A Triangle Inequality for p-Resistance
Mark Herbster
In: NIPS Workshop 2010: Networks Across Disciplines: Theory and Applications, December 11, 2010, Whistler, Canada.

Abstract

The geodesic distance (path length) and effective resistance are both metrics defined on the vertices of a graph. The effective resistance is a more refined measure of connectivity than the geodesic distance. For example if there are $k$ edge disjoint paths of geodesic distance $d$ between two vertices, then the effective resistance is no more than $\frac{d}{k}$. Thus, the more paths, the closer the vertices. We continue the study of the recently introduced p-effective resistance [HL09]. The main technical contribution of this note is to prove that the p-effective resistance is a metric for p in (1,2] and obeys a strong triangle inequality. Given an efficient method to compute the p-effective resistance then an easy consequence of this inequality is that we may efficiently find a k-center clustering within a factor of 2^(p-1) from the optimal clustering with respect to p-effective resistance.

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