Project a hypergraph onto a weighted graph

Collapses every hyperedge into pairwise vertex relations, giving the weighted graph that a graph engine (paths, betweenness, communities) can consume. Two weightings are available. "clique" is plain co-occurrence, the clique expansion: a hyperedge of size $|e|$ contributes the same amount to each of its $|e|(|e|-1)/2$ pairs, so a single large hyperedge can dominate every downstream measure. "association" divides each hyperedge's contribution by $|e|-1$, following Coupette et al. (2024):

Usage

hg_project(
  hg,
  method = c("clique", "association", "citation"),
  weighted = TRUE,
  what = c("edges", "matrix"),
  duplicate_edges = c("count", "collapse"),
  self_association = FALSE,
  edge_source = NULL,
  directed = FALSE
)

Arguments

hg: A text_hypergraph(), knn_hypergraph(), or any hypernets net_hg.
method: Weighting. "clique" (default) sums incidence products; "association" applies the $1/(|e|-1)$ normalisation above; "citation" joins each hyperedge's source to its members.
weighted: method = "clique" only. TRUE (default) uses the incidence weights, FALSE their membership pattern. Setting it together with method = "association" is an error, because the association weighting is defined on hyperedge cardinality and never on the incidence weights.
what: "edges" (default) for the tidy edge list, or "matrix" for the symmetric weight matrix to hand to a graph engine.
duplicate_edges: For method = "association", "count" (default) lets repeated hyperedges contribute repeatedly (the paper's multi-hypergraph representation); "collapse" lets each distinct member set contribute once (its binary-hypergraph representation).
self_association: Add the paper's source-to-member association term? Requires one source vertex per hyperedge through edge_source or hg$edge_data$source. For source u, every membership occurrence of target v contributes 1 / sum_e |e| over hyperedges sourced by u, so the added incident weight from all of u's citations sums to one.
edge_source: Which node each hyperedge comes from: the name of a column of the hyperedge attributes (hg$edge_data, e.g. "citing"), a vector of length n_hyperedges, a named vector keyed by hyperedge, or a two-column data frame named edge and source. NULL uses an attribute column named source. Used with self_association = TRUE and method = "citation".
directed: For method = "citation": return the directed source-to-member matrix (rows cite columns)? Default FALSE.

Value

With what = "edges", a base data.frame with one row per unordered vertex pair of non-zero weight, sorted by from then to, with columns from, to (vertex names) and weight (numeric). Vertices sharing no hyperedge do not appear, so an edgeless hypergraph yields a zero-row data.frame with those columns. With what = "matrix", the symmetric n_nodes x n_nodes weight matrix with zero diagonal and vertex names as dimnames, sparse if hg's incidence is sparse.

Details

$$w(\{u,v\}) = \sum_{e \supseteq \{u,v\}} \frac{1}{|e| - 1}$$

so the total weight a hyperedge adds around any one of its members is exactly 1, and each vertex's weighted degree in the projection equals the number of hyperedges of size at least two that contain it. That normalisation is what makes the projection a legitimate random-walk operator rather than an arbitrary co-occurrence count.

A hyperedge of size one has no pairs, so it contributes nothing and is skipped rather than dividing by zero. Such hyperedges are common in text: a word used in exactly one document is a singleton hyperedge in the document orientation, which is why the degree identity above counts only hyperedges of size at least two.

A third projection, "citation", is the classic graph a hypergraph with sources reduces to: one edge from the source of every hyperedge to each of its members, so a citation-block hypergraph becomes the ordinary citation graph of the paper's Table 2. duplicate_edges = "count" weights an edge by how many blocks repeat it (the multi-graph, mg); "collapse" keeps the binary graph (bg). directed = TRUE keeps the source-to-member orientation; the default symmetrises.

References

Coupette, C., Hartung, D., & Katz, D. M. (2024). Legal hypergraphs. Philosophical Transactions of the Royal Society A, 382(2270), 20230141. doi:10.1098/rsta.2023.0141

Zhou, D., Huang, J., & Schoelkopf, B. (2006). Learning with hypergraphs: clustering, classification, and embedding. NeurIPS 19, 1601-1608.

Examples