Transport and Percolation Theory in Weighted Networks

Abstract

We study the distribution $P(σ)$ of the equivalent conductance $σ$ for Erdős-Rényi (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 $P(σ)$ and compare it with the traditional algorithm of solving Kirchhoff equations. We find, both analytically and numerically, that $P(σ)$ for ER networks exhibits two regimes. (i) A low conductance regime for $σ< e^{-ap_c}$ 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 $P(σ)$ is independent of $N$ and follows the power law $P(σ) \sim σ^{-α}$, where $α=1-\av{k}/a$. (ii) A high conductance regime for $σ>e^{-ap_c}$ in which we find that $P(σ)$ has strong $N$ dependence and scales as $P(σ) \sim f(σ,ap_c/N^{1/3})$. For SF networks with degree distribution $P(k)\sim k^{-λ}$, $k_{min} \le k \le k_{max}$, we find numerically also two regimes, similar to those found for ER networks.