OLLIVIER-RICCI CURVATURE FOR HYPERGRAPHS: A UNIFIED FRAMEWORK

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.

1. INTRODUCTION

Hypergraphs generalize graphs by allowing any number of nodes to participate in an edge. They enable us to faithfully represent complex relations, such as co-authorship of scientific papers, multilateral interactions between chemicals, or group conversations, which cannot be adequately captured by graphs. While hypergraphs are more expressive than graphs and other relational objects like simplicial complexes, they are harder to analyze both theoretically and empirically, and many concepts that have proven useful for understanding graphs have yet to be transferred to the hypergraph setting. Curvature has established itself as a powerful characteristic of Riemannian manifolds, as it permits the description of global properties through local measurements by harmonizing ideas from geometry and topology. For graphs, graph curvature measures to what extent the neighborhood of an edge deviates from certain idealized model spaces, such as cliques, grids, or trees. It has proven helpful, for example, in assessing differences between real-world networks (Samal et al., 2018) , identifying bottlenecks in real-world networks (Gosztolai & Arnaudon, 2021), and alleviating oversquashing in graph neural networks (Topping et al., 2022) . One prominent notion of graph curvature is Ollivier-Ricci curvature (ORC). ORC compares random walks based at specific nodes, revealing differences in the information diffusion behavior in the graph. As the sizes of edges and edge intersections can vary in hypergraphs, there are many ways to generalize ORC to hypergraphs. While some notions of hypergraph ORC have been previously studied in isolation (e.g., Asoodeh et al., 2018; Eidi & Jost, 2020; Leal et al., 2020) , a unified framework for their definition and computation is still lacking. Contributions. We introduce ORCHID, a unified framework for Ollivier-Ricci curvature on hypergraphs. ORCHID integrates and generalizes existing approaches to hypergraph ORC. Our work is the first to identify the individual building blocks shared by all notions of hypergraph ORC, and to perform a rigorous theoretical and empirical analysis of the resulting curvature formulations. We develop hypergraph ORC notions that are aligned with our geometric intuition while still efficient to compute, and we demonstrate the utility of these notions in practice through extensive experiments.

