Anomalous Transport in Scale-Free Networks

To study transport properties of scale-free and Erd {{o}} {o}{}s-R\'enyi networks, we analyze the conductance G between two arbitrarily chosen nodes of random scale-free networks with degree distribution P(k)~k^-λ in which all links have unit resistance. We predict a broad range of values of G, with a power-law tail distribution ΦSF(G)~G^-gG, where gG=2λ-1, and confirm our predictions by simulations. The power-law tail in ΦSF(G) leads to large values of G, signaling better transport in scale-free networks compared to Erd {{o}} {o}{}s-R\'enyi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical ``transport backbone'' picture we show that the conductances of scale-free and Erd {{o}} {o}{}s-R\'enyi networks are well approximated by ckAkB/(kA+kB) for any pair of nodes A and B with degrees kA and kB, where c emerges as the main parameter characterizing network transport.