Curvature on Graphs via Equilibrium Measures

TL;DR

This paper introduces a new way to measure curvature on finite graphs using equilibrium measures, establishing geometric and spectral bounds, and demonstrating its computation and relation to existing curvature notions.

Contribution

It proposes a novel curvature concept for graphs, proves key geometric and spectral theorems, and compares it with existing curvature measures.

Findings

Graphs with positive curvature have bounded diameter.

Constant curvature graphs are characterized by diameter equality.

Spectral gap bounds are established based on curvature.

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.