Anomalous Transport in Complex Networks

TL;DR

This paper investigates the transport properties of scale-free networks, revealing a power-law distribution of conductance that enhances transport efficiency compared to random graphs, and introduces a simple model to approximate conductance between node pairs.

Contribution

It predicts the conductance distribution in scale-free networks, confirms it through simulations, and proposes a simple physical model to approximate conductance between nodes.

Findings

Conductance distribution follows a power-law tail with exponent 2λ-1.

Scale-free networks exhibit significantly improved transport due to large conductance values.

A simple formula c k_A k_B / (k_A + k_B) effectively approximates node-to-node conductance.

Abstract

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 $P(k)\sim k^{-\lambda}$ in which each link has the same unit resistance. We predict a broad range of values of $G$, with a power-law tail distribution $\Phi_{\rm SF}(G)\sim G^{-g_G}$, where $g_G=2\lambda -1$, and confirm our predictions by simulations. The power-law tail in $\Phi_{\rm SF}(G)$ leads to large values of $G$, thereby significantly improving the transport in scale-free networks, compared to Erd\H{o}s-R\'{e}nyi 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.