Stability of Hypergraph Invariants and Transformations

TL;DR

This paper develops a new metric for hypergraphs, analyzes the stability of hypergraph transformations, and explores the robustness of hypergraph invariants using tools from topology and optimal transport.

Contribution

It introduces a hypergraph metric inspired by Gromov-Hausdorff distance, studies Lipschitz properties of hypergraph-to-graph transformations, and derives bounds for hypergraph invariants.

Findings

Hypergraph metric exhibits Lipschitz continuity under certain transformations.

Graphification method relates hypergraphs to hierarchical clustering.

Lower bounds for hypergraph distances derived from topological invariants.

Abstract

Graphs are fundamental tools for modeling pairwise interactions in complex systems. However, many real-world systems involve multi-way interactions that cannot be fully captured by standard graphs. Hypergraphs, which generalize graphs by allowing edges to connect any number of vertices, offer a more expressive framework. In this paper, we introduce a new metric on the space of hypergraphs, inspired by the Gromov-Hausdorff distance for metric spaces. We establish Lipschitz properties of common hypergraph transformations, which send hypergraphs to graphs, including a novel graphification method with ties to single linkage hierarchical clustering. Additionally, we derive lower bounds for the hypergraph distance via invariants coming from basic summary statistics and from topological data analysis techniques. Finally, we explore stability properties of cost functions in the context of optimal transport. Our results in this direction consider Lipschitzness of the Hausdorff map and conservation of the non-negative cross curvature property under limits of cost functions.