First Passage Percolation on the Erdős–Rényi Random Graph

Abstract

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 → 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 → ∞. We find that although the base graph is a 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 → 1 in probability and, more precisely, (H n −β n log n)/ √ 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 ≫ √ log n, a case that has appeared in the literature (under stronger conditions on λ n ) in [2,12]. We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated in [2] and [3]. Compared to the setting on the configuration model studied in [3], 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.

First Passage Percolation on the Newman–Watts Small World Model

“…Note that Theorems 1.1 and 1.2 are analogous to similar results in the sequence of papers [11][12][13]24], while Theorem 1.3 is analogous to the results in [9,14]. The intuitive message of Theorem 1.3 is that a linear proportion of infected vertices can be observed after a time that is proportional to the logarithm of the size of the population.…”

“…[11][12][13]23,24]) van der Hofstad et al investigated FPP on random graphs. Their aim was to determine universality classes for the shortest path metric for weighted random graphs without 'extrinsic' geometry (e.g.…”

Competing first passage percolation on random regular graphs

“…Often the distribution is assumed to be exponential and then the ball of a radius t (from a fixed vertex) is a Markov set process R, in which new vertices are occupied at a rate proportional to the number of their neighbors already in R(t). Apart from the classical shape problem on infinite transitive graphs (see [14]), recently there was substantial interest in estimating diameter, typical distance, flooding times and related quantities for the process on large finite (and possibly random) graphs [33,44,4,6,7,5].…”

The behavior of epidemics under bounded susceptibility

“…Most of our arguments involve standard Markov chain machinery, along with careful use of measure concentration. Some of the ideas we use are of fairly recent origin, in particular, the sharp expressions for extinction time in SIS models via embedding in an ergodic Markov chain (Lemma 1, which we adapt from Lemma 8 in [10]), and characterization of the SI spreading time in clique-like graphs, which is similar to results for first-passage percola-tion in random graphs [4]. However, unlike [4], our results (Theorems 1,3, 4 and 6) do not focus on a particular graph or generative model, but rather, relate the epidemic behavior to structural properties of the network.…”

“…The SI or 'Susceptible-Infected' dynamics [1,2,3,4] is the most common model for one-way dissemination. Nodes exist in 2 states -'susceptible' (S) and 'infected' (I).…”