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Discrete time random walks on a finite set naturally translate via aone-to-one correspondence to discrete Laplace operators. Typically, Olliviercurvature has been investigated via random walks. We first extend thedefinition of Ollivier curvature to general weighted graphs and then give astrikingly simple representation of Ollivier curvature using the graphLaplacian. Using the Laplacian as a generator of a continuous time Markovchain, we connect Ollivier curvature with the heat equation which is stronglyrelated to continuous time random walks. In particular, we prove that a lowerbound on the Ollivier curvature is equivalent to a certain Lipschitz decay ofsolutions to the heat equation. This is a discrete analogue to a celebratedRicci curvature lower bound characterization by Renesse and Sturm. Ourrepresentation of Ollivier curvature via the Laplacian allows us to deduce aLaplacian comparison principle by which we prove non-explosion and improveddiameter bounds.

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