Curvature on Graphs via Equilibrium Measures

Abstract

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by $K>0$ have diameter bounded by $\mbox{diam}(G) \leq 2/K$ (a Bonnet-Myers theorem), that $\mbox{diam}(G) = 2/K$ implies that $G$ has constant curvature (a Cheng theorem) and that there is a spectral gap $\lambda_1 \geq K/(2n)$ (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin-Lu-Yau curvature. The von Neumann minimax theorem features prominently in the proofs.