Topological Simplifications of Hypergraphs

TL;DR

This paper introduces a topological framework for simplifying hypergraphs by transforming them into graph representations and merging similar vertices and hyperedges, aiding visualization and analysis.

Contribution

It proposes a unified, mathematically justified method for hypergraph simplification using topological data analysis tools, including line graphs and clique expansions.

Findings

Effective hypergraph simplification via topological methods

Unification of vertex and hyperedge simplification approaches

Mathematically rigorous framework for hypergraph visualization

Abstract

We study hypergraph visualization via its topological simplification. We explore both vertex simplification and hyperedge simplification of hypergraphs using tools from topological data analysis. In particular, we transform a hypergraph to its graph representations known as the line graph and clique expansion. A topological simplification of such a graph representation induces a simplification of the hypergraph. In simplifying a hypergraph, we allow vertices to be combined if they belong to almost the same set of hyperedges, and hyperedges to be merged if they share almost the same set of vertices. Our proposed approaches are general, mathematically justifiable, and they put vertex simplification and hyperedge simplification in a unifying framework.