Short paths for first passage percolation on the complete graph

TL;DR

This paper analyzes the minimal weight paths in first passage percolation on complete graphs with i.i.d. edge weights, establishing asymptotics and universality classes based on extreme-value behavior of the weights.

Contribution

It introduces novel first and second moment methods for analyzing first passage percolation, identifying different universality classes depending on edge weight distributions.

Findings

Asymptotic behavior of minimal path weights and lengths is characterized.

Universality classes depend on the extreme-value behavior of edge weights.

Results include both n-independent and n-dependent edge weight distributions.

Abstract

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}.