A Characterization of Effective Resistance Metrics

TL;DR

This paper characterizes finite metric spaces derived from effective resistance in graphs and explores their limits, connecting resistance metrics with graph convergence and random walk behavior.

Contribution

It provides a new characterization of resistance metrics and analyzes their limits in infinite graph sequences, linking them to Kigami's resistance metrics.

Findings

Finite resistance metrics correspond to specific graph structures.

Sequences of finite graphs with increasing resistance metrics converge to a limit graph.

Random walks on these graphs converge to the walk on the limit graph.

Abstract

We produce a characterization of finite metric spaces which are given by the effective resistance of a graph. This characterization is applied to the more general context of resistance metrics defined by Kigami. A countably infinite resistance metric gives rise to a sequence of finite, increasing graphs with invariant effective resistance. We show that these graphs have a unique limit graph in terms of the convergence of edge weights and that their associated random walks converge weakly to the random walk on the limit graph. If the limit graph is recurrent, its effective resistance is identified as the initial resistance metric.