The pairwise network of a hypergraph

Projects a hypergraph onto a network of pairs of its nodes. Two nodes are joined when they belong to a common hyperedge, and the weight of the edge depends on type. "clique" is the clique expansion: a hyperedge of size $|e|$ contributes the same amount to each of its $|e|(|e|-1)/2$ pairs, so the weight of a pair is the number of hyperedges that contain both nodes (Zhou et al. 2006). "association" divides the contribution of each hyperedge by $|e|-1$, following Coupette et al. (2024):

Usage

pairwise_network(
  hg,
  type = c("clique", "association", "citation"),
  weighted = NULL,
  duplicate_edges = c("count", "collapse"),
  self_association = FALSE,
  edge_source = NULL,
  directed = FALSE
)

Arguments

hg: A hypergraph (net_hg), such as one built by hypergraph() or text_hypergraph().
type: The projection: "clique" (default), "association" or "citation".
weighted: type = "clique" only. TRUE (default) uses the incidence weights, FALSE their membership pattern. Setting it with another type is an error, because those projections are defined on membership.
duplicate_edges: For type = "association" or "citation": "count" (default) lets repeated hyperedges contribute repeatedly (the multi-hypergraph); "collapse" lets each distinct member set contribute once (the binary hypergraph).
self_association: For type = "association": add the source-to-member association term of Coupette et al. (2024). Requires one source node per hyperedge through edge_source or hg$edge_data$source. For source u, every membership of target v contributes 1 / sum_e |e| over the hyperedges sourced by u.
edge_source: The node each hyperedge comes from: the name of a column of the hyperedge attributes, a vector of length n_hyperedges, a named vector keyed by hyperedge, or a data frame with columns edge and source (a hyperedge named twice with different sources raises hypergraphs_bad_input). NULL uses an attribute column named source. Used with self_association = TRUE and type = "citation".
directed: For type = "citation": keep the source-to-member direction. Default FALSE.

Value

A net_hg_pairwise, which is also a netobject and a cograph_network: the weighted adjacency matrix of the projection (weights, zero diagonal, symmetric unless directed = TRUE), its nodes and edges. hg_get(x) returns the edges as a data frame with one row per pair of non-zero weight and columns from, to and weight, sorted by from then to.

Details

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

so the weight a hyperedge adds around each of its members is 1, and the weighted degree of a node equals the number of hyperedges of size at least two that contain it. A hyperedge of size one has no pairs and contributes nothing.

"citation" is the graph of a hypergraph whose hyperedges have sources: one edge from the source of every hyperedge to each of its members, so a hypergraph of citation blocks becomes the ordinary citation graph. duplicate_edges = "count" weights an edge by the number of blocks that repeat it (the multi-graph); "collapse" keeps the binary graph. directed = TRUE keeps the source-to-member orientation; the default symmetrises.

The result is a network object that cograph plots and that hypergraph() reads as a network, so the cliques of the projection can be promoted back to hyperedges. Its edges are read with hg_get().

Conditions

hypergraphs_bad_input for an argument that does not apply to type, and for self_association or type = "citation" without a source per hyperedge.

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. doi:10.7551/mitpress/7503.003.0205

Examples

meetings <- data.frame(
  person = c("Alice", "Bob", "Carol", "Alice", "Bob", "Dave", "Eve"),
  meeting = c("m1", "m1", "m1", "m2", "m2", "m3", "m3"))
meeting_hg <- hypergraph(meetings, node = "person", hyperedge = "meeting")
meeting_network <- pairwise_network(meeting_hg)
hg_get(meeting_network)
#>    from    to weight
#> 1 Alice   Bob      2
#> 2 Alice Carol      1
#> 3   Bob Carol      1
#> 4  Dave   Eve      1
association_network <- pairwise_network(meeting_hg, type = "association")
hg_get(association_network)
#>    from    to weight
#> 1 Alice   Bob    1.5
#> 2 Alice Carol    0.5
#> 3   Bob Carol    0.5
#> 4  Dave   Eve    1.0