Universal behavior of optimal paths in weighted networks with general disorder

We study the statistics of the optimal path in both random and scale-free networks, where weights w are taken from a general distribution P(w). We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter (S≡AL-1/ Î1/2 for d-dimensional lattices, and S≡AN-1/3 for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here Î1/2 is the percolation connectivity exponent, and A depends on the percolation threshold and P(w). We show that for a uniform P(w), Poisson or Gaussian, the crossover from weak to strong does not occur, and only weak disorder exists.