A note on a Bonnet-Myers type diameter bound for graphs with positive entropic Ricci curvature (2003.01160v1)

Abstract: An equivalent definition of entropic Ricci curvature on discrete spaces was given in terms of the global gradient estimate. With a particular choice of the density function $\rho$, we obtain a localized gradient estimate, which in turns allow us to derive a Bonnet-Myers type diameter bound for graphs with positive entropic Ricci curvature. However, the case of the hypercubes indicates that the bound may be not optimal (where $\theta$ is chosen to be logarithmic mean by default). If $\theta$ is arithmetic mean, the Bakry-\'Emery criterion can be recovered and the diameter bound is optimal as it can be attained by the hypercubes.