Transition between strong and weak disorder regimes for the optimal path

We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path ℓopt in a disordered Erdős–Rényi (ER) random network and scale-free (SF) network. Each link i is associated with a weight τi≡exp(ari), where ri is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder. We find that for any finite a, there is a crossover network size N*(a) such that for N⪡N*(a) the scaling behavior of ℓopt is in the strong disorder regime, while for N⪢N*(a) the scaling behavior is in the weak disorder regime. We derive the scaling relation between N*(a) and a with the help of simulations and also present an analytic derivation of the relation.