  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.
