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

TL;DR

This paper introduces a mathematically tractable model for weak disorder in the stochastic mean-field model of distance, showing that the hopcount scales logarithmically with network size and follows a central limit theorem.

Contribution

It provides the first explicit analysis of weak disorder in this model, demonstrating no finite-temperature transition and deriving asymptotic distributions for path length and weight.

Findings

Hopcount scales as Θ(log n) for all finite temperatures.

Hopcount satisfies a CLT with mean and variance proportional to log n.

Asymptotic distribution of minimal path weight relates to extreme value distributions.

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 $\Theta(\log{n})$ 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.