Local Algorithms for Estimating Effective Resistance

TL;DR

This paper introduces local algorithms that efficiently estimate effective resistance between vertices in large graphs, with provable guarantees and practical validation on benchmark datasets.

Contribution

It presents novel local algorithms for approximating effective resistance with small additive error and theoretical performance guarantees, especially on graphs with bounded mixing time.

Findings

Algorithm approximates effective resistance in polylogarithmic time

Empirical validation confirms practical efficiency and accuracy

Performance guarantees hold for graphs with bounded mixing time

Abstract

Effective resistance is an important metric that measures the similarity of two vertices in a graph. It has found applications in graph clustering, recommendation systems and network reliability, among others. In spite of the importance of the effective resistances, we still lack efficient algorithms to exactly compute or approximate them on massive graphs. In this work, we design several \emph{local algorithms} for estimating effective resistances, which are algorithms that only read a small portion of the input while still having provable performance guarantees. To illustrate, our main algorithm approximates the effective resistance between any vertex pair $s,t$ with an arbitrarily small additive error $\varepsilon$ in time $O(\mathrm{poly}(\log n/\varepsilon))$, whenever the underlying graph has bounded mixing time. We perform an extensive empirical study on several benchmark datasets, validating the performance of our algorithms.