First passage percolation on Erd\H{o}s-R\'{e}nyi graphs with general weights

TL;DR

This paper studies the minimal-weight paths in Erdős-Rényi graphs with general edge weights, revealing their asymptotic distribution and the joint behavior of hopcounts and total weights, extending previous models with continuous weights.

Contribution

It generalizes earlier models by analyzing paths with arbitrary weight distributions, deriving the limiting distribution of minimal weights and hopcounts, and describing their joint behavior via Cox processes.

Findings

Joint distribution of hopcounts and total weights converges to a Cox process.

Derived the limiting distribution for minimal total weight and hopcounts.

Different behaviors observed for arithmetic and non-arithmetic weight distributions.

Abstract

We consider first passage percolation on the Erd\H{o}s--R\'{e}nyi 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.