Optimal Paths in Disordered Complex Networks

We study the optimal distance $\ell_{opt}$ in small-world and scale-free (SF) networks in the presence of disorder implemented by assigning random weights to the links or nodes. We define $\ell_{opt}$ as the length of the path that minimizes the sum of the weights along the path. For strong disorder, where the maximal weight along the path dominates the sum, we find that $\ell_{opt}\sim N^{1/3}$ in both Erdhos-Renyi (ER) and Watts-Strogatz (WS) networks. For SF networks with degree distribution $P(k) \sim k^{-\lambda}$, we find that $\ell_{opt}$ scales as $N^{(\lambda -3)/(\lambda - 1)}$ for $3<\lambda<4$ and as $N^{1/3}$ for $\lambda\geq 4$. Thus, for these networks, the small-world nature is destroyed. For $2 < \lambda < 3$, our numerical results suggest that $\ell_{opt}$ scales as $\ln^{\lambda-1}N$. We also find numerically that for weak disorder $\ell_{opt}\sim\ln N$ for both the ER and WS models as well as for SF networks.