Transport and Percolation Theory in Weighted Networks

Abstract

We study the distribution $P(\sigma )$ of the equivalent conductance $\sigma$ 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 $P(\sigma )$ 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 $\sigma <e^{-a{p}_c}$ 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 $α=1-\av{k}/a$. (ii) A high conductance regime for $\sigma >e^{-a{p}_c}$ in which we find that $P(\sigma )$ has strong $N$ dependence and scales as $P(\sigma )\sim f(\sigma ,a{p}_c/N^{1/3})$. 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.

arXiv comment4 pages, 8 figures

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• The architecture of complex weighted networks
• Optimal Paths in Disordered Complex Networks
• Anomalous Transport in Complex Networks
• Current Flow in Random Resistor Networks: The Role of Percolation in Weak and Strong Disorder
• Universal behavior of optimal paths in weighted networks with general disorder
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Cited by in corpus (8)

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• Transport in networks with multiple sources and sinks
• Optimizing transport efficiency on scale-free networks through assortative or dissortative topology
• Transport properties of nonhomogeneous segregated composites
• Condensation phenomena of conserved-mass aggregation model on weighted complex networks