  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.
