Short paths for first passage percolation on the complete graph

Maren Eckhoff, Jesse Goodman, Remco van der Hofstad, Francesca R. Nardi

Published 2012-11-19Version 1

We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct vertices in the weak disorder regime. We establish novel and simple first and second moment methods using path counting to derive first order asymptotics for the considered quantities. Our results are stated in terms of a sequence of parameters (s_n) that quantifies the extreme-value behaviour of the edge weights, and that describes different universality classes for first passage percolation on the complete graph. These classes contain both n-independent and n-dependent edge weight distributions. The method is most effective for the universality class containing the edge weights E^{s_n}, where E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n -> 0. We discuss two types of examples from this class in detail. In addition, the class where s_n log n stays finite is studied. This article is a contribution to the program initiated in \cite{BhaHof12}.