Ollivier-Ricci Curvature for Hypergraphs: A Unified Framework
This work addresses a gap in applying geometric invariants to hypergraphs, potentially benefiting researchers in network analysis and machine learning, though it is incremental as it extends an existing method to a new structure.
The authors tackled the problem of generalizing curvature to hypergraphs, which had been unexplored, by developing ORCHID, a framework that extends Ollivier-Ricci curvature to hypergraphs and demonstrates scalability and utility in experiments on synthetic and real-world data.
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.