robot planning using hypergraph-based MR-TP.", "source": "Arxiv", "date": "2024-09-16", "funding_agencies": []}}, {"edge": "The number of cliques in hypergraphs with forbidden subgraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We study the maximum number of $r$-vertex cliques in $(r-1)$-uniform hypergraphs not containing complete $r$-partite hypergraphs $K_r^{(r-1)}(a_1, \\dots, a_r)$. By using the hypergraph removal lemma, we show that this maximum is $o( n^{r - 1/(a_1 \\cdots a_{r-1})} )$. This immediately implies the corresponding results of Mubayi and Mukherjee and of Balogh, Jiang, and Luo for graphs. We also provide a lower bound by using hypergraph Tur\\'an numbers.", "source": "Arxiv", "date": "2024-05-13", "funding_agencies": []}}, {"edge": "Generating hypergraphs of finite groups", "weight": 1, "attrs": {"tags": ["Group Theory"], "abstract": "In a recent paper Cameron, Lakshmanan and Ajith began an exploration of hypergraphs defined on algebraic structures, especially groups, to investigate whether this can add a new perspective. Following their suggestions, we consider suitable hypergraphs encoding the generating properties of a finite group. In particular, answering a question asked in their paper, we classified the finite solvable groups whose generating hypergraph is the basis hypergraph of a matroid.", "source": "Arxiv", "date": "2024-04-17", "funding_agencies": []}}, {"edge": "Training-Free Message Passing for Learning on Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence", "Signal Processing", "Machine Learning"], "abstract": "Hypergraphs are crucial for modelling higher-order interactions in real-world data. Hypergraph neural networks (HNNs) effectively utilise these structures by message passing to generate informative node features for various downstream tasks like node classification. However, the message passing module in existing HNNs typically requires a computationally intensive training process, which limits their practical use. To tackle this challenge, we propose an alternative approach by decoupling the usage of hypergraph structural information from the model learning stage. This leads to a novel training-free message passing module, named TF-MP-Module, which can be precomputed in the data preprocessing stage, thereby reducing the computational burden. We refer to the hypergraph neural network equipped with our TF-MP-Module as TF-HNN. We theoretically support the efficiency and effectiveness of TF-HNN by showing that: 1) It is more training-efficient compared to existing HNNs; 2) It utilises as much information as existing HNNs for node feature generation; and 3) It is robust against the oversmoothing issue while using long-range interactions. Experiments based on seven real-world hypergraph benchmarks in node classification and hyperlink prediction show that, compared to state-of-the-art HNNs, TF-HNN exhibits both competitive performance and superior training efficiency. Specifically, on the large-scale benchmark, Trivago, TF-HNN outperforms the node classification accuracy of the best baseline by 10% with just 1% of the training time of that baseline.", "source": "Arxiv", "date": "2024-02-08", "funding_agencies": []}}, {"edge": "\\v{S}olt\\'es' hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Distance in graphs", "Hypergraphs"], "abstract": "More than $30$ years ago, \\v{S}olt\\'es observed that the total distance of the graph $C_{11}$ does not change by deleting a vertex, and wondered about the existence of other such graphs, called \\v{S}olt\\'es graphs. We extend the definition of \\v{S}olt\\'es' graphs to \\v{S}olt\\'es' hypergraphs, determine all orders for which a \\v{S}olt\\'es' hypergraph exists, observe infinitely many uniform \\v{S}olt\\'es' hypergraphs, and find the \\v{S}olt\\'es' hypergraph with minimum size (spoiler: it is not $C_{11}$).", "source": "Arxiv", "date": "2024-06-03", "funding_agencies": []}}, {"edge": "Hypergraph Horn Functions", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A survey of simplicial, relative, and chain complex homology theories for hypergraphs", "weight": 1, "attrs": {"tags": ["Algebraic Topology", "Other homology theories"], "abstract": "Hypergraphs have seen widespread applications in network and data science communities in recent years. We present a survey of recent work to define topological objects from hypergraphs -- specifically simplicial, relative, and chain complexes -- that can be used to build homology theories for hypergraphs. We define and describe nine different homology theories and their relevant topological objects. We discuss some interesting properties of each method to show how the hypergraph structures are preserved or destroyed by modifying a hypergraph. Finally, we provide a series of illustrative examples by computing many of these homology theories for small hypergraphs to show the variability of the methods and build intuition.", "source": "Arxiv", "date": "2024-09-26", "funding_agencies": []}}, {"edge": "Normal approximation for subgraph count in random hypergraphs", "weight": 1, "attrs": {"tags": ["Probability", "Combinatorics", "Hypergraphs", "Random graphs", "Central limit and other weak theorems"], "abstract": "A non-uniform and inhomogeneous random hypergraph model is considered, which is a straightforward extension of the celebrated binomial random graph model $\\mathbb G(n, p)$. We establish necessary and sufficient conditions for small hypergraph count to be asymptotically normal, and complement them with convergence rate in both the Wasserstein and Kolmogorov distances. Next we narrow our attention to the homogeneous model and relate the obtained results to the fourth moment phenomenon. Additionally, a short proof of necessity of aforementioned conditions is presented, which seems to be absent in the literature even in the context of the model $\\mathbb G(n, p)$.", "source": "Arxiv", "date": "2024-08-12", "funding_agencies": []}}, {"edge": "Generalized Ramsey numbers of cycles, paths, and hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Given a $k$-uniform hypergraph $G$ and a set of $k$-uniform hypergraphs $\\mathcal{H}$, the generalized Ramsey number $f(G,\\mathcal{H},q)$ is the minimum number of colors needed to edge-color $G$ so that every copy of every hypergraph $H\\in \\mathcal{H}$ in $G$ receives at least $q$ different colors. In this note we obtain bounds, some asymptotically sharp, on several generalized Ramsey numbers, when $G=K_n$ or $G=K_{n,n}$ and $\\mathcal{H}$ is a set of cycles or paths, and when $G=K_n^k$ and $\\mathcal{H}$ contains a clique on $k+2$ vertices or a tight cycle.", "source": "Arxiv", "date": "2024-05-24", "funding_agencies": []}}, {"edge": "Frustrated Random Walks: A Fast Method to Compute Node Distances on Hypergraphs", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Discrete Mathematics", "Machine Learning"], "abstract": "A hypergraph is a generalization of a graph that arises naturally when attribute-sharing among entities is considered. Compared to graphs, hypergraphs have the distinct advantage that they contain explicit communities and are more convenient to manipulate. An open problem in hypergraph research is how to accurately and efficiently calculate node distances on hypergraphs. Estimating node distances enables us to find a node's nearest neighbors, which has important applications in such areas as recommender system, targeted advertising, etc. In this paper, we propose using expected hitting times of random walks to compute hypergraph node distances. We note that simple random walks (SRW) cannot accurately compute node distances on highly complex real-world hypergraphs, which motivates us to introduce frustrated random walks (FRW) for this task. We further benchmark our method against DeepWalk, and show that while the latter can achieve comparable results, FRW has a distinct computational advantage in cases where the number of targets is fairly small. For such cases, we show that FRW runs in significantly shorter time than DeepWalk. Finally, we analyze the time complexity of our method, and show that for large and sparse hypergraphs, the complexity is approximately linear, rendering it superior to the DeepWalk alternative.", "source": "Arxiv", "date": "2024-01-23", "funding_agencies": []}}, {"edge": "Hypergraphs as Weighted Directed Self-Looped Graphs: Spectral Properties, Clustering, Cheeger Inequality", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Discrete Mathematics", "Data Structures and Algorithms", "Machine Learning"], "abstract": "Hypergraphs naturally arise when studying group relations and have been widely used in the field of machine learning. There has not been a unified formulation of hypergraphs, yet the recently proposed edge-dependent vertex weights (EDVW) modeling is one of the most generalized modeling methods of hypergraphs, i.e., most existing hypergraphs can be formulated as EDVW hypergraphs without any information loss to the best of our knowledge. However, the relevant algorithmic developments on EDVW hypergraphs remain nascent: compared to spectral graph theories, the formulations are incomplete, the spectral clustering algorithms are not well-developed, and one result regarding hypergraph Cheeger Inequality is even incorrect. To this end, deriving a unified random walk-based formulation, we propose our definitions of hypergraph Rayleigh Quotient, NCut, boundary/cut, volume, and conductance, which are consistent with the corresponding definitions on graphs. Then, we prove that the normalized hypergraph Laplacian is associated with the NCut value, which inspires our HyperClus-G algorithm for spectral clustering on EDVW hypergraphs. Finally, we prove that HyperClus-G can always find an approximately linearly optimal partitioning in terms of Both NCut and conductance. Additionally, we provide extensive experiments to validate our theoretical findings from an empirical perspective.", "source": "Arxiv", "date": "2024-10-23", "funding_agencies": []}}, {"edge": "Matching stability for 3-partite 3-uniform hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Let $n,k,s$ be three integers such that $k\\geq 2$ and $n\\geq s\\geq 1$. Let $H$ be a $k$-partite $k$-uniform hypergraph with $n$ vertices in each class. Aharoni (2017) showed that if $e(H)>(s-1)n^{k-1}$, then $H$ has a matching of size $s$. In this paper, we give a stability result for 3-partite 3-uniform hypergraphs: if $G$ is a $3$-partite $3$-uniform hypergraph with $n\\geq 162$ vertices in each class, $e(G)\\geq (s-1)n^2+3n-s$ and $G$ contains no matching of size $s+1$, then $G$ has a vertex cover of size $s$. Our bound is also tight.", "source": "Arxiv", "date": "2024-10-21", "funding_agencies": []}}, {"edge": "On the Construction of Singular and Cospectral Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "In this paper, we define two operations, neighbourhood m-splitting hypergraph $NS_m(\\mathscr{G}^*)$ and non-neighbourhood splitting hypergraph $NNS(\\mathscr{G}^*)$, and obtain several properties of their adjacency spectrum. We also estimate the energies of $NS_m(\\mathscr{G}^*)$ and $NNS(\\mathscr{G}^*)$. Moreover, we introduce two new join operations on $k$-uniform hypergraphs: the neighbourhood splitting V-vertex join $\\mathscr{G}_1^*\\veebar \\mathscr{G}_2^*$ and the S-vertex join $\\mathscr{G}_1^*\\barwedge \\mathscr{G}_2^*$ of hypergraphs $\\mathscr{G}_1^*$ and $\\mathscr{G}_2^*$, and determine their adjacency spectrum. As an application, we obtain infinite families of singular hypergraphs and infinite pairs of non-regular non-isomorphic cospectral hypergraphs.", "source": "Arxiv", "date": "2024-05-27", "funding_agencies": []}}, {"edge": "Hypergraph Dynamic System", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "On the tractability and approximability of non-submodular cardinality-based $s$-$t$ cut problems in hypergraphs", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms", "Computational Complexity", "Discrete Mathematics"], "abstract": "A minimum $s$-$t$ cut in a hypergraph is a bipartition of vertices that separates two nodes $s$ and $t$ while minimizing a hypergraph cut function. The cardinality-based hypergraph cut function assigns a cut penalty to each hyperedge based on the number of nodes in the hyperedge that are on each side of the split. Previous work has shown that when hyperedge cut penalties are submodular, this problem can be reduced to a graph $s$-$t$ cut problem and hence solved in polynomial time. NP-hardness results are also known for a certain class of non-submodular penalties, though the complexity remained open in many parameter regimes. In this paper we highlight and leverage a connection to Valued Constraint Satisfaction Problems to show that the problem is NP-hard for all non-submodular hyperedge cut penalty, except for one trivial case where a 0-cost solution is always possible. We then turn our attention to approximation strategies and approximation hardness results in the non-submodular case. We design a strategy for projecting non-submodular penalties to the submodular region, which we prove gives the optimal approximation among all such projection strategies. We also show that alternative approaches are unlikely to provide improved guarantees, by showing it is UGC-hard to obtain a better approximation in the simplest setting where all hyperedges have exactly 4 nodes.", "source": "Arxiv", "date": "2024-09-24", "funding_agencies": []}}, {"edge": "Hypergraph Isomorphism Computation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Hypergraphs and political structures", "weight": 1, "attrs": {"tags": ["Physics and Society", "Social and behavioral sciences", "History, political science"], "abstract": "Building on previous work, this paper extends the modeling of political structures from simplicial complexes to hypergraphs. This allows the analysis of more complex political dynamics where agents who are willing to form coalitions contain subsets that would not necessarily form coalitions themselves. We extend topological constructions such as wedge, cone, and collapse from simplicial complexes to hypergraphs and use them to study mergers, mediators, and power delegation in political structures. Concepts such as agent viability and system stability are generalized to the hypergraph context, alongside the introduction of the notion of local viability. Additionally, we use embedded homology of hypergraphs to analyze power concentration within political systems. Along the way, we introduce some new notions within the hypergraph framework that are of independent interest.", "source": "Arxiv", "date": "2024-04-22", "funding_agencies": []}}, {"edge": "Scalable Hypergraph Visualization", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Counting simplicial pairs in hypergraphs", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Discrete Mathematics", "Combinatorics"], "abstract": "We present two ways to measure the simplicial nature of a hypergraph: the simplicial ratio and the simplicial matrix. We show that the simplicial ratio captures the frequency, as well as the rarity, of simplicial interactions in a hypergraph while the simplicial matrix provides more fine-grained details. We then compute the simplicial ratio, as well as the simplicial matrix, for 10 real-world hypergraphs and, from the data collected, hypothesize that simplicial interactions are more and more deliberate as edge size increases. We then present a new Chung-Lu model that includes a parameter controlling (in expectation) the frequency of simplicial interactions. We use this new model, as well as the real-world hypergraphs, to show that multiple stochastic processes exhibit different behaviour when performed on simplicial hypergraphs vs. non-simplicial hypergraphs.", "source": "Arxiv", "date": "2024-08-21", "funding_agencies": []}}, {"edge": "Influence Maximization in Hypergraphs Using A Genetic Algorithm with New Initialization and Evaluation Methods", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Neural and Evolutionary Computing"], "abstract": "Influence maximization (IM) is a crucial optimization task related to analyzing complex networks in the real world, such as social networks, disease propagation networks, and marketing networks. Publications to date about the IM problem focus mainly on graphs, which fail to capture high-order interaction relationships from the real world. Therefore, the use of hypergraphs for addressing the IM problem has been receiving increasing attention. However, identifying the most influential nodes in hypergraphs remains challenging, mainly because nodes and hyperedges are often strongly coupled and correlated. In this paper, to effectively identify the most influential nodes, we first propose a novel hypergraph-independent cascade model that integrates the influences of both node and hyperedge failures. Afterward, we introduce genetic algorithms (GA) to identify the most influential nodes that leverage hypergraph collective influences. In the GA-based method, the hypergraph collective influence is effectively used to initialize the population, thereby enhancing the quality of initial candidate solutions. The designed fitness function considers the joint influences of both nodes and hyperedges. This ensures the optimal set of nodes with the best influence on both nodes and hyperedges to be evaluated accurately. Moreover, a new mutation operator is designed by introducing factors, i.e., the collective influence and overlapping effects of nodes in hypergraphs, to breed high-quality offspring. In the experiments, several simulations on both synthetic and real hypergraphs have been conducted, and the results demonstrate that the proposed method outperforms the compared methods.", "source": "Arxiv", "date": "2024-05-15", "funding_agencies": []}}, {"edge": "A classification model based on a population of hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Combinatorics"], "abstract": "This paper introduces a novel hypergraph classification algorithm. The use of hypergraphs in this framework has been widely studied. In previous