Curvature of Hypergraphs via Multi-Marginal Optimal Transport

TL;DR

This paper defines a new curvature measure for hypergraphs using multi-marginal optimal transport, extending Ricci curvature concepts to hypergraph structures and demonstrating its ability to identify bridges in hypergraphs.

Contribution

It introduces the coarse scalar curvature for hypergraphs, generalizing Ricci curvature for Markov chains and connecting to Riemannian geometry, with theoretical bounds and empirical validation.

Findings

Curvature detects bridges across hypergraph components.

The new curvature generalizes Ricci curvature to hypergraphs.

Empirical results support the effectiveness of the proposed measure.

Abstract

We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for Markov chains on metric spaces by Ollivier [Journal of Functional Analysis 256 (2009) 810-864], and is related to the scalar curvature when the hypergraph arises naturally from a Riemannian manifold. We investigate basic properties of the coarse scalar curvature and obtain several bounds. Empirical experiments indicate that coarse scalar curvatures are capable of detecting "bridges" across connected components in hypergraphs, suggesting it is an appropriate generalization of curvature on simple graphs.