# Large scale Ricci curvature on graphs

## TL;DR

This paper introduces a new hybrid Ricci curvature concept for graphs that combines Ollivier and Bakry-Emery notions, leading to significant geometric and analytic inequalities.

## Contribution

It defines a novel curvature notion on graphs depending on variable neighborhoods, extending classical results like diameter bounds and eigenvalue estimates.

## Key findings

- Hexagonal lattice is non-negatively curved under the new curvature.
- Derived diameter bounds and eigenvalue estimates analogous to classical results.
- Established gradient, Harnack, and Buser inequalities for the new curvature.

## Abstract

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.