work, hypergraph models are typically constructed using distance or attribute based methods. That is, hyperedges are generated by connecting a set of samples which are within a certain distance or have a common attribute. These methods however, do not often focus on multi-way interactions directly. The algorithm provided in this paper looks to address this problem by constructing hypergraphs which explore multi-way interactions of any order. We also increase the performance and robustness of the algorithm by using a population of hypergraphs. The algorithm is evaluated on two datasets, demonstrating promising performance compared to a generic random forest classification algorithm.", "source": "Arxiv", "date": "2024-05-23", "funding_agencies": []}}, {"edge": "Tensorized Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph: A Unified and Uniform Definition with Application to Chemical Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Spectral Tur\\'an problem for hypergraphs with bipartite or multipartite pattern", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "General criteria on spectral extremal problems for hypergraphs were developed by Keevash, Lenz, and Mubayi in their seminal work (SIAM J. Discrete Math., 2014), in which extremal results on \\alpha-spectral radius of hypergraphs for \\alpha>1 may be deduced from the corresponding hypergraph Tur\\'an problem which has the stability property and whose extremal construction satisfies some continuity assumptions. Using this criterion, we give two general spectral Tur\\'an results for hypergraphs with bipartite or mulitpartite pattern, transform corresponding the spectral Tur\\'an problems into pure combinatorial problems with respect to degree-stability of a nondegenerate k-graph family. As an application, we determine the maximum \\alpha-spectral radius for some classes of hypergraphs and characterize the corresponding extremal hypergraphs, such as the expansion of complete graphs, the generalized Fans, the cancellative hypergraphs, the generalized triangles, and a special book hypergraph.", "source": "Arxiv", "date": "2024-09-26", "funding_agencies": []}}, {"edge": "HygHD: Hyperdimensional Hypergraph Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Beyond Graphs: Can Large Language Models Comprehend Hypergraphs?", "weight": 1, "attrs": {"tags": ["Artificial Intelligence"], "abstract": "Existing benchmarks like NLGraph and GraphQA evaluate LLMs on graphs by focusing mainly on pairwise relationships, overlooking the high-order correlations found in real-world data. Hypergraphs, which can model complex beyond-pairwise relationships, offer a more robust framework but are still underexplored in the context of LLMs. To address this gap, we introduce LLM4Hypergraph, the first comprehensive benchmark comprising 21,500 problems across eight low-order, five high-order, and two isomorphism tasks, utilizing both synthetic and real-world hypergraphs from citation networks and protein structures. We evaluate six prominent LLMs, including GPT-4o, demonstrating our benchmark's effectiveness in identifying model strengths and weaknesses. Our specialized prompting framework incorporates seven hypergraph languages and introduces two novel techniques, Hyper-BAG and Hyper-COT, which enhance high-order reasoning and achieve an average 4% (up to 9%) performance improvement on structure classification tasks. This work establishes a foundational testbed for integrating hypergraph computational capabilities into LLMs, advancing their comprehension. The source codes are at https://github.com/iMoonLab/LLM4Hypergraph.", "source": "Arxiv", "date": "2024-10-13", "funding_agencies": []}}, {"edge": "Hypergraph reconstruction from dynamics", "weight": 1, "attrs": {"tags": ["Physics and Society", "Adaptation and Self-Organizing Systems", "Data Analysis, Statistics and Probability"], "abstract": "A plethora of methods have been developed in the past two decades to infer the underlying network structure of an interconnected system from its collective dynamics. However, methods capable of inferring nonpairwise interactions are only starting to appear. Here, we develop an inference algorithm based on sparse identification of nonlinear dynamics (SINDy) to reconstruct hypergraphs and simplicial complexes from time-series data. Our model-free method does not require information about node dynamics or coupling functions, making it applicable to complex systems that do not have a reliable mathematical description. We first benchmark the new method on synthetic data generated from Kuramoto and Lorenz dynamics. We then use it to infer the effective connectivity in the brain from resting-state EEG data, which reveals significant contributions from non-pairwise interactions in shaping the macroscopic brain dynamics.", "source": "Arxiv", "date": "2024-01-30", "funding_agencies": []}}, {"edge": "Hypergraph modeling and hypergraph multi-view attention neural network for link prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph Neural Architecture Search", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Engineering Hypergraph $b$-Matching Algorithms", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "Recently, researchers have extended the concept of matchings to the more general problem of finding $b$-matchings in hypergraphs broadening the scope of potential applications and challenges. The concept of $b$-matchings, where $b$ is a function that assigns positive integers to the vertices of the graph, is a natural extension of matchings in graphs, where each vertex $v$ is allowed to be matched to up to $b(v)$ edges, rather than just one. The weighted $b$-matching problem then seeks to select a subset of the hyperedges that fulfills the constraint and maximizes the weight. In this work, we engineer novel algorithms for this generalized problem. More precisely, we introduce exact data reductions for the problem as well as a novel greedy initial solution and local search algorithms. These data reductions allow us to significantly shrink the input size. This is done by either determining if a hyperedge is guaranteed to be in an optimum $b$-matching and thus can be added to our solution or if it can be safely ignored. Our iterated local search algorithm provides a framework for finding suitable improvement swaps of edges. Experiments on a wide range of real-world hypergraphs show that our new set of data reductions are highly practical, and our initial solutions are competitive for graphs and hypergraphs as well.", "source": "Arxiv", "date": "2024-08-13", "funding_agencies": []}}, {"edge": "Hypergraphs with uniform Tur\\'an density equal to 8/27", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "In the 1980s, Erd\\H{o}s and S\\'os initiated the study of Tur\\'an problems with a uniformity condition on the distribution of edges: the uniform Tur\\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least $d$ contains $H$. In particular, they asked to determine the uniform Tur\\'an densities of $K_4^{(3)-}$ and $K_4^{(3)}$. After more than 30 years, the former was solved in [Israel J. Math. 211 (2016), 349-366] and [J. Eur. Math. Soc. 20 (2018), 1139-1159], while the latter still remains open. Till today, there are known constructions of 3-uniform hypergraphs with uniform Tur\\'an density equal to 0, 1/27, 4/27 and 1/4 only. We extend this list by a fifth value: we prove an easy to verify condition for the uniform Tur\\'an density to be equal to 8/27 and identify hypergraphs satisfying this condition.", "source": "Arxiv", "date": "2024-07-08", "funding_agencies": []}}, {"edge": "Improved linearly ordered colorings of hypergraphs via SDP rounding", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "We consider the problem of linearly ordered (LO) coloring of hypergraphs. A hypergraph has an LO coloring if there is a vertex coloring, using a set of ordered colors, so that (i) no edge is monochromatic, and (ii) each edge has a unique maximum color. It is an open question as to whether or not a 2-LO colorable 3-uniform hypergraph can be LO colored with 3 colors in polynomial time. Nakajima and \\v{Z}ivn\\'{y} recently gave a polynomial-time algorithm to color such hypergraphs with $\\widetilde{O}(n^{1/3})$ colors and asked if SDP methods can be used directly to obtain improved bounds. Our main result is to show how to use SDP-based rounding methods to produce an LO coloring with $\\widetilde{O}(n^{1/5})$ colors for such hypergraphs. We show how to reduce the problem to cases with highly structured SDP solutions, which we call balanced hypergraphs. Then, we discuss how to apply classic SDP-rounding tools to obtain improved bounds.", "source": "Arxiv", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Cascading failures on interdependent hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Convolutional Signal Propagation: A Simple Scalable Algorithm for Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning"], "abstract": "Last decade has seen the emergence of numerous methods for learning on graphs, particularly Graph Neural Networks (GNNs). These methods, however, are often not directly applicable to more complex structures like bipartite graphs (equivalent to hypergraphs), which represent interactions among two entity types (e.g. a user liking a movie). This paper proposes Convolutional Signal Propagation (CSP), a non-parametric simple and scalable method that natively operates on bipartite graphs (hypergraphs) and can be implemented with just a few lines of code. After defining CSP, we demonstrate its relationship with well-established methods like label propagation, Naive Bayes, and Hypergraph Convolutional Networks. We evaluate CSP against several reference methods on real-world datasets from multiple domains, focusing on retrieval and classification tasks. Our results show that CSP offers competitive performance while maintaining low computational complexity, making it an ideal first choice as a baseline for hypergraph node classification and retrieval. Moreover, despite operating on hypergraphs, CSP achieves good results in tasks typically not associated with hypergraphs, such as natural language processing.", "source": "Arxiv", "date": "2024-09-26", "funding_agencies": []}}, {"edge": "Hypergraph Transformer for Knowledge Tracing", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Colour-bias perfect matchings in hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We study conditions under which an edge-coloured hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock and Sgueglia, and of Balogh, Treglown and Z\\'arate-Guer\\'en.", "source": "Arxiv", "date": "2024-08-20", "funding_agencies": []}}, {"edge": "Evolutionary game on any hypergraph", "weight": 1, "attrs": {"tags": ["Adaptation and Self-Organizing Systems"], "abstract": "Cooperation plays a fundamental role in societal and biological domains, and the population structure profoundly shapes the dynamics of evolution. Practically, individuals behave either altruistically or egoistically in multiple groups, such as relatives, friends and colleagues, and feedbacks from these groupwise interactions will contribute to one's cognition and behavior. Due to the intricacy within and between groups, exploration of evolutionary dynamics over hypergraphs is relatively limited to date. To uncover this conundrum, we develop a higher-order random walk framework for five distinct updating rules, thus establishing explicit conditions for cooperation emergence on hypergraphs, and finding the overlaps between groups tend to foster cooperative behaviors. Our systematic analysis quantifies how the order and hyperdegree govern evolutionary outcomes. We also discover that whenever following a group wisdom update protocol, choosing a high-fitness group to interact equally within its members, cooperators will significantly prevail throughout the community. These findings underscore a crucial role of higher-order interaction and interdisciplinary collaboration throughout a broad range of living systems, favoring social prosperity.", "source": "Arxiv", "date": "2024-04-04", "funding_agencies": []}}, {"edge": "Penalized Flow Hypergraph Local Clustering", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "A simple model of global cascades on random hypergraphs", "weight": 1, "attrs": {"tags": ["Physics and Society"], "abstract": "This study introduces a comprehensive framework that situates information cascades within the domain of higher-order interactions, utilizing a double-threshold hypergraph model. We propose that individuals (nodes) gain awareness of information through each communication channel (hyperedge) once the number of information adopters surpasses a threshold $\\phi_m$. However, actual adoption of the information only occurs when the cumulative influence across all communication channels exceeds a second threshold, $\\phi_k$. We analytically derive the cascade condition for both the case of a single seed node using percolation methods and the case of any seed size employing mean-field approximation. Our findings underscore that when considering the fractional seed size, $r_0 \\in (0,1]$, the connectivity pattern of the random hypergraph, characterized by the hyperdegree, $k$, and cardinality, $m$, distributions, exerts an asymmetric impact on the global cascade boundary. This asymmetry manifests in the observed differences in the boundaries of the global cascade within the $(\\phi_m, \\langle m \\rangle)$ and $(\\phi_k, \\langle k \\rangle)$ planes. However, as $r_0 \\to 0$, this asymmetric effect gradually diminishes. Overall, by elucidating the mechanisms driving information cascades within a broader context of higher-order interactions, our research contributes to theoretical advancements in complex systems theory.", "source": "Arxiv", "date": "2024-02-28", "funding_agencies": []}}, {"edge": "Engineering Hypergraph b-Matching Algorithms", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Uniquely colorable hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \\ge r \\ge 2$, let $\\Phi_{k,r}$ denote the minimum real number such that every $n$-vertex $k$-partite $r$-uniform hypergraph with positive codegree greater than $\\Phi_{k,r} \\cdot n$ and no isolated vertices is uniquely $k$-colorable. A classic result by of Bollob\\'{a}s\\cite{Bol78} established that $\\Phi_{k,2} = \\frac{3k-5}{3k-2}$ for every $k \\ge 2$. We consider the uniquely colorable problem for hypergraphs. Our main result determines the precise value of $\\Phi_{k,r}$ for all $k \\ge r \\ge 3$. In particular, we show that $\\Phi_{k,r}$ exhibits a phase transition at approximately $k = \\frac{4r-2}{3}$, a phenomenon not seen in the graph case. As an application of the main result, combined with a classic theorem by Frankl--F\\`{u}redi--Kalai, we derive general bounds for the analogous problem on minimum positive $i$-degrees for all $1\\leq i k\\geq3$.", "source": "Arxiv", "date": "2024-10-11", "funding_agencies": []}}, {"edge": "Scalable High-Quality Hypergraph Partitioning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Matroidal Cycles and Hypergraph Families", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We propose a novel definition of hypergraphical matroids, defined for arbitrary hypergraphs, simultaneously generalizing previous definitions for regular hypergraphs (Main, 1978), and for the hypergraphs of circuits of a matroid (Freij-Hollanti, Jurrius, Kuznetsova, 2023). As a consequence, we obtain a new notion of cycles in hypergraphs, and hypertrees. We give an equivalence relation on hypergraphs, according to when their so-called matroidal closures agree. Finally, we characterize hypergraphs that are isomorphic to the circuit hypergraphs of the associated matroids.", "source": "Arxiv", "date": "2024-10-31", "funding_agencies": []}}, {"edge": "DHMConv: Directed Hypergraph Momentum Convolution Framework", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Attention-Based Hypergraph Knowledge Tracing", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph operations preserving sc-greediness", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Comparison of modularity-based approaches for nodes clustering in hypergraphs", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Combinatorics", "Data Analysis, Statistics and Probability", "Applications"], "abstract": "Statistical analysis and node clustering in hypergraphs constitute an emerging topic suffering from a lack of standardization. In contrast to the case of graphs, the concept of nodes' community in hypergraphs is not unique and encompasses various distinct situations. In this work, we conducted a comparative analysis of the performance of modularity-based methods for clustering nodes in binary hypergraphs. To address this, we begin by presenting, within a unified framework, the various hypergraph modularity criteria proposed in the literature, emphasizing their differences and respective focuses. Subsequently, we provide an overview of the state-of-the-art codes available to maximize hypergraph modularities for detecting node communities in binary hypergraphs. Through exploration of various simulation settings with controlled ground truth clustering, we offer a comparison of these methods using different quality measures, including true clustering recovery, running time, (local) maximization of the objective, and the number of clusters detected. Our contribution marks the first attempt to clarify the advantages and drawbacks of these newly available methods. This effort lays the foundation for a better understanding of the primary objectives of modularity-based node clustering methods for binary hypergraphs.", "source": "Arxiv", "date": "2024-01-25", "funding_agencies": []}}, {"edge": "Hypergraph Neural Networks with Logic Clauses", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Minimizing External Vertices in Hypergraph Orientations", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph Matching for 3D Point Set", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Large Cuts in Hypergraphs via Energy", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is $r=2$, the resulting bound $m/2$ can be improved to $m/2+\\Omega(m^{1/2})$, and this is sharp. We prove that if $r\\geq 3$, then there is an $r$-partite subhypergraph with at least $\\frac{r!