Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework

TL;DR

This paper introduces ORCHID, a novel framework that extends Ollivier-Ricci curvature to hypergraphs, providing a scalable and theoretically grounded tool for analyzing complex hypergraph structures in various applications.

Contribution

The paper develops a new framework, ORCHID, for defining Ollivier-Ricci curvature on hypergraphs, with proven theoretical properties and practical effectiveness demonstrated through experiments.

Findings

ORCHID curvature is scalable to large hypergraphs.

It effectively captures geometric properties of hypergraphs.

Experimental results show utility in multiple hypergraph tasks.

Abstract

Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging domain of hypergraphs has remained largely unexplored. On graphs, the Ollivier-Ricci curvature measures differences between random walks via Wasserstein distances, thus grounding a geometric concept in ideas from probability theory and optimal transport. We develop ORCHID, a flexible framework generalizing Ollivier-Ricci curvature to hypergraphs, and prove that the resulting curvatures have favorable theoretical properties. Through extensive experiments on synthetic and real-world hypergraphs from different domains, we demonstrate that ORCHID curvatures are both scalable and useful to perform a variety of hypergraph tasks in practice.