Random walks on the Apollonian network with a single trap

TL;DR

This paper derives an exact formula for the mean first-passage time of random walks with a trap on the Apollonian network, revealing its high efficiency for diffusion transport in complex networks.

Contribution

It provides the first explicit analytic expression for MFPT on the Apollonian network with a trap at a hub node, demonstrating its unique transport efficiency.

Findings

MFPT grows as a power-law with system size, exponent less than 1

Apollonian network is the most efficient structure for diffusion transport

Analytic results confirmed by numerical calculations

Abstract

Explicit determination of the mean first-passage time (MFPT) for 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 previously studied structure.