}{r^r} m+m^{3/5-o(1)}$ edges. Moreover, if the hypergraph is linear, this can be improved to $\\frac{r!}{r^r} m+m^{3/4-o(1)},$ which is tight up to the $o(1)$ term. These improve results of Conlon, Fox, Kwan, and Sudakov. Our proof is based on a combination of probabilistic, combinatorial, and linear algebraic techniques, and semidefinite programming. A key part of our argument is relating the energy $\\mathcal{E}(G)$ of a graph $G$ (i.e. the sum of absolute values of eigenvalues of the adjacency matrix) to its maximum cut. We prove that every $m$ edge multigraph $G$ has a cut of size at least $m/2+\\Omega(\\frac{\\mathcal{E}(G)}{\\log m})$, which might be of independent interest.", "source": "Arxiv", "date": "2024-10-02", "funding_agencies": []}}, {"edge": "On Metzler positive systems on hypergraphs", "weight": 1, "attrs": {"tags": ["Systems and Control", "Systems and Control"], "abstract": "In graph-theoretical terms, an edge in a graph connects two vertices while a hyperedge of a hypergraph connects any more than one vertices. If the hypergraph's hyperedges further connect the same number of vertices, it is said to be uniform. In algebraic graph theory, a graph can be characterized by an adjacency matrix, and similarly, a uniform hypergraph can be characterized by an adjacency tensor. This similarity enables us to extend existing tools of matrix analysis for studying dynamical systems evolving on graphs to the study of a class of polynomial dynamical systems evolving on hypergraphs utilizing the properties of tensors. To be more precise, in this paper, we first extend the concept of a Metzler matrix to a Metzler tensor and then describe some useful properties of such tensors. Next, we focus on positive systems on hypergraphs associated with Metzler tensors. More importantly, we design control laws to stabilize the origin of this class of Metzler positive systems on hypergraphs. In the end, we apply our findings to two classic dynamical systems: a higher-order Lotka-Volterra population dynamics system and a higher-order SIS epidemic dynamic process. The corresponding novel stability results are accompanied by ample numerical examples.", "source": "Arxiv", "date": "2024-01-07", "funding_agencies": []}}, {"edge": "A Survey on Hypergraph Representation Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Reordering and Compression for Hypergraph Processing", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Purity Skeleton Dynamic Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "The spectral property of hypergraph coverings", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Towards Multi-agent Reinforcement Learning based Traffic Signal Control through Spatio-temporal Hypergraphs", "weight": 1, "attrs": {"tags": ["Multiagent Systems", "Artificial Intelligence"], "abstract": "Traffic signal control systems (TSCSs) are integral to intelligent traffic management, fostering efficient vehicle flow. Traditional approaches often simplify road networks into standard graphs, which results in a failure to consider the dynamic nature of traffic data at neighboring intersections, thereby neglecting higher-order interconnections necessary for real-time control. To address this, we propose a novel TSCS framework to realize intelligent traffic control. This framework collaborates with multiple neighboring edge computing servers to collect traffic information across the road network. To elevate the efficiency of traffic signal control, we have crafted a multi-agent soft actor-critic (MA-SAC) reinforcement learning algorithm. Within this algorithm, individual agents are deployed at each intersection with a mandate to optimize traffic flow across the entire road network collectively. Furthermore, we introduce hypergraph learning into the critic network of MA-SAC to enable the spatio-temporal interactions from multiple intersections in the road network. This method fuses hypergraph and spatio-temporal graph structures to encode traffic data and capture the complex spatial and temporal correlations between multiple intersections. Our empirical evaluation, tested on varied datasets, demonstrates the superiority of our framework in minimizing average vehicle travel times and sustaining high-throughput performance. This work facilitates the development of more intelligent and reactive urban traffic management solutions.", "source": "Arxiv", "date": "2024-04-16", "funding_agencies": []}}, {"edge": "Realizability of hypergraphs and high-dimensional contingency tables with random degrees and marginals", "weight": 1, "attrs": {"tags": ["Combinatorics", "Probability"], "abstract": "A result of Deza, Levin, Meesum, and Onn shows that the problem of deciding if a given sequence is the degree sequence of a 3-uniform hypergraph is NP complete. We tackle this problem in the random case and show that a random integer partition can be realized as the degree sequence of a $3$-uniform hypergraph with high probability. These results are in stark contrast with the case of graphs, where a classical result of Erd\\H{o}s and Gallai provides an efficient algorithm for checking if a sequence is a degree sequence of a graph and a result of Pittel shows that with high probability a random partition is not the degree sequence of a graph. By the same method, we address analogous realizability problems about high-dimensional binary contingency tables. We prove that if $(\\lambda,\\mu,\\nu)$ are three independent random partitions then with high probability one can construct a three-dimensional binary contingency table with marginals $(\\lambda,\\mu,\\nu)$. Conversely, if one insists that the contingency table forms a pyramid shape, then we show that with high probability one cannot construct such a contingency table. These two results confirm two conjectures of Pak and Panova.", "source": "Arxiv", "date": "2024-08-20", "funding_agencies": []}}, {"edge": "Independent Sets in Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Ajtai, Koml\\'{o}s, Pintz, Spencer and Szemer\\'{e}di proved that every $(r + 1)$-uniform ``locally sparse'' hypergraph $H$ of average degree $d \\geq 1$ has an independent set of size \\[ \\Omega_r\\Bigl(\\frac{\\log d}{d}\\Bigr)^{\\frac{1}{r}} \\cdot |V(H)| \\] as $d \\rightarrow \\infty$. We give a short proof inspired by an approach of Shearer.", "source": "Arxiv", "date": "2024-09-29", "funding_agencies": []}}, {"edge": "The edge code of hypergraphs", "weight": 1, "attrs": {"tags": ["Commutative Algebra", "Information Theory", "Combinatorics", "Information Theory", "Hypergraphs", "Applications to coding theory and cryptography", "Geometric methods (including applications of algebraic geometry)"], "abstract": "Given a hypergraph $\\mathcal{H}$, we introduce a new class of evaluation toric codes called edge codes derived from $\\mathcal{H}$. We analyze these codes, focusing on determining their basic parameters. We provide estimations for the minimum distance, particularly in scenarios involving $d$-uniform clutters. Additionally, we demonstrate that these codes exhibit self-orthogonality. Furthermore, we compute the minimum distances of edge codes for all graphs with five vertices.", "source": "Arxiv", "date": "2024-04-02", "funding_agencies": []}}, {"edge": "Hypergraph Clustering with Path-Length Awareness", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Online Algorithms for Spectral Hypergraph Sparsification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Cascading failures with group support in interdependent hypergraphs", "weight": 1, "attrs": {"tags": ["Physics and Society"], "abstract": "The functionality of an entity frequently necessitates the support of a group situated in another layer of the system. To unravel the profound impact of such group support on a system's resilience against cascading failures, we devise a framework comprising a double-layer interdependent hypergraph system, wherein nodes are capable of receiving support via hyperedges. Our central hypothesis posits that the failure may transcend to another layer when all support groups of each dependent node fail, thereby initiating a potentially iterative cascade across layers. Through rigorous analytical methods, we derive the critical threshold for the initial node survival probability that marks the second-order phase transition point. A salient discovery is that as the prevalence of dependent nodes escalates, the system dynamics shift from a second-order to a first-order phase transition. Notably, irrespective of the collapse pattern, systems characterized by scale-free hyperdegree distributions within both hypergraph layers consistently demonstrate superior robustness compared to those adhering to Poisson hyperdegree distributions. In summary, our research underscores the paramount significance of group support mechanisms and intricate network topologies in determining the resilience of interconnected systems against the propagation of cascading failures. By exploring the interplay between these factors, we have gained insights into how systems can be designed or optimized to mitigate the risk of widespread disruptions, ensuring their continued functionality and stability in the face of adverse events.", "source": "Arxiv", "date": "2024-08-02", "funding_agencies": []}}, {"edge": "Geometric Aspects of Observability of Hypergraphs", "weight": 1, "attrs": {"tags": ["Dynamical Systems"], "abstract": "In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and unambiguously represent multi-way relationships which are ubiquitous in many real-world networks including those that arise in biology. We consider polynomial dynamical systems with linear outputs defined according to hypergraph structure, and we propose methods to evaluate local, weak observability.", "source": "Arxiv", "date": "2024-04-11", "funding_agencies": []}}, {"edge": "Hypergraph p-Laplacians and Scale Spaces", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "About Berge-F\\`uredi's conjecture on the chromatic index of hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Coloring of graphs and hypergraphs"], "abstract": "We show that the chromatic index of a hypergraph $\\mathcal{H}$ satisfies Berge-F\\`uredi conjectured bound $\\mathrm{q}(\\mathcal{H})\\le \\Delta([\\mathcal{H}]_2)+1$ under certain hypotheses on the antirank $\\mathrm{ar}(\\mathcal{H})$ or on the maximum degree $\\Delta(\\mathcal{H})$. This provides sharp information in connection with Erd\\H{o}s-Faber-Lov\\'asz Conjecture which deals with the coloring of a family of cliques that intersect pairwise in at most one vertex.", "source": "Arxiv", "date": "2024-03-14", "funding_agencies": []}}, {"edge": "Hypergraph Transformer for Semi-Supervised Classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HGRec: Group Recommendation With Hypergraph Convolutional Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Hypergraph network embedding for community detection", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Multi-Linear Pseudo-PageRank for Hypergraph Partitioning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Almost tight bounds for online hypergraph matching", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Long running times for hypergraph bootstrap percolation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Bisection Width, Discrepancy, and Eigenvalues of Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Computational Complexity"], "abstract": "A celebrated result of Alon from 1993 states that any $d$-regular graph on $n$ vertices (where $d=O(n^{1/9})$) has a bisection with at most $\\frac{dn}{2}(\\frac{1}{2}-\\Omega(\\frac{1}{\\sqrt{d}}))$ edges, and this is optimal. Recently, this result was greatly extended by R\\`aty, Sudakov, and Tomon. We build on the ideas of the latter, and use a semidefinite programming inspired approach to prove the following variant for hypergraphs: every $r$-uniform $d$-regular hypergraph on $n$ vertices (where $d\\ll n^{1/2}$) has a bisection of size at most $$\\frac{dn}{r}\\left(1-\\frac{1}{2^{r-1}}-\\frac{c}{\\sqrt{d}}\\right),$$ for some $c=c(r)>0$. This bound is the best possible up to the precise value of $c$. Moreover, a bisection achieving this bound can be found by a polynomial-time randomized algorithm. The minimum bisection is closely related to discrepancy. We also prove sharp bounds on the discrepancy and so called positive discrepancy of hypergraphs, extending results of Bollob\\'as and Scott. Furthermore, we discuss implications about Alon-Boppana type bounds. We show that if $H$ is an $r$-uniform $d$-regular hypergraph, then certain notions of second largest eigenvalue $\\lambda_2$ associated with the adjacency tensor satisfy $\\lambda_2\\geq \\Omega_r(\\sqrt{d})$, improving results of Li and Mohar.", "source": "Arxiv", "date": "2024-09-23", "funding_agencies": []}}, {"edge": "Isomorphisms between random $d$-hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Probability", "Combinatorial probability", "Random graphs", "Hypergraphs"], "abstract": "We characterize the size of the largest common induced subgraph of two independent random uniform $d$-hypergraphs of different sizes with $d\\geq 3$. More precisely, its distribution is asymptotically concentrated on two points, and we obtain as a consequence a phase transition for the inclusion of the smallest hypergraph in the largest one. This generalizes to uniform random $d$-hypergraphs the results of Chatterjee and Diaconis for uniform random graphs. Our proofs rely on the first and second moment methods.", "source": "Arxiv", "date": "2024-05-07", "funding_agencies": []}}, {"edge": "Quantum algorithms for hypergraph simplex finding", "weight": 1, "attrs": {"tags": ["Quantum Physics"], "abstract": "We study the quantum query algorithms for simplex finding, a generalization of triangle finding to hypergraphs. This problem satisfies a rank-reduction property: a quantum query algorithm for finding simplices in rank-$r$ hypergraphs can be turned into a faster algorithm for finding simplices in rank-$(r-1)$ hypergraphs. We then show that every nested Johnson graph quantum walk (with any constant number of nested levels) can be converted into an adaptive learning graph. Then, we introduce the concept of $\\alpha$-symmetric learning graphs, which is a useful framework for designing and analyzing complex quantum search algorithms. Inspired by the work of Le Gall, Nishimura, and Tani (2016) on $3$-simplex finding, we use our new technique to obtain an algorithm for $4$-simplex finding in rank-$4$ hypergraphs with $O(n^{2.46})$ quantum query cost, improving the trivial $O(n^{2.5})$ algorithm.", "source": "Arxiv", "date": "2024-08-30", "funding_agencies": []}}, {"edge": "Tur\\'{a}n problems for star-path forests in hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "An $r$-uniform hypergraph ($r$-graph) is linear if any two edges intersect at most one vertex. Let $\\mathcal{F}$ be a given family of $r$-graphs. A hypergraph $H$ is called $\\mathcal{F}$-free if $H$ does not contain any hypergraphs in $\\mathcal{F}$. The Tur\\'{a}n number ${\\rm{ex}}_r(n,\\mathcal{F})$ of $\\mathcal{F}$ is defined as the maximum number of edges of all $\\mathcal{F}$-free $r$-graphs on $n$ vertices, and the linear Tur\\'{a}n number ${\\rm{ex}}^{\\rm{lin}}_r(n,\\mathcal{F})$ of $\\mathcal{F}$ is defined as the Tur\\'{a}n number of $\\mathcal{F}$ in linear host hypergraphs. An $r$-uniform linear path $P^r_\\ell$ of length $\\ell$ is an $r$-graph with edges $e_1,\\cdots,e_\\ell$ such that $|V(e_i)\\cap V(e_j)|=1$ if $|i-j|=1$, and $V(e_i)\\cap V(e_j)=\\emptyset$ for $i\\neq j$ otherwise. Gy\\'{a}rf\\'{a}s et al. [Linear Tur\\'{a}n numbers of acyclic triple systems, European J. Combin., 2022, 103435] obtained an upper bound for the linear Tur\\'{a}n number of $P_\\ell^3$. In this paper, an upper bound for the linear Tur\\'{a}n number of $P_\\ell^r$ is obtained, which generalizes the result of $P_\\ell^3$ to any $P_\\ell^r$. Furthermore, some results for the linear Tur\\'{a}n number and Tur\\'{a}n number of some linear star-path forests are obtained.", "source": "Arxiv", "date": "2024-03-11", "funding_agencies": []}}, {"edge": "The structural evolution of temporal hypergraphs through the lens of hyper-cores", "weight": 1, "attrs": {"tags": ["Physics and Society", "Social and Information Networks"], "abstract": "The richness of many complex systems stems from the interactions among their components. The higher-order nature of these interactions, involving many units at once, and their temporal dynamics constitute crucial properties that shape the behaviour of the system itself. An adequate description of these systems is offered by temporal hypergraphs, that integrate these features within the same framework. However, tools for their temporal and topological characterization are still scarce. Here we develop a series of methods specifically designed to analyse the structural properties of temporal hypergraphs at multiple scales. Leveraging the hyper-core decomposition of hypergraphs, we follow the evolution of the hyper-cores through time, characterizing the hypergraph structure and its temporal dynamics at different topological scales, and quantifying the multi-scale structural stability of the system. We also define two static hypercoreness centrality measures that provide an overall description of the nodes aggregated structural behaviour. We apply the characterization methods to several data sets, establishing connections between structural properties and specific activities within the systems. Finally, we show how the proposed method can be used as a model-validation tool for synthetic temporal hypergraphs, distinguishing the higher-order structures and dynamics generated by different models from the empirical ones, and thus identifying the essential model mechanisms to reproduce the empirical hypergraph structure and evolution. Our work opens