Heat flow and concentration of measure on directed graphs with a lower Ricci curvature bound

TL;DR

This paper characterizes lower Ricci curvature bounds on directed graphs using heat semigroup gradient estimates and transportation inequalities, leading to measure concentration results for graphs with positive Ricci curvature.

Contribution

It provides a von Renesse-Sturm type characterization of Ricci curvature bounds for directed graphs and applies this to derive measure concentration inequalities.

Findings

Gradient estimates for heat semigroup on directed graphs

Transportation inequalities along heat flow

Concentration of measure for graphs with positive Ricci curvature

Abstract

In a previous work, the authors introduced a Lin-Lu-Yau type Ricci curvature for directed graphs referring to the formulation of the Chung Laplacian. The aim of this note is to provide a von Renesse-Sturm type characterization of our lower Ricci curvature bound via a gradient estimate for the heat semigroup, and a transportation inequality along the heat flow. As an application, we will conclude a concentration of measure inequality for directed graphs of positive Ricci curvature.