Effect of disorder strength on optimal paths in complex networks

We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path [Formula presented] in a disordered Erdős-Rényi (ER) random network and scale-free (SF) network. Each link [Formula presented] is associated with a weight [Formula presented], where [Formula presented] is a random number taken from a uniform distribution between 0 and 1 and the parameter [Formula presented] controls the strength of the disorder. We find that for any finite [Formula presented], there is a crossover network size [Formula presented] at which the transition occurs. For [Formula presented] the scaling behavior of [Formula presented] is in the strong disorder regime, with [Formula presented] for ER networks and for SF networks with [Formula presented], and [Formula presented] for SF networks with [Formula presented]. For [Formula presented] the scaling behavior is in the weak disorder regime, with [Formula presented] for ER networks and SF networks with [Formula presented]. In order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder. We propose a scaling ansatz for this measure and demonstrate its validity. We proceed to derive the scaling relation between [Formula presented] and [Formula presented]. We find that [Formula presented] for ER networks and for SF networks with [Formula presented], and [Formula presented] for SF networks with [Formula presented].