Optimal Paths in Disordered Complex Networks

We study the optimal distance in networks, ${\ell }_{\mathrm{o}\mathrm{p}\mathrm{t}}$, defined as the length of the path minimizing the total weight, in the presence of disorder. Disorder is introduced by assigning random weights to the links or nodes. For strong disorder, where the maximal weight along the path dominates the sum, we find that ${\ell }_{\mathrm{o}\mathrm{p}\mathrm{t}}\sim N^{1/3}$ in both Erdős-Rényi (ER) and Watts-Strogatz (WS) networks. For scale-free (SF) networks, with degree distribution $P(k)\sim k^{-\lambda }$, we find that ${\ell }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ scales as $N^{(\lambda -3)/(\lambda -1)}$ for $3<\lambda <4$ and as $N^{1/3}$ for $\lambda \ge 4$. Thus, for these networks, the small-world nature is destroyed. For $2<\lambda <3$, our numerical results suggest that ${\ell }_{\mathrm{o}\mathrm{p}\mathrm{t}}$ scales as ${ln}^{\lambda -1}N$. We also find numerically that for weak disorder ${\ell }_{\mathrm{o}\mathrm{p}\mathrm{t}}\sim \ln N$ for both the ER and WS models as well as for SF networks.