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An equivalent definition of entropic Ricci curvature on discrete spaces wasgiven in terms of the global gradient estimate. With a particular choice of thedensity function $\rho$, we obtain a localized gradient estimate, which inturns allow us to derive a Bonnet-Myers type diameter bound for graphs withpositive entropic Ricci curvature. However, the case of the hypercubesindicates that the bound may be not optimal (where $\theta$ is chosen to belogarithmic mean by default). If $\theta$ is arithmetic mean, the Bakry-\'Emerycriterion can be recovered and the diameter bound is optimal as it can beattained by the hypercubes.

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