Optimal paths in disordered complex networks

We study the optimal distance in networks, [Formula presented], 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 [Formula presented] in both Erdős-Rényi (ER) and Watts-Strogatz (WS) networks. For scale-free (SF) networks, with degree distribution [Formula presented], we find that [Formula presented] scales as [Formula presented] for [Formula presented] and as [Formula presented] for [Formula presented]. Thus, for these networks, the small-world nature is destroyed. For [Formula presented], our numerical results suggest that [Formula presented] scales as [Formula presented]. We also find numerically that for weak disorder [Formula presented] for both the ER and WS models as well as for SF networks.