First passage percolation on the Erdős-Rényi random graph

In this paper we explore first passage percolation (FPP) on the Erdős-Rényi 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 $np_n\to λ>1,$ 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 $λ/(λ-1)\log{n}$. Furthermore, we prove that the minimal weight centered by $\log{n}/(λ-1)$ 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, $(H_n-β_n\log{n})/\sqrt{\log{n}}$, where $β_n=λ_n/(λ_n-1)$, has a limiting standard normal distribution. The constant $β_n$ can be replaced by 1 precisely when $λ_n\gg \sqrt{\log{n}}$, a case that has appeared in the literature (under stronger conditions on $λ_n$). 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 Erdős-Rényi random graph and thinned continuous-time branching processes.

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