Graph curvature via resistance distance

TL;DR

This paper introduces a new notion of curvature for finite graphs based on resistance distance, establishing its key properties and implications for graph diameter, spectral gap, commute times, and Markov chain mixing times.

Contribution

It defines a curvature measure on graph vertices using resistance distance and proves its fundamental properties and applications to graph analysis.

Findings

Graphs with positive curvature have bounded diameter.

Spectral gap is at least twice the curvature lower bound.

Bounds on commute times and mixing times are derived.

Abstract

Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of curvature on the vertices $V$ via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by $K>0$ have diameter bounded from above. The Laplacian $L=D-A$ satisfies a Lichnerowicz estimate, there is a spectral gap $\lambda_2 \geq 2K$. We obtain matching two-sided bounds on the maximal commute time between any two vertices in terms of $|E| \cdot |V|^{-1} \cdot K^{-1}$. Moreover, we derive quantitative rates for the mixing time of the corresponding Markov chain and prove a general equilibrium result.