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arXiv · 2302.06021

Graph curvature via resistance distance

Karel Devriendt·Andrea Ottolini·Stefan Steinerberger
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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.

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Karel Devriendt, Andrea Ottolini, Stefan Steinerberger. 2023-02-12. Graph curvature via resistance distance. https://arxiv.org/abs/2302.06021

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