several research directions, from the understanding of dynamic processes on temporal higher-order networks to the design of new models of time-varying hypergraphs.", "source": "Arxiv", "date": "2024-02-09", "funding_agencies": []}}, {"edge": "Heterogeneous Hypergraph Structure Learning for Multimedia Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph motifs and their extensions beyond binary", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Quantitative Helly-type Theorems via Hypergraph Chains", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Counting Perfect Matchings In Dirac Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs"], "abstract": "One of the foundational theorems of extremal graph theory is Dirac's theorem, which says that if an n-vertex graph G has minimum degree at least n/2, then G has a Hamilton cycle, and therefore a perfect matching (if n is even). Later work by S\\'arkozy, Selkow and Szemer\\'edi showed that in fact Dirac graphs have many Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph G (in terms of an entropy-like parameter of G). In this paper we extend Cuckler and Kahn's result to perfect matchings in hypergraphs. For positive integers d < k, and for n divisible by k, let $m_{d}(k,n)$ be the minimum d-degree that ensures the existence of a perfect matching in an n-vertex k-uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of $m_{d}(k,n)$, but we are nonetheless able to prove an analogue of the Cuckler-Kahn theorem, showing that if an n-vertex k-uniform hypergraph G has minimum d-degree at least $(1+\\gamma)m_{d}(k,n)$ (for any constant $\\gamma>0$), then the number of perfect matchings in G is controlled by an entropy-like parameter of G. This strengthens cruder estimates arising from work of Kang-Kelly-K\\`uhn-Osthus-Pfenninger and Pham-Sah-Sawhney-Simkin.", "source": "Arxiv", "date": "2024-08-18", "funding_agencies": []}}, {"edge": "Collaborative contrastive learning for hypergraph node classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "On the zeroes of hypergraph independence polynomials", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A note on colour-bias perfect matchings in hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "A result of Balogh, Csaba, Jing and Pluh\\'ar yields the minimum degree threshold that ensures a $2$-coloured graph contains a perfect matching of significant colour-bias (i.e., a perfect matching that contains significantly more than half of its edges in one colour). In this note we prove an analogous result for perfect matchings in $k$-uniform hypergraphs. More precisely, for each $2\\leq \\ell \\frac{m-1}{1-2^{\\frac{1}{1-h}}}+h-1$, a partial $\\textbf{r}$-factorization of $\\lambda K_m^h$ can be extended to an $\\textbf{r}$-factorization of $\\lambda K_n^h$ if and only if the obvious necessary conditions are satisfied.", "source": "Arxiv", "date": "2024-09-17", "funding_agencies": []}}, {"edge": "A Hypergraph-based Model for Cyberincident-related Data Analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Text-Video Retrieval via Multi-Modal Hypergraph Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Self-Supervised Hypergraph Learning for Enhanced Multimodal Representation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Distance Enhanced Hypergraph Learning for Dynamic Node Classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-10-01", "funding_agencies": []}}, {"edge": "MedPart: A Multi-Level Evolutionary Differentiable Hypergraph Partitioner", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Generalized Ramsey numbers via conflict-free hypergraph matchings", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Given graphs $G, H$ and an integer $q \\ge 2$, the generalized Ramsey number, denoted $r(G,H,q)$, is the minimum number of colours needed to edge-colour $G$ such that every copy of $H$ receives at least $q$ colours. In this paper, we prove that for a fixed integer $k \\ge 3$, we have $r(K_n,C_k,3) = n/(k-2)+o(n)$. This generalises work of Joos and Muybayi, who proved $r(K_n,C_4,3) = n/2+o(n)$. We also provide an upper bound on $r(K_{n,n}, C_k, 3)$, which generalises a result of Joos and Mubayi that $r(K_{n,n},C_4,3) = 2n/3+o(n)$. Both of our results are in fact specific cases of more general theorems concerning families of cycles.", "source": "Arxiv", "date": "2024-05-26", "funding_agencies": []}}, {"edge": "Course contrastive recommendation algorithm based on hypergraph convolution", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "CINA: Curvature-Based Integrated Network Alignment with Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Windowed hypergraph Fourier transform and vertex-frequency representation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Mean-field limit of non-exchangeable multi-agent systems over hypergraphs with unbounded rank", "weight": 1, "attrs": {"tags": ["Analysis of PDEs"], "abstract": "Interacting particle systems are known for their ability to generate large-scale self-organized structures from simple local interaction rules between each agent and its neighbors. In addition to studying their emergent behavior, a main focus of the mathematical community has been concentrated on deriving their large-population limit. In particular, the mean-field limit consists of describing the limit system by its population density in the product space of positions and labels. The strategy to derive such limits is often based on a careful combination of methods ranging from analysis of PDEs and stochastic analysis, to kinetic equations and graph theory. In this article, we focus on a generalization of multi-agent systems that includes higher-order interactions, which has largely captured the attention of the applied community in the last years. In such models, interactions between individuals are no longer assumed to be binary (i.e. between a pair of particles). Instead, individuals are allowed to interact by groups so that a full group jointly generates a non-linear force on any given individual. The underlying graph of connections is then replaced by a hypergraph, which we assume to be dense, but possibly non uniform and of unbounded rank. For the first time in the literature, we show that when the interaction kernels are regular enough, then the mean-field limit is determined by a limiting Vlasov-type equation, where the hypergraph limit is encoded by a so-called UR-hypergraphon (unbounded-rank hypergraphon), and where the resulting mean-field force admits infinitely-many orders of interactions.", "source": "Arxiv", "date": "2024-06-07", "funding_agencies": []}}, {"edge": "Hypergraph-Mlp: Learning on Hypergraphs Without Message Passing", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "The Maximal Running Time of Hypergraph Bootstrap Percolation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multilevel polynomial partitioning and semialgebraic hypergraphs: regularity, Tur\\'an, and Zarankiewicz results", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "We prove three main results about semialgebraic hypergraphs. First, we prove an optimal and oblivious regularity lemma. Fox, Pach, and Suk proved that the class of $k$-uniform semialgebraic hypergraphs satisfies a very strong regularity lemma where the vertex set can be partitioned into $\\mathrm{poly}(1/\\varepsilon)$ parts so that all but an $\\varepsilon$-fraction of $k$-tuples of parts are homogeneous (either complete or empty). Our result improves the number of parts in the partition to $O_{d,k}((D/\\varepsilon)^{d})$ where $d$ is the dimension of the ambient space and $D$ is a measure of the complexity of the hypergraph; additionally, the partition is oblivious to the edge set of the hypergraph. We give examples that show that the dependence on both $\\varepsilon$ and $D$ is optimal. From this regularity lemma we deduce the best-known Tur\\'an-type result for semialgebraic hypergraphs. Third, we prove a Zarankiewicz-type result for semialgebraic hypergraphs. Previously Fox, Pach, Sheffer, Suk, and Zahl showed that a $K_{u,u}$-free semialgebraic graph on $N$ vertices has at most $O_{d,D,u}(N^{2d/(d+1)+o(1)})$ edges and Do extended this result to $K_{u,\\ldots,u}^{(k)}$-free semialgebraic hypergraphs. We improve upon both of these results by removing the $o(1)$ in the exponent and making the dependence on $D$ and $u$ explicit and polynomial. All three of these results follow from a novel ``multilevel polynomial partitioning scheme'' that efficiently partitions a point set $P\\subset\\mathbb{R}^d$ via low-complexity semialgebraic pieces. We prove this result using the polynomial method over varieties as developed by Walsh which extends the real polynomial partitioning technique of Guth and Katz. We give additional applications to the unit distance problem, the Erd\\H{o}s--Hajnal problem for semialgebraic graphs, and property testing of semialgebraic hypergraphs.", "source": "Arxiv", "date": "2024-07-29", "funding_agencies": []}}, {"edge": "Navigating Social Networks: A Hypergraph Approach to Influence Optimization", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Distributed constrained combinatorial optimization leveraging hypergraph neural networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Dynamic hypergraph convolutional network for multimodal sentiment analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Tur\u00e1n theorems for even cycles in random hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Dual-view hypergraph attention network for news recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph rewriting and Causal structure of \u03bb-calculus", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "On the number of H-free hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Two central problems in extremal combinatorics are concerned with estimating the number $ex(n,H)$, the size of the largest $H$-free hypergraph on $n$ vertices, and the number $forb(n,H)$ of $H$-free hypergraph on $n$ vertices. While it is known that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ for $k$-uniform hypergraphs that are not $k$-partite, estimates for hypergraphs that are $k$-partite (or degenerate) are not nearly as tight. In a recent breakthrough, Ferber, McKinley, and Samotij proved that for many degenerate hypergraphs $H$, $forb(n, H) = 2^{O(ex(n,H))}$. However, there are few known instances of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. In this paper, we show that $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds for a wide class of degenerate hypergraphs known as $2$-contractible hypertrees. This is the first known infinite family of degenerate hypergraphs $H$ for which $forb(n,H)=2^{(1+o(1))ex(n,H)}$ holds. As a corollary of our main results, we obtain a surprisingly sharp estimate of $forb(n,C^{(k)}_\\ell)=2^{(\\lfloor\\frac{\\ell-1}{2}\\rfloor+o(1))\\binom{n}{k-1}}$ for the $k$-uniform linear $\\ell$-cycle, for all pairs $k\\geq 5, \\ell\\geq 3$, thus settling a question of Balogh, Narayanan, and Skokan affirmatively for all $k\\geq 5, \\ell\\geq 3$. Our methods also lead to some related sharp results on the corresponding random Turan problem. As a key ingredient of our proofs, we develop a novel supersaturation variant of the delta systems method for set systems, which may be of independent interest.", "source": "Arxiv", "date": "2024-09-10", "funding_agencies": []}}, {"edge": "EFBH: Collaborative Filtering Model Based on Multi-Hypergraph Encoder", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Disentangled Contrastive Hypergraph Learning for Next POI Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph convolutional network for longitudinal data analysis in Alzheimer's disease", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "PhGraph: A High-Performance ReRAM-Based Accelerator for Hypergraph Applications", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Hypergraph-Enhanced Self-Supervised Robust Graph Learning for Social Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "An Efficient Hypergraph Partitioner under Inter - Block Interconnection Constraints", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Dual-level Hypergraph Contrastive Learning with Adaptive Temperature Enhancement", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "On hypergraph Tur\\'an problems with bounded matching number", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Very recently, Alon and Frankl, and Gerbner studied the maximum number of edges in $n$-vertex $F$-free graphs with bounded matching number, respectively. We consider the analogous Tur\\'{a}n problems on hypergraphs with bounded matching number, and we obtain some exact results.", "source": "Arxiv", "date": "2024-10-09", "funding_agencies": []}}, {"edge": "HGNN4Perf: Detecting Performance Optimization Opportunities via Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "From Graphs to Hypergraphs: Hypergraph Projection and its Reconstruction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HRNN: Hypergraph Recurrent Neural Network for Network Intrusion Detection", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Dynamic Hypergraph Structure Learning for Multivariate Time Series Forecasting", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Retrieval-Augmented Hypergraph for Multimodal Social Media Popularity Prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Cross-view hypergraph contrastive learning for attribute-aware recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "EHGNN: Enhanced Hypergraph Neural Network for Hyperspectral Image Classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "FCHG: Fuzzy Cognitive Hypergraph for interpretable fault detection", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Spatio-Temporal Hypergraph Convolutional Network Based Network Traffic Prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multiple stocks recommendation: a spatio-temporal hypergraph learning approach", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Central-Smoothing Hypergraph Neural Networks for Predicting Drug-Drug Interactions", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Multi-Modal Temporal Hypergraph Neural Network for Flotation Condition Recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Spanning spheres in Dirac hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal problems", "Hypergraphs", "Factorization, matching, covering and packing"], "abstract": "We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups.", "source": "Arxiv", "date": "2024-07-08", "funding_agencies": []}}, {"edge": "On discrete-time polynomial dynamical systems on hypergraphs", "weight": 1, "attrs": {"tags": ["Systems and Control", "Systems and Control"], "abstract": "This paper studies the stability of discrete-time polynomial dynamical systems on hypergraphs by utilizing the Perron-Frobenius theorem for nonnegative tensors with respect to the tensors Z-eigenvalues and Z-eigenvectors. Firstly, for a multilinear polynomial system on a uniform hypergraph, we study the stability of the origin of the corresponding systems. Next, we extend our results to non-homogeneous polynomial systems on non-uniform hypergraphs. We confirm that the local stability of any discrete-time polynomial system is in general dominated by pairwise terms. Assuming that the origin is locally stable, we construct a conservative (but explicit) region of attraction from the system parameters. Finally, we validate our results via some numerical examples.", "source": "Arxiv", "date": "2024-03-05", "funding_agencies": []}}, {"edge": "Hypergraph-Based Multi-View Action Recognition Using Event Cameras", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-10-01", "funding_agencies": []}}, {"edge": "Scalable Algorithms for Hypergraph Analytics using Symmetric Tensor Decompositions", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperMuse: Hypergraph rewriting for real-time musical visualisation (HRRTMV)", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Attribute-Enhanced Hypergraph Neural Networks for Session-based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperComm: Hypergraph-based communication in multi-agent reinforcement learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-scale hypergraph-based feature alignment network for cell localization", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Inhomogeneous Interest Modeling via Hypergraph Convolutional Networks for Social Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Integrating adaptive fuzzy embedding with topology and property hypergraphs: Enhancing membership degree-aware knowledge graph reasoning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Automatic Hypergraph Generation for Enhancing Recommendation With Sparse Optimization", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HGPflow: Extending Hypergraph Particle Flow to Collider Event Reconstruction", "weight": 1, "attrs": {"tags": ["High Energy Physics - Experiment", "Instrumentation and Detectors"], "abstract": "In high energy physics, the ability to reconstruct particles based on their detector signatures is essential for downstream data analyses. A particle reconstruction algorithm based on learning hypergraphs (HGPflow) has previously been explored in the context of single jets. In this paper, we expand the scope to full proton-proton and electron-positron collision events and study reconstruction quality using metrics at the particle, jet, and event levels. Rather than operating on the entire event in a single pass, we train HGPflow on smaller partitions to avoid potentially learning long-range correlations related to the physics process. We demonstrate that this approach is feasible and that on most metrics, HGPflow outperforms both traditional particle flow algorithms and a machine learning-based benchmark model.", "source": "Arxiv", "date": "2024-10-30", "funding_agencies": []}}, {"edge": "Self-Supervised Masked Hypergraph Autoencoders for Spatio-Temporal Forecasting", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A Mixed Hypergraph Convolutional Network for