A Characterization of Effective Resistance Metrics

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.Mathematics Subject Classification (2010) Primary 05C12 · 05C81 · 31E05; Secondary 05C50 · 54E70 · 60J45

On the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs

“…Recurrent graphs, however, have a property which is often referred to as unique currents [6] and consequently also have one unique effective resistance. In this case, the above repre-sentation holds [1,8]. Indeed, [1, Lemma 2.61] states the more general inequalities…”

“…Remark 5. 8 An equivalent approach would be to consider a lazy random walk on G n which has the same transition probabilities p(v, w) as P x for v = w but stays at v with probability…”

On the Probabilistic Representation of the Free Effective Resistance of Infinite Graphs

“…Recurrent graphs, however, have a property which is often referred to as unique currents [LP16] and consequently also have one unique effective resistance. In this case, the above representation holds [Bar17,Wei18]. Indeed, [Bar17, Theorem 2.61] states the more general inequalities…”