Phase transition in the family of p-resistances
Morteza Alamgir, Ulrike von Luxburg · MPG.PuRe (Max Planck Society) · 2011
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 =2it coincides with the standard resistance distance, and for p!1it 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 ⇤ ⇤ such that if p p ⇤ ⇤ , 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 ⇤ ⇤ = 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 ⇤ ⇤ 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 ⇤ +1/q =1. 1