Phase transition in the family of p-resistances

Part of Advances in Neural Information Processing Systems 24 (NIPS 2011)

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Authors

Morteza Alamgir, Ulrike Luxburg

Abstract

<p>We study the family of p-resistances on graphs for p ≥ 1. This family generalizes the standard resistance distance. We prove that for any fixed graph, for p=1, the p-resistance coincides with the shortest path distance, for p=2 it coincides with the standard resistance distance, and for p → ∞ it converges to the inverse of the minimal s-t-cut in the graph. Secondly, we consider the special case of random geometric graphs (such as k-nearest neighbor graphs) when the number n of vertices in the graph tends to infinity. We prove that an interesting phase-transition takes place. There exist two critical thresholds p^* and p^<em>* such that if p &lt; p^</em>, then the p-resistance depends on meaningful global properties of the graph, whereas if p &gt; p^<strong>, it only depends on trivial local quantities and does not convey any useful information. We can explicitly compute the critical values: p^* = 1 + 1/(d-1) and p^</strong> = 1 + 1/(d-2) where d is the dimension of the underlying space (we believe that the fact that there is a small gap between p^* and p^<em>* is an artifact of our proofs. We also relate our findings to Laplacian regularization and suggest to use q-Laplacians as regularizers, where q satisfies 1/p^</em> + 1/q = 1.</p>