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.

翻译：连接几何与拓扑，曲率是一种强大且富有表现力的不变量。尽管曲率在流形和图上的实用性已在理论和实证层面得到验证，但其在新兴的超图领域中的泛化仍鲜有探索。在图论中，Ollivier-Ricci曲率通过Wasserstein距离度量随机游走之间的差异，从而将几何概念植根于概率论和最优传输的思想。我们开发了ORCHID，一个将Ollivier-Ricci曲率泛化到超图的灵活框架，并证明由此产生的曲率具有优越的理论性质。通过在不同领域的合成超图和真实超图上进行大量实验，我们证明ORCHID曲率兼具可扩展性，并能有效实践多种超图任务。