Phase transition in the family of p-resistances
published: Sept. 6, 2012, recorded: December 2011, views: 3040
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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.
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