Length of optimal path in random networks with strong disorder

We study the optimal distance ℓ_opt in random networks in the presence of disorder implemented by assigning random weights to the links. The optimal distance between two nodes is the length of the path for which the sum of weights along the path ("cost") is a minimum. We study the case of strong disorder for which the distribution of weights is so broad that its sum along any path is dominated by the largest link weight in the path. We find that in random graphs, ℓ_opt scales as N^1/3, where N is the number of nodes in the network. Thus, ℓ_opt increases dramatically compared to the known small-world result for the minimum distance ℓ_min, which scales as log N. We also study, theoretically and by simulations, scale-free networks characterized by a power law distribution for the number of links, P(k) ∼ k^-λ, and find that ℓ_opt scales as N^1/3 for λ > 4 and as N^{(λ-3)/(λ-1)} for 3 < λ < 4. For 2 < λ < 3, our numerical results suggest that ℓ_opt scales logarithmically with N.