The Annals of Applied Probability

Weak disorder asymptotics in the stochastic mean-field model of distance

Shankar Bhamidi and Remco van der Hofstad

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Abstract

In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is Θ(log n) and satisfies a central limit theorem with asymptotic means and variances of order Θ(log n), with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution Es for some parameter s > 0, where E is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are s log n and s2 log n, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.

Article information

Source
Ann. Appl. Probab., Volume 22, Number 1 (2012), 29-69.

Dates
First available in Project Euclid: 7 February 2012

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1328623695

Digital Object Identifier
doi:10.1214/10-AAP753

Mathematical Reviews number (MathSciNet)
MR2932542

Zentralblatt MATH identifier
1248.60012

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

Keywords
Flows random graphs first passage percolation hopcount central limit theorem weak disorder continuous-time branching process stable-age distribution theory mean-field model of distance Cox point processes

Citation

Bhamidi, Shankar; van der Hofstad, Remco. Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Probab. 22 (2012), no. 1, 29--69. doi:10.1214/10-AAP753. https://projecteuclid.org/euclid.aoap/1328623695


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The Annals of Applied Probability

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