First passage percolation on random graphs with finite mean degrees

Abstract

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount. We analyze the configuration model with degree power-law exponent $\tau>2$, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent $\tau-1>1$, or has even thinner tails ($\tau=\infty$). In this model, the degrees have a finite first moment, while the variance is finite for $\tau>3$, but infinite for $\tau\in(2,3)$. We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to $\alpha\log{n}$, where $\alpha\in(0,1)$ for $\tau\in(2,3)$, while $\alpha>1$ for $\tau>3$. Here $n$ denotes the size of the graph. For $\tau\in (2,3)$, it is known that the graph distance between two randomly chosen connected vertices is proportional to $\log \log{n}$ [Electron. J. Probab. 12 (2007) 703--766], that is, distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [J. Math. Phys. 49 (2008) 125218] of showing that $\log{n}$ is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees ($\tau\in[1,2)$) is studied in [Extreme value theory, Poisson--Dirichlet distributions and first passage percolation on random networks (2009) Preprint] where it is proved that the hopcount remains uniformly bounded and converges in distribution.Comment: Published in at http://dx.doi.org/10.1214/09-AAP666 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

First Passage Percolation on the Erdős–Rényi Random 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. …”

“…(a) The degree sequence in [3] was assumed to satisfy F (x) = 0, x < 2, so that all degrees are at least 2, with probability 1. As said this implies that uniformly chosen vertices are whp connected.…”

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].…”

When is a Scale-Free Graph Ultra-Small?

“…So, before stating the result, we have to do a small excursion into defining these objects. In particular, Lemma 2.2 and Corollary 2.3 below, based on [7,Proposition 4.7], states that under Assumption 1.3, whp, the number of vertices and their forward degrees in an exploration of the neighborhood of v r , v b can be coupled to two independent branching processes (that are embedded in the graph disjointly, and have offspring distribution F for the second and further generations, and with offspring distribution given by F for the first generation), as long as the total number of vertices of the explored clusters does not exceed n for some small > 0. Let us do the exploration in a breadth-first-search manner, and then the exploration at time t contains all the vertices with at most distance t away from C (r) 0 := {v r } and C (b) 0 := {v b }.…”

“…Let Z k denote the k-th generation of a branching process with offspring distribution given by a distribution function G, that can be written in the form 7 as in (1.7) and (1.5) for some τ ∈ (2, 3) and some x 0 > 0 for all x ≥ x 0 . Then (τ − 2) k log Z k ∨ 1 converges almost surely to a random variable Y .…”