Hypergraph

A hypergraph consists of a set of vertices and a family of edges, each of which can contain any number of vertices. Unlike ordinary graph edges, hyperedges can connect three or more vertices at once.

Graph theory: The mathematical study of objects made from vertices and edges. Hypergraphs extend graph theory by allowing edges to join more than two vertices.
Incidence matrix: A matrix recording which objects are related, commonly with entries indicating incidence. Its rows and columns encode vertices and hyperedges, exposing their membership pattern.
Simplicial complex: A collection of finite vertex sets closed under taking subsets. It models higher-order relations too, but requires every subset of a face to be included.
Graph: A mathematical structure consisting of vertices joined by edges, each connecting a pair of vertices. A graph is the special case where every hyperedge has size two.
Set: A collection of distinct objects treated as a single mathematical object. A hyperedge is defined as a subset of the hypergraph’s vertex set.
Dual hypergraph: A hypergraph formed by exchanging the roles of vertices and edges through incidence. Duality swaps the two sides of the hypergraph’s incidence relation.
Glynn–Kahn–Saks hypergraph model: A hypergraph-based model used to represent cooperative interactions among multiple participants. It illustrates how one hyperedge can represent a group interaction rather than pairwise ties.
Directed hypergraph: A hypergraph whose edges distinguish source vertices from target vertices. Direction adds ordered roles to the vertices joined by each edge.
Incidence relation: A relation that records which objects are incident with which others. It records which vertices belong to each hyperedge.
Hypergraph connectivity: The study of connectedness and separating sets in hypergraphs. Connectivity notions must account for edges that can join many vertices.

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