Session-Based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "PN-HGNN: Precipitation Nowcasting Network Via Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperMatch: long-form text matching via hypergraph convolutional networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-11-01", "funding_agencies": []}}, {"edge": "DHMAE: A Disentangled Hypergraph Masked Autoencoder for Group Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Tri-directional Hypergraph Contrastive Learning for Session-based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Putting Sense into Incomplete Heterogeneous Data with Hypergraph Clustering Analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Intent Enhanced Self-supervised Hypergraph Learning for Session-Based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Research on migraine classification model based on hypergraph neural network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-11-01", "funding_agencies": []}}, {"edge": "Estimating package arrival time via heterogeneous hypergraph neural network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Divide-Aggregate Heterogeneous Hypergraph for large-scale user intention detection", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Parallel Set Cover and Hypergraph Matching via Uniform Random Sampling", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph Representation Learning for Remote Sensing Image Change Detection", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-09-01", "funding_agencies": []}}, {"edge": "LightHGNN: Distilling Hypergraph Neural Networks into MLPs for 100x Faster Inference", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph Dualization with sfFPT-delay Parameterized by the Degeneracy and Dimension", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Spatial-Temporal Dynamic Hypergraph Information Bottleneck for Brain Network Classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-10-01", "funding_agencies": []}}, {"edge": "Focal-free uniform hypergraphs and codes", "weight": 1, "attrs": {"tags": ["Combinatorics", "Information Theory", "Information Theory"], "abstract": "Motivated by the study of a variant of sunflowers, Alon and Holzman recently introduced focal-free hypergraphs. In this paper, we show that there is an interesting connection between the maximum size of focal-free hypergraphs and the renowned Erd\\H{o}s Matching Conjecture on the maximum number of edges that can be contained in a uniform hypergraph with bounded matching number. As a consequence, we give asymptotically optimal bounds on the maximum sizes of focal-free uniform hypergraphs and codes, thereby significantly improving the previous results of Alon and Holzman. Moreover, by using the existentce results of combinatorial designs and orthogonal arrays, we are able to explicitly determine the exact sizes of maximum focal-free uniform hypergraphs and codes for a wide range of parameters.", "source": "Arxiv", "date": "2024-10-30", "funding_agencies": []}}, {"edge": "Dirac's theorem for linear hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Dirac's theorem states that any $n$-vertex graph $G$ with even integer $n$ satisfying $\\delta(G) \\geq n/2$ contains a perfect matching. We generalize this to $k$-uniform linear hypergraphs by proving the following. Any $n$-vertex $k$-uniform linear hypergraph $H$ with minimum degree at least $\\frac{n}{k} + \\Omega(1)$ contains a matching that covers at least $(1-o(1))n$ vertices. This minimum degree condition is asymptotically tight and obtaining perfect matching is impossible with any degree condition. Furthermore, we show that if $\\delta(H) \\geq (\\frac{1}{k}+o(1))n$, then $H$ contains almost spanning linear cycles, almost spanning hypertrees with $o(n)$ leaves, and ``long subdivisions'' of any $o(\\sqrt{n})$-vertex graphs.", "source": "Arxiv", "date": "2024-03-21", "funding_agencies": []}}, {"edge": "Hypergraph-Based Unified Model Development for Active Battery Equalization Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HHGNN: Hyperbolic Hypergraph Convolutional Neural Network based on variational autoencoder", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "AHCL-TC: Adaptive Hypergraph Contrastive Learning Networks for Text Classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-View Time-Series Hypergraph Neural Network for Action Recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "An Efficient Hypergraph-Based Routing Algorithm in Time-Sensitive Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Framelet-based dual hypergraph neural networks for student performance prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-09-01", "funding_agencies": []}}, {"edge": "Aspect-based sentiment classification with aspect-specific hypergraph attention networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "AHD-SLE: Anomalous Hyperedge Detection on Hypergraph Symmetric Line Expansion", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Hamilton Cycles in the Line Graph of a Random Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Adaptive Spatial-Temporal Hypergraph Fusion Learning for Next POI Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HGSNet: A hypergraph network for subtle lesions segmentation in medical imaging", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "A Spatial-Temporal Gated Hypergraph Convolution Network for Traffic Prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "A decentralized control approach in hypergraph distributed optimization decomposition cases", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-12-01", "funding_agencies": []}}, {"edge": "Heterogeneous hypergraph learning for literature retrieval based on citation intents", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Traffic Origin-Destination Demand Prediction via Multichannel Hypergraph Convolutional Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Hypergraph p-Laplacian regularization on point clouds for data interpolation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-based Truth Discovery for Sparse Data in Mobile Crowdsensing", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Multi-aspect Knowledge-enhanced Hypergraph Attention Network for Conversational Recommendation Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Inter- and intra-hypergraph regularized nonnegative matrix factorization with hybrid constraints", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-Channel Hypergraph Network for Sequential Diagnosis Prediction in Healthcare", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Perfect stable regularity lemma and slice-wise stable hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Discrete Mathematics", "Logic", "Classification theory, stability and related concepts", "Extremal problems", "Hypergraphs", "Structural characterization of types of graphs"], "abstract": "We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.", "source": "Arxiv", "date": "2024-02-12", "funding_agencies": []}}, {"edge": "Interval hypergraphic lattices", "weight": 1, "attrs": {"tags": ["Combinatorics", "Lattices", "Distributive lattices", "$n$-dimensional polytopes", "Special polytopes (linear programming, centrally symmetric, etc.)"], "abstract": "For a hypergraph $\\mathbb{H}$ on $[n]$, the hypergraphic poset $P_\\mathbb{H}$ is the transitive closure of the oriented skeleton of the hypergraphic polytope $\\triangle_\\mathbb{H}$ (the Minkowski sum of the standard simplices $\\triangle_H$ for all $H \\in \\mathbb{H}$). Hypergraphic posets include the weak order for the permutahedron (when $\\mathbb{H}$ is the complete graph on $[n]$) and the Tamari lattice for the associahedron (when $\\mathbb{H}$ is the set of all intervals of $[n]$), which motivates the study of lattice properties of hypergraphic posets. In this paper, we focus on interval hypergraphs, where all hyperedges are intervals of $[n]$. We characterize the interval hypergraphs $\\mathbb{I}$ for which $P_\\mathbb{I}$ is a lattice, a distributive lattice, a semidistributive lattice, and a lattice quotient of the weak order.", "source": "Arxiv", "date": "2024-11-14", "funding_agencies": []}}, {"edge": "Hypergraph-enhanced multi-interest learning for multi-behavior sequential recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-Based Multi-Modal Representation for Open-Set 3D Object Retrieval", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Complex hypergraph analysis of Australian MPs' professional connections, 1947-2019", "weight": 1, "attrs": {"tags": ["Social and Information Networks"], "abstract": "We propose a suit of methods to analyse the professional networks of MPs, showing how to analyse weak-tie connections between legislators and the connections between background charactersitic attributes. Applied to a novel dataset on the backgrounds of Australian MPs in the Australian Labor Party and the Liberal Party of Australia (1947-2019), we show that our approach can help to describe and explain the decline in working-class and trade unionist MPs from the Labor Party, the homogeneous elitism of the mid-20 century Liberal Party, and the increasing similarity of both parties' professional networks, occuring in the period of party cartellisation from the 1980s onward. Our paper's finding show that our method has clear potential for broader applications in the study of political representation, diversity, and elite political networks.", "source": "Arxiv", "date": "2024-04-17", "funding_agencies": []}}, {"edge": "Multiple hypergraph convolutional network social recommendation using dual contrastive learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Czekanowsky Hypergraph-Based Deep Learning Classifier for Precision Cyclone Forecasting", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-session aware hypergraph neural network for session-based recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Hypergraph Convolutional Network for User-Oriented Fairness in Recommender Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Stock trend prediction based on dynamic hypergraph spatio-temporal network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Coupling Fault Diagnosis of Bearings Based on Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-10-01", "funding_agencies": []}}, {"edge": "Bi-preference Learning Heterogeneous Hypergraph Networks for Session-based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Higher-order knowledge-enhanced recommendation with heterogeneous hypergraph multi-attention", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "State-element-aware syndrome classification based on hypergraph convolutional network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperGALE: ASD Classification via Hypergraph Gated Attention with Learnable HyperEdges", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Learning higher-order features for relation prediction in knowledge hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-view heterogeneous graph learning with compressed hypergraph neural networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Unsupervised hyperspectral images classification using hypergraph convolutional extreme learning machines", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "MaPart: An Efficient Multi-FPGA System-Aware Hypergraph Partitioning Framework", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-10-01", "funding_agencies": []}}, {"edge": "Auto-adjustable hypergraph regularized non-negative matrix factorization for image clustering", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Adaptive hypergraph regularized logistic regression model for bioinformatic selection and classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Prediction of miRNA-disease associations based on strengthened hypergraph convolutional autoencoder", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "The spectrum of the Corona of Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Hypergraphs", "Graphs and matrices", "Eigenvalues, singular values, and eigenvectors"], "abstract": "The corona of hypergraphs is an extension of the corona operation applied to graphs. The corona $G_0^* \\odot_1^n G_1^*$ of two hypergraphs is obtained by taking $n$ copies of $G_1^*$ (where $n$ is the order of $G_0^*$) and by joining the $i$-th vertex of $G_0^*$ with the $i$-th copy of $G_1^*$. In this paper, we estimate the complete spectrum(adjacency and Seidel) and eigenvectors of the corona $G_0^* \\odot_1^n G_1^*$ of two hypergraphs when $G_1^*$ is regular. Additionally, we define the corona hypergraph $G_0^{*(m)}=G_0^{*(m-1)} \\odot_1^n G_0^*$ and determined its adjacency spectrum. Also, we extend the definition coronal of the adjacency matrix. Moreover, we estimate the characteristic polynomial of Seidel matrix of the generalised corona of hypergraphs. Applying these results, we obtain infinitely many non-regular non-isomorphic adjacency and Seidel cospectral hypergraphs.", "source": "Arxiv", "date": "2024-03-04", "funding_agencies": []}}, {"edge": "Towards Unified Representation Learning for Career Mobility Analysis with Trajectory Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Exploiting Spatial-Temporal Data for Sleep Stage Classification via Hypergraph Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A Hierarchical Hypergraph Attention Network for Survival Analysis from Pathological Images", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Masked hypergraph learning for weakly supervised histopathology whole slide image classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Exploiting local detail in single image super-resolution via hypergraph convolution", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Trajectory set Empowered Hypergraph Transformer for Mobile Sensor Based Traffic Prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A finite element-inspired hypergraph neural network: Application to fluid dynamics simulations", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-based convex semi-supervised unconstraint symmetric matrix factorization for image clustering", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "H2GCN: A hybrid hypergraph convolution network for skeleton-based action recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph Joint Representation Learning for Hypervertices and Hyperedges via Cross Expansion", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Online multi-hypergraph fusion learning for cross-subject emotion recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hawkes-Enhanced Spatial-Temporal Hypergraph Contrastive Learning Based on Criminal Correlations", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-Based Numerical Spiking Neural Membrane Systems with Novel Repartition Protocols", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Link prediction in social networks using hyper-motif representation on hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Fast Algorithms for Hypergraph PageRank with Applications to Semi-Supervised Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Constructing Knowledge Hypergraph of Liver Diseases based on Tibetan Medicinal Materials", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "DyHGDAT: Dynamic Hypergraph Dual Attention Network for multi-agent trajectory prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HECS: A Hypergraph Learning-Based System for Detecting Extract Class Refactoring Opportunities", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Re-HGNM: a repeat aware hypergraph neural machine for session-based recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-Guided Disentangled Spectrum Transformer Networks for Near-Infrared Facial Expression Recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "High-Accuracy Anxiety Disorder Identification Through Subspace-Enhanced Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Open-domain event schema induction via weighted attentive hypergraph neural network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Unifying multimodal interactions for rumor diffusion prediction with global hypergraph modeling", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyCoRec: Hypergraph-Enhanced Multi-Preference Learning for Alleviating Matthew Effect in Conversational Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A General Latent Embedding Approach for Modeling Non-uniform High-dimensional Sparse Hypergraphs with Multiplicity", "weight": 1, "attrs": {"tags": ["Methodology"], "abstract": "Recent research has shown growing interest in modeling hypergraphs, which capture polyadic interactions among entities beyond traditional dyadic relations. However, most existing methodologies for hypergraphs face significant limitations, including their heavy reliance on uniformity restrictions for hyperlink orders and their inability to account for repeated observations of identical hyperlinks. In this work, we introduce a novel and general latent embedding approach that addresses these challenges through the integration of latent embeddings, vertex degree heterogeneity parameters, and an order-adjusting parameter. Theoretically, we investigate the identifiability conditions for the latent embeddings and associated parameters, and we establish the convergence rates of their estimators along with asymptotic distributions. Computationally, we employ a projected gradient ascent algorithm for parameter estimation. Comprehensive simulation studies demonstrate the effectiveness of the algorithm and validate the theoretical findings. Moreover, an application to a co-citation hypergraph illustrates the advantages of the proposed method.", "source": "Arxiv", "date": "2024-10-15", "funding_agencies": []}}, {"edge": "Scalable Tensor Methods for Nonuniform Hypergraphs", "weight": 1, "attrs": {"tags": ["97 MATHEMATICS AND COMPUTING", "hypergraph", "adjacency tensor", "tensor times same vector", "tensor-free methods", "centrality", "clustering", "AC05-76RL01830", "PNNL-SA-186918"], "abstract": "While multilinear algebra appears natural for studying the multiway interactions modeled by hypergraphs, tensor methods