First passage percolation on the Erd\H{o}s-R\'enyi random graph

TL;DR

This paper analyzes first passage percolation on Erdős-Rényi graphs, revealing asymptotic behaviors of minimal path weights and hopcounts in both sparse and dense regimes, and demonstrating universal properties of network distances under random edge weights.

Contribution

The paper provides refined asymptotics and central limit theorems for FPP on Erdős-Rényi graphs, extending understanding of network geometry with random weights in different regimes.

Findings

Central limit theorem for hopcount in sparse regime

Convergence in distribution of minimal weight after centering

Hopcount scales as log(n) with universal behavior in dense regime

Abstract

In this paper we explore first passage percolation (FPP) on the Erd\H{o}s-R\'enyi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to \lambda>1,$ we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to $\lambda/(\lambda-1)\log{n}$. Furthermore, we prove that the minimal weight centered by $\log{n}/(\lambda-1)$ converges in distribution. We also investigate the dense regime, where $np_n \to \infty$. We find that although the base graph is a {\it ultra small} (meaning that graph distances between uniformly chosen vertices are $o(\log{n})$), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount $H_n$ satisfies the universality property that whatever be the value of $p_n$, \ $H_n/\log{n}\to 1$ in probability and, more precisely, $(H_n-\beta_n\log{n})/\sqrt{\log{n}}$, where $\beta_n=\lambda_n/(\lambda_n-1)$, has a limiting standard normal distribution. The constant $\beta_n$ can be replaced by 1 precisely when $\lambda_n\gg \sqrt{\log{n}}$, a case that has appeared in the literature (under stronger conditions on $\lambda_n$). We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated by the authors. Compared to the setting on the configuration model studied in \cite{BHHS08}, the proofs presented here are much simpler due to a direct relation between FPP on the Erd\H{o}s-R\'enyi random graph and thinned continuous-time branching processes.