  We define a hybrid between Ollvier 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.
