Optimal path in random networks with disorder: A mini review

We review the analysis of the length of the optimal path ℓopt 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 ℓopt increases dramatically compared to the known small-world result for the minimum distance ℓmin: for Erdős–Rényi (ER) networks ℓopt∼N1/3, while for scale-free (SF) networks, with degree distribution P(k)∼k-λ, we find that ℓopt scales as N(λ-3)/(λ-1) for 3<λ<4 and as N1/3 for λ⩾4. Thus, for these networks, the small-world nature is destroyed. For 2<λ<3, our numerical results suggest that ℓopt scales as lnλ-1N. We also find numerically that for weak disorder ℓopt∼lnN 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 ℓopt in ER and SF networks.