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

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.