Coarse Ricci curvature of hypergraphs and its generalization
MasaHiro Ikeda, Yu Kitabeppu, Yuuki Takai, Takato Uehara
TL;DR
The paper introduces a coarse Ricci curvature for finite hypergraphs defined via a nonlinear Kantorovich difference tied to the resolvent of a submodular hypergraph Laplacian. This curvature extends Lin-Lu-Yau’s graph curvature to hypergraphs and yields concrete analytic and geometric consequences, including eigenvalue bounds, gradient estimates for the heat flow, and Bonnet-Myers-type diameter bounds, highlighting how nonlinearity influences curvature. Existence of the curvature is established through a combination of piecewise-linear and linear-programming arguments, and the framework naturally extends to submodular transformations beyond hypergraphs. The authors also verify consistency with the classical LL-Y curvature on graphs and provide worked examples illustrating both positive and negative curvature regimes. The work lays a foundation for applying synthetic curvature concepts to nonlinear discrete structures and broad submodular models.
Abstract
In the present paper, we introduce a concept of Ricci curvature on hypergraphs for a nonlinear Laplacian. We prove that our definition of the Ricci curvature is a generalization of Lin-Lu-Yau coarse Ricci curvature for graphs to hypergraphs. We also show a lower bound of nonzero eigenvalues of Laplacian, gradient estimate of heat flow, and diameter bound of Bonnet-Myers type for our curvature notion. This research leads to understanding how nonlinearity of Laplacian causes complexity of curvatures.