  We study the distribution of optimal path lengths in random graphs with
random weights associated with each link (``disorder''). With each link $i$ we
associate a weight $\tau_i = \exp(ar_i)$ where $r_i$ is a random number taken
from a uniform distribution between 0 and 1, and the parameter $a$ controls the
strength of the disorder. We suggest, in analogy with the average length of the
optimal path, that the distribution of optimal path lengths has a universal
form which is controlled by the expression
$\frac{1}{p_c}\frac{\ell_{\infty}}{a}$, where $\ell_{\infty}$ is the optimal
path length in strong disorder ($a \to \infty$) and $p_c$ is the percolation
threshold. This relation is supported by numerical simulations for
Erd\H{o}s-R\'enyi and scale-free graphs. We explain this phenomenon by showing
explicitly the transition between strong disorder and weak disorder at
different length scales in a single network.
