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We review results on the scaling of the optimal path length in randomnetworks with weighted links or nodes. In strong disorder we find that thelength of the optimal path increases dramatically compared to the known smallworld result for the minimum distance. For Erd\H{o}s-R\'enyi (ER) and scalefree networks (SF), with parameter $\lambda$ ($\lambda >3$), we find that thesmall-world nature is destroyed. We also find numerically that for weakdisorder the length of the optimal path scales logaritmically with the size ofthe networks studied. We also review the transition between the strong and weakdisorder regimes in the scaling properties of the length of the optimal pathfor ER and SF networks and for a general distribution of weights, and suggestthat for any distribution of weigths, the distribution of optimal path lengthshas a universal form which is controlled by the scaling parameter$Z=\ell_{\infty}/A$ where $A$ plays the role of the disorder strength, and$\ell_{\infty}$ is the length of the optimal path in strong disorder. Therelation for $A$ is derived analytically and supported by numericalsimulations. We then study the minimum spanning tree (MST) and show that it iscomposed of percolation clusters, which we regard as "super-nodes", connectedby a scale-free tree. We furthermore show that the MST can be partitioned intotwo distinct components. One component the {\it superhighways}, for which thenodes with high centrality dominate, corresponds to the largest cluster at thepercolation threshold which is a subset of the MST. In the other component,{\it roads}, low centrality nodes dominate. We demonstrate the significanceidentifying the superhighways by showing that one can improve significantly theglobal transport by improving a very small fraction of the network.Comment: review, accepted at IJB

OPTIMAL PATH AND MINIMAL SPANNING TREES IN RANDOM WEIGHTED NETWORKS