Anomalous electrical and frictionless flow conductance in complex networks

[摘要] 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) similar to G(-gG), where gG = 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(A)k(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) over bar of the network, characterizes transport on both scale-free and Erdes-Renyi networks. We determine that c tends to 1 for increasing (k) over bar, 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. Published by Elsevier B.V.

[发布日期] 2006-12-01

[关键词] complex networks;transport;diffusion;conductance;scaling