Transport and Percolation Theory in Weighted Networks

We study the distribution $P(\sigma)$ of the equivalent conductance $\sigma$ for Erd\H{o}s-R\'enyi (ER) and scale-free (SF) weighted resistor networks with $N$ nodes. Each link has conductance $g\equiv e^{-ax}$, 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 $\sigma < e^{-ap_c}$ where $p_c=1/\av{k}$ is the critical percolation threshold of the network and $\av{k}$ is average degree of the network. In this regime $P(\sigma)$ is independent of $N$ and follows the power law $P(\sigma) \sim \sigma^{-\alpha}$, where $\alpha=1-\av{k}/a$. (ii) A high conductance regime for $\sigma >e^{-ap_c}$ in which we find that $P(\sigma)$ has strong $N$ dependence and scales as $P(\sigma) \sim f(\sigma,ap_c/N^{1/3})$. For SF networks with degree distribution $P(k)\sim k^{-\lambda}$, $k_{min} \le k \le k_{max}$, we find numerically also two regimes, similar to those found for ER networks.