Curvature of Hypergraphs via Multi-Marginal Optimal Transport
This work addresses a theoretical gap in extending geometric concepts to hypergraphs, which is incremental but useful for researchers in network analysis and discrete geometry.
The paper tackles the problem of defining curvature for hypergraphs by introducing a novel coarse scalar curvature based on multi-marginal optimal transport, generalizing Ricci curvature for Markov chains and showing it can detect bridges across connected components in empirical experiments.
We introduce a novel definition of curvature for hypergraphs, a natural generalization of graphs, by introducing a multi-marginal optimal transport problem for a naturally defined random walk on the hypergraph. This curvature, termed \emph{coarse scalar curvature}, generalizes a recent definition of Ricci curvature for Markov chains on metric spaces by Ollivier [Journal of Functional Analysis 256 (2009) 810-864], and is related to the scalar curvature when the hypergraph arises naturally from a Riemannian manifold. We investigate basic properties of the coarse scalar curvature and obtain several bounds. Empirical experiments indicate that coarse scalar curvatures are capable of detecting "bridges" across connected components in hypergraphs, suggesting it is an appropriate generalization of curvature on simple graphs.