  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.
