Random walks on the Apollonian network with a single trap

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

Corresponding authors: zhangzz@fudan.edu.cn jhguan@tongji.edu.cn sgzhou@fudan.edu.cn

Received: 22 December 2008
Accepted: 16 March 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.