Transport and percolation theory in weighted networks

We study the distribution P(sigma) of the equivalent conductance sigma for Erdos-Renyi (ER) and scale-free (SF) weighted resistor networks with N nodes. Each link has conductance g equivalent to e(-ax), where x is a random number taken from a uniform distribution between 0 and 1 and the parameter a represents the strength of the disorder. We provide an iterative fast algorithm to obtain P(sigma) and compare it with the traditional algorithm of solving Kirchhoff equations. We find, both analytically and numerically, that P(sigma) for ER networks exhibits two regimes: (i) A low conductance regime for sigma < e(-apc), where p(c)=1/< k > is the critical percolation threshold of the network and < k > is the average degree of the network. In this regime P(sigma) is independent of N and follows the power law P(sigma)similar to sigma(-alpha), where alpha=1-< k >/a. (ii) A high conductance regime for sigma>e(-apc) in which we find that P(sigma) has strong N dependence and scales as P(sigma)similar to f(sigma,ap(c)/N-1/3). For SF networks with degree distribution P(k)similar to k(-lambda), k(min)<= k <= k(max), we find numerically also two regimes, similar to those found for ER networks.