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 lₒₚₜ in a disordered Erd {{o}} {o}{}s-R\'enyi (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 $N*(a)$ at which the transition occurs. For $N⪡N*(a)$ the scaling behavior of lₒₚₜ is in the strong disorder regime, with lₒₚₜ~N1∕3 for ER networks and for SF networks with λ4, and lₒₚₜ~N^(λ-3)∕(λ-1) for SF networks with 3<λ<4. For $N⪢N*(a)$ the scaling behavior is in the weak disorder regime, with lₒₚₜ~ln0.2em0exN for ER networks and SF networks with λ>3. 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 $N*(a)$ and a. We find that N*(a)~a³ for ER networks and for SF networks with λ4, and N*(a)~a^(λ-1)∕(λ-3) for SF networks with 3<λ<4.