Large scale Ricci curvature on graphs

Abstract

We define a hybrid between Ollivier and Bakry-Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue estimates follow from the standard arguments. We prove gradient estimates similar to the ones obtained from Bakry-Emery curvature allowing us to prove Harnack and Buser inequalities.