Volume 88, Number 1, October 2009
Article Number 10001
Number of page(s) 6
Section General
Published online 16 October 2009
EPL, 88 (2009) 10001
DOI: 10.1209/0295-5075/88/10001

Anomalous behavior of trapping on a fractal scale-free network

Zhongzhi Zhang1, 2, Wenlei Xie1, 2, Shuigeng Zhou1, 2, Shuyang Gao1, 2 and Jihong Guan3

1   School of Computer Science, Fudan University - Shanghai 200433, China
2   Shanghai Key Lab of Intelligent Information Processing, Fudan University - Shanghai 200433, China
3   Department of Computer Science and Technology, Tongji University - 4800 Cao'an Road, Shanghai 201804, China

received 29 April 2009; accepted in final form 16 September 2009; published October 2009
published online 16 October 2009

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 $\langle T\rangle $ behaves superlinearly as $\langle T \rangle \sim V^{\frac{3}{2}} $ with an exponent $\frac{3}{2} $ much larger than 1, which is in sharp contrast to the scaling $\langle T\rangle \sim V^{\theta}$ with $\theta \leqslant 1$, 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.

05.40.Fb - Random walks and Levy flights.
89.75.Hc - Networks and genealogical trees.
05.60.Cd - Classical transport.

© EPLA 2009