Higher-order hypergraph from a network's clique structure

Takes a network and produces a hypergraph by promoting k-cliques (k >= 3) to k-hyperedges. Each k-clique is independently included as a k-hyperedge with probability p. Optionally retains the underlying pairwise edges as 2-hyperedges. Foundation for higher-order analyses.

Usage

network_hypergraph(
  net,
  p = 1,
  type = c("clique", "vr", "rips"),
  include_pairwise = TRUE,
  max_size = 3L,
  threshold = 0,
  seed = NULL
)

# S3 method for class 'net_hg'
print(x, n = 10L, ...)

# S3 method for class 'net_hg'
summary(object, ...)

Arguments

net: A netobject, cograph_network, simplicial_complex, or numeric adjacency / weight matrix. Directed inputs are symmetrised by the underlying clique enumerator.
p: Probability in [0, 1] that each k-clique with k >= 3 becomes a k-hyperedge. Default 1 (deterministic - every found clique is included).
type: Hyperedge enumeration. "clique" (default) promotes k-cliques in the binarised adjacency to k-hyperedges. A metric Vietoris-Rips construction is not implemented; "vr" / "rips" are accepted by match.arg but raise an error rather than silently aliasing "clique".
include_pairwise: Logical. Include 2-edges from the input network as 2-hyperedges. Default TRUE. Set FALSE for a "fully higher-order" hypergraph containing only k-hyperedges with k >= 3.
max_size: Integer >= 2. Maximum hyperedge size to extract. Default 3L (triangles only). 4L also includes 4-cliques as 4-hyperedges, etc.
threshold: Numeric. Edge weight cutoff used to binarise the adjacency for clique enumeration. Default 0 (any non-zero weight is an edge).
seed: Optional integer for reproducible Bernoulli sampling when 0 < p < 1.
x: A net_hg object (for print).
n: Number of rows of the default table to print. Default 10.
...: Additional arguments (ignored).
object: A net_hg object (for summary).

Value

A net_hg object: a list with components

hyperedges: List of integer vectors. Each entry is a hyperedge given as the sorted node indices it spans.
incidence: Numeric matrix of size n_nodes x n_hyperedges. incidence[i, j] = 1 iff node i belongs to hyperedge j. Row names are node names; column names are h1, h2, ...
nodes: Character vector of node names.
n_nodes, n_hyperedges: Scalar counts.
size_distribution: Named integer vector: number of hyperedges of each size, named size_2, size_3, ...
params: Recorded call parameters: source, type, p, include_pairwise, max_size, threshold, seed.

The input x invisibly.

A data.frame, one row per node: node, degree (the number of hyperedges the node belongs to). Returned invisibly: summary(x) prints the summary and nothing else; assign the result to keep the table.

Details

The construction follows Burgio, Matamalas, Gomez & Arenas (2020) on simplicial / hypergraph contagion. For each k-clique with k >= 3 found in the underlying graph (via simplicial()), an independent Bernoulli(p) trial decides whether that clique becomes a k-hyperedge. Underlying pairwise edges are always retained when include_pairwise = TRUE, so the resulting hypergraph contains both the original 2-edge structure and the sampled higher-order interactions.

At the limits:

p = 0 with include_pairwise = TRUE reproduces the input pairwise network as a hypergraph of size-2 edges.
p = 1 with include_pairwise = FALSE returns a fully higher-order hypergraph containing only the k-hyperedges (k >= 3) found in the network's clique complex.

References

Burgio, G., Matamalas, J. T., Gomez, S., & Arenas, A. (2020). Evolution of cooperation in the presence of higher-order interactions: from networks to hypergraphs. Entropy 22(7), 744. doi:10.3390/e22070744

Examples

set.seed(1)
n <- 8
adj <- matrix(stats::rbinom(n * n, 1, 0.5), n, n)
diag(adj) <- 0
adj <- (adj + t(adj)) > 0
rownames(adj) <- colnames(adj) <- LETTERS[seq_len(n)]
hg <- network_hypergraph(adj, p = 1, max_size = 3L)
print(hg)
#> Hypergraph: 8 nodes, 51 hyperedges (sizes 2: 23, 3: 28)
#> Source: network cliques (p = 1.00, include_pairwise = TRUE, max_size = 3)
#>  hyperedge size members weight
#>         h1    2    C, E     NA
#>         h2    2    A, C     NA
#>         h3    2    A, D     NA
#>         h4    2    D, E     NA
#>         h5    2    C, D     NA
#>         h6    2    A, B     NA
#>         h7    2    B, E     NA
#>         h8    2    B, C     NA
#>         h9    2    B, H     NA
#>        h10    2    D, H     NA
#> ... 41 more rows
summary(hg)
#> Hypergraph summary
#>   Nodes:         8
#>   Hyperedges:    51
#>   Mean size:     2.55
#>   Max size:      3