Optimal path in random networks with disorder: A mini review

We review the analysis of the length of the optimal path l opt in random networks with disorder (i.e., random weights on the links). In the case of strong disorder, in which the maximal weight along the path dominates the sum, we find that l opt increases dramatically compared to the known small-world result for the minimum distance l min : for Erdős–Renyi (ER) networks l opt ∼ N 1 / 3 , while for scale-free (SF) networks, with degree distribution P ( k ) ∼ k - λ , we find that l opt scales as N ( λ - 3 ) / ( λ - 1 ) for 3 λ 4 and as N 1 / 3 for λ ⩾ 4 . Thus, for these networks, the small-world nature is destroyed. For 2 λ 3 , our numerical results suggest that l opt scales as ln λ - 1 N . We also find numerically that for weak disorder l opt ∼ ln N for ER models as well as for SF networks. We also study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path l opt in ER and SF networks.