Anomalous conductance and diffusion in complex networks
(2005)Abstract
We study transport properties such as conductance and diffusion of
complex networks such as scale-free and Erdos-Renyi networks. We
consider the equivalent conoductance G between two arbitrarily chosen
nodes of random scale-free networks with degree distribution P (k) ∼
k−λ and Erd˝ s-R´ nyi networks in which each link has the o e 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 ΦSF (G) ∼ G−gG , where
gG = 2λ − 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 ΦSF (G) leads to large values of
G, thereby significantly improving the transport in scale-free
networks, compared to Erd˝ s-R´ nyi o e 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 Erd˝ s-R´ nyi networks can be approximated by ckA kB
/(kA + kB ) o e 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 Erd˝ s-R´ nyi networks. o e