Issue
EPL
Volume 86, Number 1, April 2009
Article Number 10006
Number of page(s) 6
Section General
DOI http://dx.doi.org/10.1209/0295-5075/86/10006
Published online 21 April 2009
EPL, 86 (2009) 10006
DOI: 10.1209/0295-5075/86/10006

Random walks on the Apollonian network with a single trap

Zhongzhi Zhang1, 2, Jihong Guan3, Wenlei Xie1, 2, Yi Qi1, 2 and Shuigeng Zhou1, 2

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

zhangzz@fudan.edu.cn
jhguan@tongji.edu.cn
sgzhou@fudan.edu.cn

received 22 December 2008; accepted in final form 16 March 2009; published April 2009
published online 21 April 2009

Abstract
Explicit determination of the mean first-passage time (MFPT) for the trapping problem on complex media is a theoretical challenge. In this paper, we study random walks on the Apollonian network with a trap fixed at a given hub node (i.e., node with the highest degree), which are simultaneously scale-free and small-world. We obtain the precise analytic expression for the MFPT that is confirmed by direct numerical calculations. In the large system size limit, the MFPT approximately grows as a power law function of the number of nodes, with the exponent much less than 1, which is significantly different from the scaling for some regular networks or fractals such as regular lattices, Sierpinski fractals, T-graph, and complete graphs. The Apollonian network is the most efficient configuration for transport by diffusion among all the previously studied structures.

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

© EPLA 2009