Phase transition in the family of p-resistances

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∗∗ such that if p < p∗, then the p-resistance depends on meaningful global properties of the graph, whereas 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.