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Hypergraph eigenvector centralities

Source: R/hypergraph_centrality.R
hypergraph_centrality.Rd

Computes one or more eigenvector-style centralities on a net_hypergraph: clique-motif (CEC), Z-eigenvector (ZEC), and H-eigenvector (HEC). Each variant captures influence differently - CEC flattens group structure via clique expansion, while ZEC and HEC propagate through the higher-order groups directly.

Usage

hypergraph_centrality(
  hg,
  type = c("clique", "Z", "H"),
  max_iter = 1000L,
  tol = 1e-08,
  normalize = TRUE
)

Arguments

hg

A net_hypergraph (from build_hypergraph() or bipartite_groups()).

type

Character vector, any subset of c("clique", "Z", "H"). Default computes all three.

max_iter

Maximum number of power-iteration steps. Default 1000.

tol

Convergence tolerance on the L1 change between successive iterates. Default 1e-8.

normalize

Logical. If TRUE (default), each returned centrality vector is L2-normalized to unit norm (compatible with igraph::eigen_centrality()'s scale for type "clique").

Value

A named list; one component per requested type. Each component is a named numeric vector of length hg$n_nodes.

Details

Clique-motif eigenvector centrality (CEC): forms the clique-expanded pairwise graph \(W\) where \(W_{ij} = |\{e : i, j \in e\}|\) and returns the leading eigenvector of \(W\). Equivalent to running igraph::eigen_centrality() on clique_expansion() output.

Z-eigenvector centrality (ZEC): solves the linear eigen-equation on the hyperedge tensor, $$\lambda\, x_i \;=\; \sum_{e \ni i}\; \prod_{j \in e,\; j \neq i} x_j,$$ via power iteration. Works for hypergraphs with mixed edge sizes.

H-eigenvector centrality (HEC): solves the power-k-1 eigen-equation, $$\lambda\, x_i^{k-1} \;=\; \sum_{e \ni i}\; \prod_{j \in e,\; j \neq i} x_j.$$ For uniform hypergraphs (all hyperedges of size \(k\)), this is equivalent to normalizing the ZEC update by the geometric-mean exponent \(1/(k-1)\). For mixed sizes, the effective exponent is taken from the largest hyperedge; expect slightly different rankings from ZEC in the mixed case.

Note

The "clique" (CEC) variant is validated against igraph::eigen_centrality (cosine ~ 1). The "Z" and "H" variants are (experimental) - validated only against a clean-room list-based tensor power iteration (same operator, different loop structure); no R package exposes tensor eigenvectors as a primitive for independent comparison.

References

Benson, A. R. (2019). Three hypergraph eigenvector centralities. SIAM Journal on Mathematics of Data Science 1(2), 293-312. arXiv:1807.09644.

See also

build_hypergraph(), clique_expansion(), hypergraph_measures().

Examples

df <- data.frame(
  player  = c("A", "B", "C", "A", "B", "D", "C", "D", "E"),
  session = c("S1", "S1", "S1", "S2", "S2", "S3", "S3", "S3", "S3")
)
hg <- bipartite_groups(df, "player", "session")
cent <- hypergraph_centrality(hg)
# Compare rankings across the three variants
do.call(cbind, cent)
#>      clique            Z         H
#> A 0.5345225 6.666667e-01 0.6147176
#> B 0.5345225 6.666667e-01 0.6147176
#> C 0.5345225 3.333333e-01 0.4216259
#> D 0.2672612 3.983626e-09 0.1823130
#> E 0.2672612 3.983626e-09 0.1823130