Anomalous Transport in Scale-Free Networks

To study transport properties of scale-free and Erdős-Rényi networks, we analyze the conductance $G$ between two arbitrarily chosen nodes of random scale-free networks with degree distribution $P(k)\sim k^{-\lambda }$ in which all links have unit resistance. We predict a broad range of values of $G$, with a power-law tail distribution ${\Phi }_{\mathrm{SF}}(G)\sim G^{-g_G}$, where $g_G=2\lambda -1$, and confirm our predictions by simulations. The power-law tail in ${\Phi }_{\mathrm{SF}}(G)$ leads to large values of $G$, signaling better transport in scale-free networks compared to Erdős-Rényi 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 Erdős-Rényi networks are well approximated by $c{k}_A{k}_B/(k_A+k_B)$ for any pair of nodes $A$ and $B$ with degrees $k_A$ and $k_B$, where $c$ emerges as the main parameter characterizing network transport.