Ollivier Ricci curvature for general graph Laplacians: Heat equation, Laplacian comparison, non-explosion and diameter bounds

TL;DR

This paper extends Ollivier Ricci curvature to general graphs, linking it with the heat equation and Laplacian, leading to new bounds on graph diameter and non-explosion properties.

Contribution

It introduces a novel Laplacian-based representation of Ollivier curvature for weighted graphs, connecting it with heat flow and geometric bounds.

Findings

Lower bounds on Ollivier curvature imply Lipschitz decay of heat solutions.

Established a Laplacian comparison principle for graphs.

Proved non-explosion and improved diameter bounds for graphs.

Abstract

Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representation of Ollivier curvature using the graph Laplacian. Using the Laplacian as a generator of a continuous time Markov chain, we connect Ollivier curvature with the heat equation which is strongly related to continuous time random walks. In particular, we prove that a lower bound on the Ollivier curvature is equivalent to a certain Lipschitz decay of solutions to the heat equation. This is a discrete analogue to a celebrated Ricci curvature lower bound characterization by Renesse and Sturm. Our representation of Ollivier curvature via the Laplacian allows us to deduce a Laplacian comparison principle by which we prove non-explosion and improved diameter bounds.