for general hypergraphs have been stymied by theoretical and practical barriers. A recently proposed adjacency tensor is applicable to nonuniform hypergraphs, but is prohibitively costly to form and analyze in practice. We develop tensor times same vector (TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to a low-degree polynomial in $r$, where $n$ is the number of vertices and $r$ is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order $r$ adjacency tensor. Here, we demonstrate the flexibility and utility of our approach in practice by developing tensor-based hypergraph centrality and clustering algorithms. We also show these tensor measures offer complementary information to analogous graph-reduction approaches on data, and are also able to detect higher-order structure that many existing matrix-based approaches provably cannot.", "source": "Osti", "date": "2024-06-11", "funding_agencies": ["USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)"]}}, {"edge": "A Novel Adaptive Hypergraph Neural Network for Enhancing Medical Image Segmentation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HGDO: An oversampling technique based on hypergraph recognition and Gaussian distribution", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "FastHGNN: A New Sampling Technique for Learning with Hypergraph Neural Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-09-01", "funding_agencies": []}}, {"edge": "AMHGCN: Adaptive multi-level hypergraph convolution network for human motion prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Random Connection Hypergraphs", "weight": 1, "attrs": {"tags": ["Probability", "Point processes", "Geometric probability, stochastic geometry, random sets", "Central limit and other weak theorems", "Simplicial sets and complexes", "Hypergraphs", "Random graphs"], "abstract": "In this paper, we introduce a novel model for random hypergraphs based on weighted random connection models. In accordance with the standard theory for hypergraphs, this model is constructed from a bipartite graph. In our stochastic model, both vertex sets of this bipartite graph form marked Poisson point processes and the connection radius is inversely proportional to a product of suitable powers of the marks. Hence, our model is a common generalization of weighted random connection models and AB random geometric graphs. For this hypergraph model, we investigate the limit theory of a variety of graph-theoretic and topological characteristics such as higher-order degree distribution, Betti numbers of the associated Dowker complex as well as simplex counts. In particular, for the latter quantity we identify regimes of convergence to normal and to stable distribution depending on the heavy-tailedness of the weight distribution. We conclude our investigation by a simulation study and an application to the collaboration network extracted from the arXiv dataset.", "source": "Arxiv", "date": "2024-07-23", "funding_agencies": []}}, {"edge": "The microscale organization of directed hypergraphs", "weight": 1, "attrs": {"tags": ["Physics and Society", "Social and Information Networks"], "abstract": "Many real-world complex systems are characterized by non-pairwise -- higher-order -- interactions among system's units, and can be effectively modeled as hypergraphs. Directed hypergraphs distinguish between source and target sets within each hyperedge, and allow to account for the directional flow of information between nodes. Here, we provide a framework to characterize the structural organization of directed higher-order networks at their microscale. First, we extract the fingerprint of a directed hypergraph, capturing the frequency of hyperedges with a certain source and target sizes, and use this information to compute differences in higher-order connectivity patterns among real-world systems. Then, we formulate reciprocity in hypergraphs, including exact, strong, and weak definitions, to measure to which extent hyperedges are reciprocated. Finally, we extend motif analysis to identify recurring interaction patterns and extract the building blocks of directed hypergraphs. We validate our framework on empirical datasets, including Bitcoin transactions, metabolic networks, and citation data, revealing structural principles behind the organization of real-world systems.", "source": "Arxiv", "date": "2024-10-21", "funding_agencies": []}}, {"edge": "Link Prediction with Relational Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "Link prediction with knowledge graphs has been thoroughly studied in graph machine learning, leading to a rich landscape of graph neural network architectures with successful applications. Nonetheless, it remains challenging to transfer the success of these architectures to relational hypergraphs, where the task of link prediction is over $k$-ary relations, which is substantially harder than link prediction with knowledge graphs. In this paper, we propose a framework for link prediction with relational hypergraphs, unlocking applications of graph neural networks to fully relational structures. Theoretically, we conduct a thorough analysis of the expressive power of the resulting model architectures via corresponding relational Weisfeiler-Leman algorithms and also via logical expressiveness. Empirically, we validate the power of the proposed model architectures on various relational hypergraph benchmarks. The resulting model architectures substantially outperform every baseline for inductive link prediction, and lead to state-of-the-art results for transductive link prediction.", "source": "Arxiv", "date": "2024-02-06", "funding_agencies": []}}, {"edge": "Attention based adaptive spatial-temporal hypergraph convolutional networks for stock price trend prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Subchannel assignment for social-assisted UAV cellular networks using dynamic hypergraph coloring", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Meta-path and hypergraph fused distillation framework for heterogeneous information networks embedding", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperMR: Hyperbolic Hypergraph Multi-hop Reasoning for Knowledge-based Visual Question Answering", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hierarchical Reinforcement Learning on Multi-Channel Hypergraph Neural Network for Course Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Faster algorithms for sparse ILP and hypergraph multi-packing/multi-cover problems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "An Efficient Hypergraph Based Clustering Technique for UAV-Enabled D2D Cellular Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Identifying cooperating cancer driver genes in individual patients through hypergraph random walk", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Improved Hypergraph Clustering with Weighted GRA for Dynamic V ANET Environment", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "H3GNN: Hybrid Hierarchical HyperGraph Neural Network for Personalized Session-based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Spectral bipartite Turan problems on linear hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal problems", "Hypergraphs"], "abstract": "Let $F$ be a graph and let $\\mathcal{B}_r(F)$ be the class of $r$-uniform Berge-$F$ hypergraphs. In this paper, by establishing a relationship between the spectral radius of the adjacency tensor of a uniform hypergraph and its local structure via walks, we give a spectral asymptotic bound for $\\mathcal{B}_{r}(C_3)$-free linear $r$-uniform hypergraphs and upper bounds for the spectral radii of $\\mathcal{B}_{r}(K_{2,t})$-free or $\\{\\mathcal{B}_{r}(K_{s,t}),\\mathcal{B}_{r}(C_{3})\\}$-free linear $r$-uniform hypergraphs, where $C_{3}$ and $K_{s,t}$ are respectively the triangle and the complete bipartite graph with one part having $s$ vertices and the other part having $t$ vertices. Our work implies an upper bound for the number of edges of $\\{\\mathcal{B}_{r}(K_{s,t}),\\mathcal{B}_{r}(C_{3})\\}$-free linear $r$-uniform hypergraphs, and extends some known work on (spectral) extreme problems of hypergraphs.", "source": "Arxiv", "date": "2024-03-04", "funding_agencies": []}}, {"edge": "The Hypergraph Tur\\'{a}n Densities of Tight Cycles Minus an Edge", "weight": 1, "attrs": {"tags": ["Combinatorics", "Extremal problems", "Hypergraphs"], "abstract": "A tight $\\ell$-cycle minus an edge $C_\\ell^-$ is the $3$-graph on the vertex set $[\\ell]$, where any three consecutive vertices in the string $123\\ldots\\ell 1$ form an edge. We show that for every $\\ell\\ge 5$, $\\ell$ not divisible by $3$, the extremal number is $ ex\\left(C_\\ell^-,n\\right)=\\tfrac1{24}n^3+O(n\\ln n)=\\left(\\tfrac14+o(1)\\right){n\\choose 3}. $ We determine the extremal graph up to $O(n)$ edge edits.", "source": "Arxiv", "date": "2024-09-21", "funding_agencies": []}}, {"edge": "Session-based recommendation with fusion of hypergraph item global and context features", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "A family of gradient methods using Householder transformation with application to hypergraph partitioning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Live streaming channel recommendation based on viewers' interaction behavior: A hypergraph approach", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph \u03b2-Model", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Extending Graph-Based LP Techniques for Enhanced Insights Into Complex Hypergraph Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Augmented skeleton sequences with hypergraph network for self-supervised group activity recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "UniG-Encoder: A universal feature encoder for graph and hypergraph node classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "CLHHN: Category-aware Lossless Heterogeneous Hypergraph Neural Network for Session-based Recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Capturing Homogeneous Influence among Students: Hypergraph Cognitive Diagnosis for Intelligent Education Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "K-SpecPart: Supervised Embedding Algorithms and Cut Overlay for Improved Hypergraph Partitioning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Multiview ensemble clustering of hypergraph p-Laplacian regularization with weighting and denoising", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Exploiting Conversation-Branch-Tweet HyperGraph Structure to Detect Misinformation on Social Media", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Hypergraph-based locality-enhancing methods for graph operations in Big Data applications", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Reinforcement Learning Based Resource Management for 6G-Enabled mIoT With Hypergraph Interference Model", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "FGeo-HyperGNet: Geometry Problem Solving Integrating Formal Symbolic System and Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multiview Hypergraph Fusion Network for Change Detection in High-Resolution Remote Sensing Images", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HyperSegRec: enhanced hypergraph-based recommendation system with user segmentation and item similarity learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-11-01", "funding_agencies": []}}, {"edge": "Hypergraph Convolutional Networks for Fine-Grained lCU Patient Similarity Analysis and Risk Prediction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Sum-Rate Optimization Algorithms for RIS Aided D2D Multicast System Based on Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "Improving speculative query execution support by the use of the hypergraph representation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-Based Session Modeling: A Multi-Collaborative Self-Supervised Approach for Enhanced Recommender Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "An Unsupervised Domain Adaption Method for Fault Diagnosis via Multichannel Variational Hypergraph Autoencoder", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A Hypergraph Analog of Dirac's Theorem for Long Cycles in 2-Connected Graphs", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Prior-Guided Adversarial Learning With Hypergraph for Predicting Abnormal Connections in Alzheimer's Disease", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "Spatial-Spectral Twin Autoencoders for Hyperspectral Unmixing Via Superpixel-Hypergraph-Augmented Feature Representation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "HHGNN: Heterogeneous Hypergraph Neural Network for Traffic Agents Trajectory Prediction in Grouping Scenarios", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Connected Tur\\'{a}n numbers for Berge paths in hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Let $\\mathcal{F}$ be a family of $r$-uniform hypergraphs. Denote by $\\ex^{\\mathrm{conn}}_r(n,\\mathcal{F})$ the maximum number of hyperedges in an $n$-vertex connected $r$-uniform hypergraph which contains no member of $\\mathcal{F}$ as a subhypergraph. Denote by $\\mathcal{B}C_k$ the Berge cycle of length $k$, and by $\\mathcal{B}P_k$ the Berge path of length $k$. F\\`{u}redi, Kostochka and Luo, and independently Gy\\H{o}ri, Salia and Zamora determined $\\ex^{\\mathrm{conn}}_r(n,\\mathcal{B}P_k)$ provided $k$ is large enough compared to $r$ and $n$ is sufficiently large. For the case $k\\le r$, Kostochka and Luo obtained an upper bound for $\\ex^{\\mathrm{conn}}_r(n,\\mathcal{B}P_k)$. In this paper, we continue investigating the case $k\\le r$. We precisely determine $\\ex^{\\mathrm{conn}}_r(n,\\mathcal{B}P_k)$ when $n$ is sufficiently large and $n$ is not a multiple of~$r$. For the case $k=r+1$, we determine $\\ex^{\\mathrm{conn}}_r(n,\\mathcal{B}P_k)$ asymptotically.", "source": "Arxiv", "date": "2024-09-05", "funding_agencies": []}}, {"edge": "Sequential Patterns Unveiled: A Novel Hypergraph Convolution Approach for Dynamic User Preference Analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Brain Function Analysis Of Insomnia Disorder Based On Hypergraph Combined With Deep Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Strong Consistency of Spectral Clustering for the Sparse Degree-Corrected Hypergraph Stochastic Block Model", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "MeshStyle: Text-driven Efficient and High-Quality 3D Mesh Stylization via Hypergraph Convolution", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Messages are Never Propagated Alone: Collaborative Hypergraph Neural Network for Time-Series Forecasting", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Forecasting the molecular interactions: A hypergraph-based neural network for molecular relational learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Low Mileage, High Fidelity: Evaluating Hypergraph Expansion Methods by Quantifying the Information Loss", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Event-triggered Reconfiguration Scheduling of Hypergraph-based System-of-systems: the Resilience Reaction", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A Hypergraph-based Formalization of Hierarchical Reactive Modules and a Compositional Verification Method", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "SKYPER: Legal case retrieval via skeleton-aware hypergraph embedding in the hyperbolic space", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Intelligent Beamforming for UAV-Assisted IIoT Based on Hypergraph Inspired Explainable Deep Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "Capturing word positions does help: A multi-element hypergraph gated attention network for document classification", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "THGFormer: Time-Aware Hypergraph Learning for Multimodal Social Media Popularity Prediction (Student Abstract)", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Dual Homogeneity Hypergraph Motifs with Cross-view Contrastive Learning for Multiple Social Recommendations", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Hypergraphs of girth 5 and 6 and coding theory", "weight": 1, "attrs": {"tags": ["Combinatorics", "Information Theory", "Information Theory", "Extremal problems", "Hypergraphs", "Bounds on codes", "Relations with coding theory"], "abstract": "In this paper, we study the maximum number of edges in an $N$-vertex $r$-uniform hypergraph with girth $g$ where $g \\in \\{5,6 \\}$. Writing $\\textrm{ex}_r ( N, \\mathcal{C}_{k- Way Hypergraph Partitioning Algorithm for Optimizing the Steiner Tree Metric", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Self-supervised hypergraph neural network for session-based recommendation supported by user continuous topic intent", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Multi-view hypergraph regularized Lp norm least squares twin support vector machines for semi-supervised learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "CI-STHPAN: Pre-trained Attention Network for Stock Selection with Channel-Independent Spatio-Temporal Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Integration of protein sequence and protein-protein interaction data by hypergraph learning to identify novel protein complexes", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Locating influential nodes in hypergraphs via fuzzy collective influence", "weight": 1, "attrs": {"tags": ["Physics and Society"], "abstract": "Complex contagion phenomena, such as the spread of information or contagious diseases, often occur among the population due to higher-order interactions between individuals. Individuals who can be represented by nodes in a network may play different roles in the spreading process, and thus finding the most influential nodes in a network has become a crucial topic in network science for applications such as viral marketing, rumor suppression, and disease control. To solve the problem of identifying nodes that have high influence in a complex system, we propose a higher-order distance-based fuzzy centrality methods (HDF and EHDF) that are customized for a