Anomalous electrical and frictionless flow conductance in complex networks

Abstract

We study transport properties such as electrical and frictionless flow conductance on scale-free and Erdős–Rényi 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 Φ SF ( G ) ∼ G − g G , where g G = 2 λ − 1 , where λ 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 Φ SF ( G ) leads to large values of G , thereby significantly improving the 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 suggest that the conductances of scale-free and Erdős–Rényi networks can be 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 . Thus, a single quantity c , which depends on the average degree k ¯ of the network, characterizes transport on both scale-free and Erdős–Rényi 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.