Graph Clustering using Effective Resistance

Abstract

We design a polynomial time algorithm that for any weighted undirected graph G = (V, E, w) and sufficiently large \delta > 1, partitions V into subsets V(1),..., V(h) for some h>= 1, such that at most \delta^{-1} fraction of the weights are between clusters, i.e.

sum(i < j) |E(V(i), V(j)| < w(E)/\delta 

and the effective resistance diameter of each of the induced subgraphs
G[V(i)] is at most \delta^3 times the inverse of the average weighted degree, i.e.

max{ Reff(u, v) :  u, v \in V(i)} < \delta^3 · |V|/w(E)

for all i = 1,..., h.  In particular, it is possible to remove one
percent of weight of edges of any given graph such that each of the
resulting connected components has effective resistance diameter at
most the inverse of the average weighted degree.  Our proof is based
on a new connection between effective resistance and low conductance
sets.  We show that if the effective resistance between two vertices u and v is large, then there must be a low conductance cut separating u from v. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.