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 ${\mathcal{l}}_{\mathrm{opt}}$ in a disordered Erdős-Rényi (ER) random network and scale-free (SF) network. Each link $i$ is associated with a weight ${\tau }_i\equiv \exp \left(a{r}_i\right)$, 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 find that for any finite $a$, there is a crossover network size $N*\left(a\right)$ at which the transition occurs. For $N⪡N*\left(a\right)$ the scaling behavior of ${\mathcal{l}}_{\mathrm{opt}}$ is in the strong disorder regime, with ${\mathcal{l}}_{\mathrm{opt}}\sim N^{1∕3}$ for ER networks and for SF networks with $\lambda ⩾4$, and ${\mathcal{l}}_{\mathrm{opt}}\sim N^{(\lambda -3)∕(\lambda -1)}$ for SF networks with $3<\lambda <4$. For $N⪢N*\left(a\right)$ the scaling behavior is in the weak disorder regime, with ${\mathcal{l}}_{\mathrm{opt}}\sim \ln N$ for ER networks and SF networks with $\lambda >3$. 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 $N*\left(a\right)$ and $a$. We find that $N*\left(a\right)\sim a^3$ for ER networks and for SF networks with $\lambda ⩾4$, and $N*\left(a\right)\sim a^{(\lambda -1)∕(\lambda -3)}$ for SF networks with $3<\lambda <4$.