The Annals of Probability

Universality for first passage percolation on sparse random graphs

Shankar Bhamidi, Remco van der Hofstad, and Gerard Hooghiemstra

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Abstract

We consider first passage percolation on the configuration model with $n$ vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform $X^{2}\log{X}$-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount.

Writing $L_{n}$ for the weight of the optimal path, we show that $L_{n}-(\log{n})/\alpha_{n}$ converges to a limiting random variable, for some sequence $\alpha_{n}$. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order $\log{n}$. The sequence $\alpha_{n}$ and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of $L_{n}-(\log{n})/\alpha_{n}$ equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. So far, for sparse random graph models, such results have only been shown for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem.

The proofs in the paper rely on a refined coupling between shortest path trees and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination.

Article information

Source
Ann. Probab., Volume 45, Number 4 (2017), 2568-2630.

Dates
Received: May 2014
Revised: September 2015
First available in Project Euclid: 11 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1502438435

Digital Object Identifier
doi:10.1214/16-AOP1120

Mathematical Reviews number (MathSciNet)
MR3693970

Zentralblatt MATH identifier
1376.60018

Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

Keywords
Central limit theorem continuous-time branching processes extreme value theory first passage percolation hopcount Malthusian rate of growth point process convergence Poisson point process stable-age distribution random graphs

Citation

Bhamidi, Shankar; van der Hofstad, Remco; Hooghiemstra, Gerard. Universality for first passage percolation on sparse random graphs. Ann. Probab. 45 (2017), no. 4, 2568--2630. doi:10.1214/16-AOP1120. https://projecteuclid.org/euclid.aop/1502438435


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