Formal Concept Lattice Representations and Algorithms for Hypergraphs

TL;DR

This paper explores the relationship between hypergraphs and formal concept lattices, introducing new algorithms to analyze hypergraph structures through lattice theory, and demonstrates their effectiveness on real-world data.

Contribution

It establishes an isomorphism between hypergraph intersection complexes and formal concept lattices, and develops algorithms for hypergraph analysis via lattice operations.

Findings

Deep lattices indicate high interconnectivity in hypergraphs

Algorithms reveal complex geometric structures of hyperedges

Application to real-world data demonstrates practical utility

Abstract

There is increasing focus on analyzing data represented as hypergraphs, which are better able to express complex relationships amongst entities than are graphs. Much of the critical information about hypergraph structure is available only in the intersection relationships of the hyperedges, and so forming the "intersection complex" of a hypergraph is quite valuable. This identifies a valuable isomorphism between the intersection complex and the "concept lattice" formed from taking the hypergraph's incidence matrix as a "formal context": hypergraphs also generalize graphs in that their incidence matrices are arbitrary Boolean matrices. This isomorphism allows connecting discrete algorithms for lattices and hypergraphs, in particular s-walks or s-paths on hypergraphs can be mapped to order theoretical operations on the concept lattice. We give new algorithms for formal concept lattices and hypergraph s-walks on concept lattices. We apply this to a large real-world dataset and find deep lattices implying high interconnectivity and complex geometry of hyperedges.