Structure from Voltage

TL;DR

This paper proposes a scaled effective resistance method to extract meaningful structural information from graphs, overcoming previous limitations, and introduces a way to compute distances from a point using a ground node.

Contribution

It introduces a scaling of resistances in graphs to obtain non-trivial limits of voltages and resistances, and adds a ground node to metric graphs for efficient distance computation.

Findings

Scaling resistances by n^2 yields meaningful limits

Adding a ground node simplifies distance calculations

Resistances provide an alternative to eigenvector methods

Abstract

Effective resistance (ER) is an attractive way to interrogate the structure of graphs. It is an alternative to computing the eigen-vectors of the graph Laplacian. Graph laplacians are used to find low dimensional structures in high dimensional data. Here too, ER based analysis has advantages over eign-vector based methods. Unfortunately Von Luxburg et al. (2010) show that, when vertices correspond to a sample from a distribution over a metric space, the limit of the ER between distant points converges to a trivial quantity that holds no information about the structure of the graph. We show that by using scaling resistances in a graph with $n$ vertices by $n^2$, one gets a meaningful limit of the voltages and of effective resistances. We also show that by adding a "ground" node to a metric graph one gets a simple and natural way to compute all of the distances from a chosen point to all other points.