Heat Flow and Concentration of Measure on Directed Graphs with a Lower Ricci Curvature Bound

Abstract

In a previous work (Ozawa et al. Calc. Var. Partial Diff. Equ. 59(4), 39 2020), 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.