hypergraph_measures: Structural measures for a hypergraph

In Nestimate: Network Estimation, Bootstrap, and Higher-Order Analysis

Structural measures for a hypergraph

Description

Computes a comprehensive structural-statistics suite for a net_hypergraph: node-level, hyperedge-level, and global measures. All measures are derived in a few BLAS calls on the incidence matrix.

Usage

hypergraph_measures(hg)

## S3 method for class 'hypergraph_measures'
print(x, ...)

Arguments

hg	A net_hypergraph (from build_hypergraph() or bipartite_groups()).
x	A hypergraph_measures object.
...	Additional arguments (ignored).

Details

All measures are computed via standard matrix operations on the binary incidence B = (b_{ij}) where b_{ij} = 1 iff node i is in hyperedge j:

• hyperdegree = rowSums(B), edge_sizes = colSums(B)

• co_degree = tcrossprod(B) (with zero diagonal)

• edge_pairwise_overlap = crossprod(B) (with zero diagonal)

• overlap_coefficient[i, j] = overlap[i, j] / min(edge_sizes[i], edge_sizes[j])

• jaccard[i, j] = overlap[i, j] / (edge_sizes[i] + edge_sizes[j] - overlap[i, j])

Empty hypergraph (n_hyperedges == 0) returns trivial zeros and empty matrices.

Value

An object of class hypergraph_measures (a named list) with components:

Node-level (length n_nodes)

hyperdegree

Number of hyperedges containing each node.

node_strength

Total participation: for node i, \sum_{e \ni i} |e|. A node in many large hyperedges has high strength.

max_edge_size

Size of the largest hyperedge containing each node.

co_degree

n_nodes x n_nodes matrix: ⁠co_degree[i, j] = |{e : i, j in e}|⁠ (number of hyperedges co-containing nodes i and j). Diagonal is zero.

Hyperedge-level (length n_hyperedges or m x m)

edge_sizes

Hyperedge sizes |e|.

edge_pairwise_overlap

m x m matrix: ⁠|e_i intersect e_j|⁠. Diagonal is zero.

overlap_coefficient

m x m: ⁠|e_i and e_j| / min(|e_i|, |e_j|)⁠. Measures how much the smaller hyperedge is contained in the larger.

jaccard

m x m: symmetric overlap index ⁠|e_i and e_j| / |e_i union e_j|⁠.

Global (scalars)

density

For k-uniform hypergraphs: m / choose(n, k). For mixed sizes: ⁠sum(|e|) / (n * m)⁠ (mean fraction of nodes per hyperedge).

avg_edge_size

Mean of edge_sizes.

size_distribution

Tabulation of hyperedge sizes (passed through from hg).

intersection_profile

Distribution of pairwise hyperedge intersection sizes - useful for spotting whether hyperedges overlap mostly trivially or share substantial cores (Do et al. 2020).

pairwise_participation

Fraction of node pairs co-appearing in at least one hyperedge.

n_nodes, n_hyperedges

Convenience scalars.

The input x invisibly.

References

Lee, G., Choe, M., & Shin, K. (2024). A survey on hypergraph representation, learning and mining. Data Mining & Knowledge Discovery 37, 1-39.

Do, M. T., Yoon, S., Hooi, B., & Shin, K. (2020). Structural patterns and generative models of real-world hypergraphs. arXiv:2006.07060.

See Also

build_hypergraph(), bipartite_groups(), clique_expansion().

Examples

df <- data.frame(
  player  = c("A", "B", "C", "A", "B", "D", "C", "D"),
  session = c("S1", "S1", "S1", "S2", "S2", "S3", "S3", "S3")
)
hg <- bipartite_groups(df, "player", "session")
m  <- hypergraph_measures(hg)
print(m)
m$hyperdegree         # how many sessions each player joined
m$co_degree           # pairwise co-membership counts
m$jaccard             # symmetric overlap between sessions

Nestimate documentation built on July 11, 2026, 1:09 a.m.