Anomalous behavior of trapping on a fractal scale-free network

Abstract

It is known that the heterogeneity of scale-free networks helps enhancing the efficiency of trapping processes performed on them. In this paper, we show that transport efficiency is much lower in a fractal scale-free network than in non-fractal networks. To this end, we examine a simple random walk with a fixed trap at a given position on a fractal scale-free network. We calculate analytically the mean first-passage time (MFPT) as a measure of the efficiency for the trapping process, and obtain a closed-form expression for MFPT, which agrees with direct numerical calculations. We find that, in the limit of a large network order $V$, the MFPT $<T>$ behaves superlinearly as $<T>\sim V^{3/2}$ with an exponent 3/2 much larger than 1, which is in sharp contrast to the scaling $<T>\sim V^{\theta }$ with $\theta \le 1$, previously obtained for non-fractal scale-free networks. Our results indicate that the degree distribution of scale-free networks is not sufficient to characterize trapping processes taking place on them. Since various real-world networks are simultaneously scale-free and fractal, our results may shed light on the understanding of trapping processes running on real-life systems.

6 pages, 5 figures; Definitive version accepted for publication in EPL (Europhysics Letters)

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• Scaling laws for diffusion on (trans)fractal scale-free networks
• Exact results for the first-passage properties in a class of fractal networks
• A general model of hierarchical fractal scale-free networks
• Effects of reciprocity on random walks in weighted networks