Anomalous electrical and frictionless flow conductance in complex networks

TL;DR

This paper investigates electrical and frictionless flow conductance in complex networks, revealing power-law distributions in scale-free networks that enhance transport efficiency compared to Erdős-Rényi networks, supported by theoretical analysis, simulations, and real-world data.

Contribution

It provides a theoretical framework and simulation validation for conductance distributions in scale-free and Erdős-Rényi networks, introducing a simple approximation model for transport properties.

Findings

Scale-free networks exhibit a power-law tail in conductance distribution.

Erdős-Rényi networks show exponential decay in conductance distribution.

The transport backbone model effectively approximates conductance between node pairs.

Abstract

We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdos-Renyi networks. We consider the conductance G between two arbitrarily chosen nodes where each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G, with a power-law tail distribution \Phi_{SF}(G) \sim G^{g_G}, where g_G = 2\lambda - 1, where \lambda is the decay exponent for the scale-free network degree distribution. 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 \Phi_{SF}(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdos-Renyi 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 Erdos-Renyi networks can be 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 quantity c, which depends on the average degree <k> of the network, characterizes transport on both scale-free and Erdos-Renyi networks. We determine that c tends to 1 for increasing <k>, and it is larger for scale-free networks. We compare the electrical results with a model for frictionless transport, where conductance is defined as the number of link-independent paths between A and B, and find that a similar picture holds. The effects of distance on the value of conductance are considered for both models, and some differences emerge. Finally, we use a recent data set for the AS (autonomous system) level of the Internet and confirm that our results are valid in this real-world example.