Ollivier-Ricci Curvature for Hypergraphs: A Unified FrameworkDownload PDF

Published: 01 Feb 2023, Last Modified: 13 Feb 2023ICLR 2023 posterReaders: Everyone
Keywords: curvature, hypergraphs, graphs, Wasserstein distance, topological data analysis, random walks, probability measure
TL;DR: We introduce a flexible framework for Ollivier-Ricci curvature on hypergraphs.
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.
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