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Anomalous Transport in Complex Networks

To study transport properties of complex networks, we analyze the equivalent conductance $G$ between two arbitrarily chosen nodes of random scale-free networks with degree distribution $P(k)\sim k^{-λ}$ in which each link has the same unit resistance. We predict a broad range of values of $G$, with a power-law tail distribution $Φ_{\rm SF}(G)\sim G^{-g_G}$, where $g_G=2λ-1$, and confirm our predictions by simulations. The power-law tail in $Φ_{\rm SF}(G)$ leads to large values of $G$, thereby significantly improving the transport in scale-free networks, compared to Erdős-Rényi random graphs where the tail of the conductivity distribution decays exponentially. Based on a simple physical ``transport backbone'' picture we show that the conductances are well approximated by $ck_Ak_B/(k_A+k_B)$ for any pair of nodes $A$ and $B$ with degrees $k_A$ and $k_B$. Thus, a single parameter $c$ characterizes transport on scale-free networks.