hypergraph which can characterize higher-order interactions between nodes via hyperedges. The methods we proposed assume that the influence of a node is reliant on the neighboring nodes with a certain higher-order distance. We compare the proposed methods with the baseline centrality methods to verify their effectiveness. Experimental results on six empirical hypergraphs show that the proposed methods could better identify influential nodes, especially showing plausible performance in finding the top influential nodes. Our proposed theoretical framework for identifying influential nodes could provide insights into how higher-order topological structure can be used for tasks such as vital node identification, influence maximization, and network dismantling.", "source": "Arxiv", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "A Synchronous Training Hypergraph Neural Network for Power Allocation in Multi-Cell Multi-User Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Routing and Charging Scheduling for EV Battery Swapping Systems: Hypergraph-Based Heterogeneous Multiagent Deep Reinforcement Learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-09-01", "funding_agencies": []}}, {"edge": "Multiview hyperedge-aware hypergraph embedding learning for multisite, multiatlas fMRI based functional connectivity network analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Spatial-temporal hypergraph based on dual-stage attention network for multi-view data lightweight action recognition", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Query-based denormalization using hypergraph (QBDNH): a schema transformation model for migrating relational to NoSQL databases", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Hypergraph-Based Interference Avoidance Resource Management in Customer-Centric Communication for Intelligent Cyber-Physical Transportation Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "A hypergraph-based hybrid graph convolutional network for intracity human activity intensity prediction and geographic relationship interpretation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "AHFormer: Hypergraph embedding coding transformer and adaptive aggregation network for intelligent fault diagnosis under noise interference", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Integrating user short-term intentions and long-term preferences in heterogeneous hypergraph networks for sequential recommendation", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "Vertex degree sums for perfect matchings in 3-uniform hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Let $n \\equiv 0\\, (\\, \\text{mod } 3\\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\\frac{1}{3}n+1$ and $\\frac{2}{3}n -1$, respectively, and whose edge set consists of all triples with at least $2$ vertices in $T$. Suppose that $n$ is sufficiently large and $H$ is a 3-uniform hypergraph of order $n$ with no isolated vertex. Zhang and Lu [Discrete Math. 341 (2018), 748--758] conjectured that if $deg(u)+deg(v) > 2(\\binom{n-1}{2}-\\binom{2n/3}{2})$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. We construct a counter-example to the conjecture. Furthermore, for all $\\gamma>0$ and let $n \\in 3 \\mathbb{Z}$ be sufficiently large, we prove that if $deg(u)+deg(v) > (3/5+\\gamma)n^2$ for any two vertices $u$ and $v$ that are contained in some edge of $H$, then $H$ contains a perfect matching or $H$ is a subgraph of $H_{n,n/3}^2$. This implies a result of Zhang, Zhao and Lu [Electron. J. Combin. 25 (3), 2018].", "source": "Arxiv", "date": "2024-01-08", "funding_agencies": []}}, {"edge": "EgoMUIL: Enhancing Spatio-Temporal User Identity Linkage in Location-Based Social Networks With Ego-Mo Hypergraph", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-08-01", "funding_agencies": []}}, {"edge": "Deep Fusion of Multi-Template Using Spatio-Temporal Weighted Multi-Hypergraph Convolutional Networks for Brain Disease Analysis", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-02-01", "funding_agencies": []}}, {"edge": "A Novel Lie Hypergraph Based Lifetime Enhancement Routing Protocol for Environmental Monitoring in Wireless Sensor Networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Influence Maximization in Hypergraphs by Stratified Sampling for Efficient Generation of Reverse Reachable Sets", "weight": 1, "attrs": {"tags": ["Social and Information Networks", "Data Structures and Algorithms"], "abstract": "Given a hypergraph, influence maximization (IM) is to discover a seed set containing $k$ vertices that have the maximal influence. Although the existing vertex-based IM algorithms perform better than the hyperedge-based algorithms by generating random reverse researchable (RR) sets, they are inefficient because (i) they ignore important structural information associated with hyperedges and thus obtain inferior results, (ii) the frequently-used sampling methods for generating RR sets have low efficiency because of a large number of required samplings along with high sampling variances, and (iii) the vertex-based IM algorithms have large overheads in terms of running time and memory costs. To overcome these shortcomings, this paper proposes a novel approach, called \\emph{HyperIM}. The key idea behind \\emph{HyperIM} is to differentiate structural information of vertices for developing stratified sampling combined with highly-efficient strategies to generate the RR sets. With theoretical guarantees, \\emph{HyperIM} is able to accelerate the influence spread, improve the sampling efficiency, and cut down the expected running time. To further reduce the running time and memory costs, we optimize \\emph{HyperIM} by inferring the bound of the required number of RR sets in conjunction with stratified sampling. Experimental results on real-world hypergraphs show that \\emph{HyperIM} is able to reduce the number of required RR sets and running time by orders of magnitude while increasing the influence spread by up to $2.73X$ on average, compared to the state-of-the-art IM algorithms.", "source": "Arxiv", "date": "2024-06-03", "funding_agencies": []}}, {"edge": "A dual-scale fused hypergraph convolution-based hyperedge prediction model for predicting missing reactions in genome-scale metabolic networks", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Improved bounds for group testing in arbitrary hypergraphs", "weight": 1, "attrs": {"tags": ["Data Structures and Algorithms"], "abstract": "Recent papers initiated the study of a generalization of group testing where the potentially contaminated sets are the members of a given hypergraph F=(V,E). This generalization finds application in contexts where contaminations can be conditioned by some kinds of social and geographical clusterings. The paper focuses on few-stage group testing algorithms, i.e., slightly adaptive algorithms where tests are performed in stages and all tests performed in the same stage should be decided at the very beginning of the stage. In particular, the paper presents the first two-stage algorithm that uses o(dlog|E|) tests for general hypergraphs with hyperedges of size at most d, and a three-stage algorithm that improves by a d^{1/6} factor on the number of tests of the best known three-stage algorithm. These algorithms are special cases of an s-stage algorithm designed for an arbitrary positive integer s<= d. The design of this algorithm resort to a new non-adaptive algorithm (one-stage algorithm), i.e., an algorithm where all tests must be decided beforehand. Further, we derive a lower bound for non-adaptive group testing. For E sufficiently large, the lower bound is very close to the upper bound on the number of tests of the best non-adaptive group testing algorithm known in the literature, and it is the first lower bound that improves on the information theoretic lower bound Omega(log |E|).", "source": "Arxiv", "date": "2024-04-29", "funding_agencies": []}}, {"edge": "Developing a novel approach in estimating urban commute traffic by integrating community detection and hypergraph representation learning", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Optimum resource allocation for D2D-assisted wireless network in industrial internet of things: a hypergraph-based clique algorithm", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-05-01", "funding_agencies": []}}, {"edge": "A Novel Variable Lie Hypergraph Technique for an Energy Aware Routing Protocol to Improve Infotainment Services in VANETs", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Model-Assisted Multi-source Fusion Hypergraph Convolutional Neural Networks for intelligent few-shot fault diagnosis to Electro-Hydrostatic Actuator", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "Breaking Down Financial News Impact: A Novel AI Approach with Geometric Hypergraphs", "weight": 1, "attrs": {"tags": ["Machine Learning", "Artificial Intelligence"], "abstract": "In the fast-paced and volatile financial markets, accurately predicting stock movements based on financial news is critical for investors and analysts. Traditional models often struggle to capture the intricate and dynamic relationships between news events and market reactions, limiting their ability to provide actionable insights. This paper introduces a novel approach leveraging Explainable Artificial Intelligence (XAI) through the development of a Geometric Hypergraph Attention Network (GHAN) to analyze the impact of financial news on market behaviours. Geometric hypergraphs extend traditional graph structures by allowing edges to connect multiple nodes, effectively modelling high-order relationships and interactions among financial entities and news events. This unique capability enables the capture of complex dependencies, such as the simultaneous impact of a single news event on multiple stocks or sectors, which traditional models frequently overlook. By incorporating attention mechanisms within hypergraphs, GHAN enhances the model's ability to focus on the most relevant information, ensuring more accurate predictions and better interpretability. Additionally, we employ BERT-based embeddings to capture the semantic richness of financial news texts, providing a nuanced understanding of the content. Using a comprehensive financial news dataset, our GHAN model addresses key challenges in financial news impact analysis, including the complexity of high-order interactions, the necessity for model interpretability, and the dynamic nature of financial markets. Integrating attention mechanisms and SHAP values within GHAN ensures transparency, highlighting the most influential factors driving market predictions. Empirical validation demonstrates the superior effectiveness of our approach over traditional sentiment analysis and time-series models.", "source": "Arxiv", "date": "2024-08-31", "funding_agencies": []}}, {"edge": "Learning Association Characteristics by Dynamic Hypergraph and Gated Convolution Enhanced Pairwise Attributes for Prediction of Disease-Related lncRNAs", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "A Sensor Placement Approach Using Multi-Objective Hypergraph Particle Swarm Optimization to Improve Effectiveness of Structural Health Monitoring Systems", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-03-01", "funding_agencies": []}}, {"edge": "A Hierarchical Graph Neural Network Framework for Predicting Protein-Protein Interaction Modulators With Functional Group Information and Hypergraph Structure", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-07-01", "funding_agencies": []}}, {"edge": "Hypergraph Structural Information Aggregation Generative Adversarial Networks for Diagnosis and Pathogenetic Factors Identification of Alzheimer's Disease With Imaging Genetic Data", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-06-01", "funding_agencies": []}}, {"edge": "A highly efficient resource slicing and scheduling optimization algorithm for power heterogeneous communication networks based on hypergraph and congruence entropy", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-12-01", "funding_agencies": []}}, {"edge": "Research on the low-dimensional visualization and identification method of the equipment's conditions by cloud-based screening and hypergraph embedding", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "One-to-One or One-to-Many? Suggesting Extract Class Refactoring Opportunities with Intra-class Dependency Hypergraph Neural Network", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Cooperation in Public Goods Games: Leveraging Other-Regarding Reinforcement Learning on Hypergraphs", "weight": 1, "attrs": {"tags": ["Physics and Society", "Adaptation and Self-Organizing Systems"], "abstract": "Cooperation as a self-organized collective behavior plays a significant role in the evolution of ecosystems and human society. Reinforcement learning (RL) offers a new perspective, distinct from imitation learning in evolutionary games, for exploring the mechanisms underlying its emergence. However, most existing studies with the public good game (PGG) employ a self-regarding setup or are on pairwise interaction networks. Players in the real world, however, optimize their policies based not only on their histories but also on the histories of their co-players, and the game is played in a group manner. In the work, we investigate the evolution of cooperation in the PGG under the other-regarding reinforcement learning evolutionary game (OR-RLEG) on hypergraph by combining the Q-learning algorithm and evolutionary game framework, where other players' action history is incorporated and the game is played on hypergraphs. Our results show that as the synergy factor increases, the parameter interval is divided into three distinct regions, the absence of cooperation (AC), medium cooperation (MC), and high cooperation (HC), accompanied by two abrupt transitions in the cooperation level near two transition points, respectively. Interestingly, we identify regular and anti-coordinated chessboard structures in the spatial pattern that positively contribute to the first cooperation transition but adversely affect the second. Furthermore, we provide a theoretical treatment for the first transition with an approximated first transition point and reveal that players with a long-sighted perspective and low exploration rate are more likely to reciprocate kindness with each other, thus facilitating the emergence of cooperation. Our findings contribute to understanding the evolution of human cooperation, where other-regarding information and group interactions are commonplace.", "source": "Arxiv", "date": "2024-10-14", "funding_agencies": []}}, {"edge": "Spectra of adjacency and Laplacian matrices of Erd\\H{o}s-R\\'{e}nyi hypergraphs", "weight": 1, "attrs": {"tags": ["Probability", "Combinatorics"], "abstract": "We study adjacency and Laplacian matrices of Erd\\H{o}s-R\\'{e}nyi $r$-uniform hypergraphs on $n$ vertices with hyperedge inclusion probability $p$, in the setting where $r$ can vary with $n$ such that $r / n \\to c \\in [0, 1)$. Adjacency matrices of hypergraphs are contractions of adjacency tensors and their entries exhibit long range correlations. We show that under the Erd\\H{o}s-R\\'{e}nyi model, the expected empirical spectral distribution of an appropriately normalised hypergraph adjacency matrix converges weakly to the semi-circle law with variance $(1 - c)^2$ as long as $\\frac{d_{\\avg}}{r^7} \\to \\infty$, where $d_{\\avg} = \\binom{n-1}{r-1} p$. In contrast with the Erd\\H{o}s-R\\'{e}nyi random graph ($r = 2$), two eigenvalues stick out of the bulk of the spectrum. When $r$ is fixed and $d_{\\avg} \\gg n^{r - 2} \\log^4 n$, we uncover an interesting Baik-Ben Arous-P\\'{e}ch\\'{e} (BBP) phase transition at the value $r = 3$. For $r \\in \\{2, 3\\}$, an appropriately scaled largest (resp. smallest) eigenvalue converges in probability to $2$ (resp. $-2$), the right (resp. left) end point of the support of the standard semi-circle law, and when $r \\ge 4$, it converges to $\\sqrt{r - 2} + \\frac{1}{\\sqrt{r - 2}}$ (resp. $-\\sqrt{r - 2} - \\frac{1}{\\sqrt{r - 2}}$). Further, in a Gaussian version of the model we show that an appropriately scaled largest (resp. smallest) eigenvalue converges in distribution to $\\frac{c}{2} \\zeta + \\big[\\frac{c^2}{4}\\zeta^2 + c(1 - c)\\big]^{1/2}$ (resp. $\\frac{c}{2} \\zeta - \\big[\\frac{c^2}{4}\\zeta^2 + c(1 - c)\\big]^{1/2}$), where $\\zeta$ is a standard Gaussian. We also establish analogous results for the bulk and edge eigenvalues of the associated Laplacian matrices.", "source": "Arxiv", "date": "2024-09-05", "funding_agencies": []}}, {"edge": "CORONet: A Cross-Sequence Joint Representation and Hypergraph Convolutional Network for Classifying Molecular Subtypes of Breast Cancer Using Incomplete DCE-MRI", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-04-01", "funding_agencies": []}}, {"edge": "The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs", "weight": 1, "attrs": {"tags": ["Computational Complexity", "Data Structures and Algorithms", "Combinatorics", "Probability", "Machine Learning"], "abstract": "We study the algorithmic task of finding large independent sets in Erdos-Renyi $r$-uniform hypergraphs on $n$ vertices having average degree $d$. Krivelevich and Sudakov showed that the maximum independent set has density $\\left(\\frac{r\\log d}{(r-1)d}\\right)^{1/(r-1)}$. We show that the class of low-degree polynomial algorithms can find independent sets of density $\\left(\\frac{\\log d}{(r-1)d}\\right)^{1/(r-1)}$ but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining $r$-partite hypergraphs. A hypergraph $H=(V,E)$ is $r$-partite if there is a partition $V=V_1\\cup\\cdots\\cup V_r$ such that each edge contains exactly one vertex from each set $V_i$. We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random $r$-partite hypergraphs with $n$ vertices in each partition and average degree $d$. We prove that the maximum balanced independent set has density $\\left(\\frac{r\\log d}{(r-1)d}\\right)^{1/(r-1)}$ asymptotically. Furthermore, we prove an analogous low-degree computational threshold of $\\left(\\frac{\\log d}{(r-1)d}\\right)^{1/(r-1)}$. Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.", "source": "Arxiv", "date": "2024-04-04", "funding_agencies": []}}, {"edge": "Spanning Euler Tours in Hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "Motivated by generalizations of de Bruijn cycles to various combinatorial structures (Chung, Diaconis, and Graham), we study various Euler tours in set systems. Let $\\mathcal{G}$ be a hypergraph whose corank and rank are $c\\geq 3$ and $k$, respetively. The minimum $t$-degree of $\\mathcal{G}$ is the fewest number of edges containing every $t$-subset of vertices. An Euler tour (family, respectively) in $\\mathcal{G}$ is a (family of, respectively) closed walk(s) that (jointly, respectively) traverses each edge of $\\mathcal{G}$ exactly once. An Euler tour is spanning if it traverses all the vertices of $\\mathcal{G}$. We show that $\\mathcal{G}$ has an Euler family if its incidence graph is $(1+\\lceil k/c \\rceil)$-edge-connected. Provided that the number of vertices of $\\mathcal{G}$ meets a reasonable lower bound, and either $2$-degree is at least $k$ or $t$-degree is at least one for $t\\geq 3$, we show that $\\mathcal{G}$ has a spanning Euler tour. To exhibit the usefulness of our results, we solve a number of open problems concerning ordering blocks of a design (these have applications in other fields such as erasure-correcting codes). Answering a question of Horan and Hurlbert, we show that a Steiner quadruple system of order $n$ has a (spanning) Euler tour if and only if $n\\geq 8$ and $n\\equiv 2,4 \\pmod 6$, and we prove a similar result for all Steiner systems, as well as all designs except for 2-designs whose index $\\lambda$ is less than the largest block size. We nearly solve a conjecture of Dewar and Stevens on the existence of universal cycles in pairwise balanced designs. Motivated by R.L. Graham's question on the existence of Hamiltonian cycles in block-intersection graphs of Steiner triple systems, we establish the Hamiltonicity of the block-intersection graph of a large family of (not necessarily uniform) designs. All our results are constructive and of polynomial time complexity.", "source": "Arxiv", "date": "2024-03-19", "funding_agencies": []}}, {"edge": "An Efficient Regularity Lemma for Semi-Algebraic Hypergraphs", "weight": 1, "attrs": {"tags": ["Computational Geometry", "Combinatorics", "Helly-type theorems and geometric transversal theory", "Arrangements of points, flats, hyperplanes", "Combinatorial complexity of geometric structures", "Ramsey theory", "Hypergraphs", "Combinatorics", "Graph Theory", "Nonnumerical Algorithms and Problems"], "abstract": "We use the polynomial method of Guth and Katz to establish stronger and {\\it more efficient} regularity and density theorems for such $k$-uniform hypergraphs $H=(P,E)$, where $P$ is a finite point set in ${\\mathbb R}^d$, and the edge set $E$ is determined by a semi-algebraic relation of bounded description complexity. In particular, for any $0<\\epsilon\\leq 1$ we show that one can construct in $O\\left(n\\log (1/\\epsilon)\\right)$ time, an equitable partition $P=U_1\\uplus \\ldots\\uplus U_K$ into $K=O(1/\\epsilon^{d+1+\\delta})$ subsets, for any $0<\\delta$, so that all but $\\epsilon$-fraction of the $k$-tuples $U_{i_1},\\ldots,U_{i_k}$ are {\\it homogeneous}: we have that either $U_{i_1}\\times\\ldots\\times U_{i_k}\\subseteq E$ or $(U_{i_1}\\times\\ldots\\times U_{i_k})\\cap E=\\emptyset$. If the points of $P$ can be perturbed in a general position, the bound improves to $O(1/\\epsilon^{d+1})$, and the partition is attained via a {\\it single partitioning polynomial} (albeit, at expense of a possible increase in worst-case running time). In contrast to the previous such regularity lemmas which were established by Fox, Gromov, Lafforgue, Naor, and Pach and, subsequently, Fox, Pach and Suk, our partition of $P$ does not depend on the edge set $E$ provided its semi-algebraic description complexity does not exceed a certain constant. As a by-product, we show that in any $k$-partite $k$-uniform hypergraph $(P_1\\uplus\\ldots\\uplus P_k,E)$ of bounded semi-algebraic description complexity in ${\\mathbb R}^d$ and with $|E|\\geq \\epsilon \\prod_{i=1}^k|P_i|$ edges, one can find, in expected time $O\\left(\\sum_{i=1}^k\\left(|P_i|+1/\\epsilon)\\right)\\log (1/\\epsilon)\\right)$, subsets $Q_i\\subseteq P_i$ of cardinality $|Q_i|\\geq |P_i|/\\epsilon^{d+1+\\delta}$, so that $Q_1\\times\\ldots\\times Q_k\\subseteq E$.", "source": "Arxiv", "date": "2024-07-22", "funding_agencies": []}}, {"edge": "Rainbow structures in properly edge-colored graphs and hypergraph systems. (Structures arc-en-ciel dans les graphes proprement ar\u00eates-color\u00e9s et les syst\u00e8mes des hypergraphes)", "weight": 1, "attrs": {"tags": [], "abstract": "", "source": "DBLP", "date": "2024-01-01", "funding_agencies": []}}, {"edge": "Asymptotically sharp bounds for cancellative and union-free hypergraphs", "weight": 1, "attrs": {"tags": ["Combinatorics"], "abstract": "An $r$-graph is called $t$-cancellative if for arbitrary $t+2$ distinct edges $A_1,\\ldots,A_t,B,C$, it holds that $(\\cup_{i=1}^t A_i)\\cup B\\neq (\\cup_{i=1}^t A_i)\\cup C$; it is called $t$-union-free if for arbitrary two distinct subsets $\\mathcal{A},\\mathcal{B}$, each consisting of at most $t$ edges, it holds that $\\cup_{A\\in\\mathcal{A}} A\\neq \\cup_{B\\in\\mathcal{B}} B$. Let $C_t(n,r)$ and $U_t(n,r)$ denote the maximum number of edges that can be contained in an $n$-vertex $t$-cancellative and $t$-union-free $r$-graph, respectively. The study of $C_t(n,r)$ and $U_t(n,r)$ has a long history, dating back to the classic works of Erd\\H{o}s and Katona, and Erd\\H{o}s and Moser in the 1970s. In 2020, Shangguan and Tamo showed that $C_{2(t-1)}(n,tk)=\\Theta(n^k)$ and $U_{t+1}(n,tk)=\\Theta(n^k)$ for all $t\\ge 2$ and $k\\ge 2$. In this paper, we determine the asymptotics of these two functions up to a lower order term, by showing that for all $t\\ge 2$ and $k\\ge 2$, \\begin{align*} \\text{$\\lim_{n\\rightarrow\\infty}\\frac{C_{2(t-1)}(n,tk)}{n^k}=\\lim_{n\\rightarrow\\infty}\\frac{U_{t+1}(n,tk)}{n^k}=\\frac{1}{k!}\\cdot \\frac{1}{\\binom{tk-1}{k-1}}$.} \\end{align*} Previously, it was only known by a result of F\\`uredi in 2012 that $\\lim_{n\\rightarrow\\infty}\\frac{C_{2}(n,4)}{n^2}=\\frac{1}{6}$. To prove the lower bounds of the limits, we utilize a powerful framework developed recently by Delcourt and Postle, and independently by Glock, Joos, Kim, K\\`uhn, and Lichev, which shows the existence of near-optimal hypergraph packings avoiding certain small configurations, and to prove the upper bounds, we apply a novel counting argument that connects $C_{2(t-1)}(n,tk)$ to a classic result of Kleitman and Frankl on a special case of the famous Erd\\H{o}s Matching Conjecture.", "source": "Arxiv", "date": "2024-11-12", "funding_agencies": []}}], "nodes": [{"node": "Daniel T. Chang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Mirjam Trieb", "weight": 1, "attrs": {"institutions": []}}, {"node": "Moritz Weber", "weight": 1, "attrs": {"institutions": []}}, {"node": "Dean Zenner", "weight": 1, "attrs": {"institutions": []}}, {"node": "R. Vishnupriya", "weight": 1, "attrs": {"institutions": []}}, {"node": "R. Rajkumar", "weight": 1, "attrs": {"institutions": []}}, {"node": "Rongping Ye", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiaobing Pei", "weight": 1, "attrs": {"institutions": []}}, {"node": "Haoran Yang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ruiqi Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Bj\u00f6rn Sch\u00e4fer", "weight": 1, "attrs": {"institutions": []}}, {"node": "Krist\u00f3f B\u00e9rczi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Karthekeyan Chandrasekaran", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tam\u00e1s Kir\u00e1ly", "weight": 1, "attrs": {"institutions": []}}, {"node": "Shubhang Kulkarni", "weight": 1, "attrs": {"institutions": []}}, {"node": "Iulia Duta", "weight": 1, "attrs": {"institutions": []}}, {"node": "Pietro Li\u00f2", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ting-Wei Chao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hung-Hsun Hans Yu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Lele Liu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhenyu Ni", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jing Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Liying Kang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ruoxu Cen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jason Li", "weight": 1, "attrs": {"institutions": []}}, {"node": "Debmalya Panigrahi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Kai Zhao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiao-Dong Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Siddhant Saxena", "weight": 1, "attrs": {"institutions": []}}, {"node": "Shounak Ghatak", "weight": 1, "attrs": {"institutions": []}}, {"node": "Raghu Kolla", "weight": 1, "attrs": {"institutions": []}}, {"node": "Debashis Mukherjee", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tanmoy Chakraborty", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tatyana Benko", "weight": 1, "attrs": {"institutions": []}}, {"node": "Martin Buck", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ilya Amburg", "weight": 1, "attrs": {"institutions": []}}, {"node": "Stephen J. Young", "weight": 1, "attrs": {"institutions": []}}, {"node": "Sinan G. Aksoy", "weight": 1, "attrs": {"institutions": []}}, {"node": "Takeshi Fukao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Masahiro Ikeda", "weight": 1, "attrs": {"institutions": []}}, {"node": "Shun Uchida", "weight": 1, "attrs": {"institutions": []}}, {"node": "Abhisek Ray", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ayush Raj", "weight": 1, "attrs": {"institutions": []}}, {"node": "Maheshkumar H. Kolekar", "weight": 1, "attrs": {"institutions": []}}, {"node": "Marcelo Campos", "weight": 1, "attrs": {"institutions": []}}, {"node": "Wojciech Samotij", "weight": 1, "attrs": {"institutions": []}}, {"node": "Loc Hoang Tran", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhao Li", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xin Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jun Zhao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Wenbin Guo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jianxin Li", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yanbang Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jon Kleinberg", "weight": 1, "attrs": {"institutions": []}}, {"node": "Anirban Banerjee", "weight": 1, "attrs": {"institutions": []}}, {"node": "Samiron Parui", "weight": 1, "attrs": {"institutions": []}}, {"node": "Simon Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Cheng Xin", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tamal K. Dey", "weight": 1, "attrs": {"institutions": []}}, {"node": "Linfeng Luo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Fengxiao Tang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiyu Liu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhiqi Guo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zihao Qiu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ming Zhao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tyler Hanks", "weight": 1, "attrs": {"institutions": []}}, {"node": "Matthew Klawonn", "weight": 1, "attrs": {"institutions": []}}, {"node": "James Fairbanks", "weight": 1, "attrs": {"institutions": []}}, {"node": "Wei Ju", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhengyang Mao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Siyu Yi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yifang Qin", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yiyang Gu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhiping Xiao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yifan Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiao Luo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ming Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Callum Birch-Sykes", "weight": 1, "attrs": {"institutions": []}}, {"node": "Brian Le", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yvonne Peters", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ethan Simpson", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zihan Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hongyu Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Dongyi Zheng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Lin Zhong", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xu Yang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jiyuan Feng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yunqing Feng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Qing Liao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Qixuan Zheng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hong Yan", "weight": 1, "attrs": {"institutions": []}}, {"node": "Natalie Behague", "weight": 1, "attrs": {"institutions": []}}, {"node": "Pawel Pralat", "weight": 1, "attrs": {"institutions": []}}, {"node": "Andrzej Rucinski", "weight": 1, "attrs": {"institutions": []}}, {"node": "Felix Christian Clemen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Kehan Shi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Martin Burger", "weight": 1, "attrs": {"institutions": []}}, {"node": "Felix Joos", "weight": 1, "attrs": {"institutions": []}}, {"node": "Dhruv Mubayi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zak Smith", "weight": 1, "attrs": {"institutions": []}}, {"node": "Geon Lee", "weight": 1, "attrs": {"institutions": []}}, {"node": "Fanchen Bu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tina Eliassi-Rad", "weight": 1, "attrs": {"institutions": []}}, {"node": "Kijung Shin", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yifan Feng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jiangang Huang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Shaoyi Du", "weight": 1, "attrs": {"institutions": []}}, {"node": "Shihui Ying", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jun-Hai Yong", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yipeng Li", "weight": 1, "attrs": {"institutions": []}}, {"node": "Guiguang Ding", "weight": 1, "attrs": {"institutions": []}}, {"node": "Rongrong Ji", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yue Gao", "weight": 1, "attrs": {"institutions": []}}, {"node": "David Conlon", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jacob Fox", "weight": 1, "attrs": {"institutions": []}}, {"node": "Benjamin Gunby", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiaoyu He", "weight": 1, "attrs": {"institutions": []}}, {"node": "Andrew Suk", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jacques Verstraete", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ziming Zhao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Tiehua Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zijian Yi", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhishu Shen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Mirko Navara", "weight": 1, "attrs": {"institutions": []}}, {"node": "Karl Svozil", "weight": 1, "attrs": {"institutions": []}}, {"node": "Darnbi Sakong", "weight": 1, "attrs": {"institutions": []}}, {"node": "Viet Hung Vu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Thanh Trung Huynh", "weight": 1, "attrs": {"institutions": []}}, {"node": "Phi Le Nguyen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hongzhi Yin", "weight": 1, "attrs": {"institutions": []}}, {"node": "Quoc Viet Hung Nguyen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Thanh Tam Nguyen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Adri\u00e1n Bazaga", "weight": 1, "attrs": {"institutions": []}}, {"node": "Gos Micklem", "weight": 1, "attrs": {"institutions": []}}, {"node": "Rajko Nenadov", "weight": 1, "attrs": {"institutions": []}}, {"node": "Huy Tuan Pham", "weight": 1, "attrs": {"institutions": []}}, {"node": "Qi Cao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yulin Shao", "weight": 1, "attrs": {"institutions": []}}, {"node": "Fan Yang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Octavia A. Dobre", "weight": 1, "attrs": {"institutions": []}}, {"node": "Dorian Gailhard", "weight": 1, "attrs": {"institutions": []}}, {"node": "Enzo Tartaglione", "weight": 1, "attrs": {"institutions": []}}, {"node": "Lirida Naviner", "weight": 1, "attrs": {"institutions": []}}, {"node": "Jhony H. Giraldo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Rongwei Xu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Guanfeng Liu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yan Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xuyun Zhang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Kai Zheng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiaofang Zhou", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hamed Sajadinia", "weight": 1, "attrs": {"institutions": []}}, {"node": "Ali Aghdaei", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhuo Feng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yihe Luo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Mingdai Yang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhiwei Liu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Liangwei Yang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Xiaolong Liu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Chen Wang", "weight": 1, "attrs": {"institutions": []}}, {"node": "Hao Peng", "weight": 1, "attrs": {"institutions": []}}, {"node": "Philip S. Yu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Chaoguang Luo", "weight": 1, "attrs": {"institutions": []}}, {"node": "Liuying Wen", "weight": 1, "attrs": {"institutions": []}}, {"node": "Yong Qin", "weight": 1, "attrs": {"institutions": []}}, {"node": "Zhineng Hu", "weight": 1, "attrs": {"institutions": []}}, {"node": "Sanjeev Khanna", "weight": 1, "attrs": {"institutions": []}}, {"node": "Aaron L. 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