WEAK DISORDER ASYMPTOTICS IN THE STOCHASTIC MEAN-FIELD MODEL OF DISTANCE

• 成果类型: Article
• 署名作者: Bhamidi, Shankar; van der Hofstad, Remco
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Eindhoven University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP753
• 发表日期: 2012
• 页码: 29-69
• 关键词: 1st passage percolation; Random graphs; variance degrees; random networks; TREE

摘要

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 Theta(log n) and satisfies a central limit theorem with asymptotic means and variances of order Theta(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 E-s 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 s(2) 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.