Topological Simplifications of Hypergraphs

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.

Hypergraph Co-Optimal Transport: Metric and Categorical Properties

“…Example 15 (Node and Edge Collapses as Basic Weak Isomorphisms). A standard way to simplify a hypergraph is via node collapse and hyperedge collapse [48]. As illustrated in Fig.…”

“…In this section, we give an example that captures changes in hypernetwork distances as we apply multi-scale hypergraph simplification following the framework of Zhou et al [48]. Visualizing large hypergraphs is a challenging task.…”

“…To reduce visual clutter, we may apply node collapse and edge collapse to reduce the size of the hypergraph, see Example 15. Zhou et al [48] relaxed these notions by allowing nodes 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 nodes; these operations are referred to as node simplification and hyperedge simplification, respectively. We focus on hyperedge simplification in this example.…”