Anomalous conductance and diffusion in complex networks

Abstract

We study transport properties such as conductance and diffusion of complex networks such as scale-free and Erdős-Rényi networks. We consider the equivalent conductance G between two arbitrarily chosen nodes of random scale-free networks with degree distribution P(k) ~ k−⋋ and Erdős-Rényi networks in which each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G (or the related diffusion constant D), with a power-law tail distribution ɸSF(G) ~ G−gG, where gG = 2⋋ − 1. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in ɸSF(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdős-R´nyi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical “transport backbone” picture we suggest that the conductances of scale-free and Erdős-Rényi networks can be approximated by ckAkB/(kA + kB) for any pair of nodes A and B with degrees kA and kB. Thus, a single parameter c characterizes transport on both scale-free and Erdős-Rényi networks.