First passage percolation on Erdős-Rényi graphs with general weights

Abstract

We consider first passage percolation on the Erdős--Rényi graph with $n$ vertices in which each pair of distinct vertices is connected independently by an edge with probability $\lambda /n$ for some $\lambda >1$. The edges of the graph are given non-negative i.i.d. weights with a non-degenerate distribution such that the probability of zero is not too large. We consider the paths with small total weight between two distinct typical vertices and analyse the joint behaviour of the numbers of edges on such paths, the so-called hopcounts, and the total weights of these paths. For $n\to \infty$, we show that, after a suitable transformation, the pairs of hopcounts and total weights of these paths converge in distribution to a Cox process, i.e., a Poisson process with a random intensity measure. The random intensity measure is controlled by two independent random variables, whose distribution is the solution of a distributional fixed point equation and is related to branching processes. For non-arithmetic and arithmetic edge-weight distributions we observe different behaviour. In particular, we derive the limiting distribution for the minimal total weight and the corresponding hopcount(s). Our results generalise earlier work of Bhamidi, van der Hofstad and Hooghiemstra, who assume that edge weights have an absolutely continuous distribution. The main tool we employ is the method of moments.

arXiv comment30 pages, 3 figures

References in corpus (9)

• First passage percolation on random graphs with finite mean degrees
• Weak disorder asymptotics in the stochastic mean-field model of distance
• First Passage Percolation on Inhomogeneous Random Graphs
• Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds
• The front of the epidemic spread and first passage percolation
• When is a scale-free graph ultra-small?
• Degree distribution of shortest path trees and bias of network sampling algorithms
• First passage percolation on the Newman-Watts small world model
• Limiting shape for First-Passage Percolation models on Random Geometric Graphs