Curvature on graphs via equilibrium measures

Abstract

AbstractWe 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 have diameter bounded by (a Bonnet–Myers theorem), that implies that has constant curvature (a Cheng theorem) and that there is a spectral gap (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.