Effective resistance in metric spaces

TL;DR

This paper introduces a region-based effective resistance approach in metric spaces that remains informative in large samples, overcoming the trivial convergence problem of point-based ER in point clouds.

Contribution

It proposes a novel region-based ER method with appropriate scaling, ensuring non-trivial limits and providing theoretical and numerical validation.

Findings

Region-based ER converges to a non-trivial limit as sample size grows.

Proper scaling of edge weights is crucial for meaningful ER analysis.

Numerical experiments support the theoretical results.

Abstract

Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigenvectors of the graph Laplacian. One attractive application of ER is to point clouds, i.e. graphs whose vertices correspond to IID samples from a distribution over a metric space. Unfortunately, it was shown that the ER between any two points converges to a trivial quantity that holds no information about the graph's structure as the size of the sample increases to infinity. In this study, we show that this trivial solution can be circumvented by considering a region-based ER between pairs of small regions rather than pairs of points and by scaling the edge weights appropriately with respect to the underlying density in each region. By keeping the regions fixed, we show analytically that the region-based ER converges to a non-trivial limit as the number of points increases to infinity. Namely the ER on a metric space. We support our theoretical findings with numerical experiments.