Coarse Ricci curvature of hypergraphs and its generalization

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.

version accepted in Osaka Journal of Mathematics