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 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 defined as AL(-1/v) for d-dimensional lattices, and S defined as AN(-1/3) for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here v 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.