Random walks on the Apollonian network with a single trap

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. Introduction. – Trapping is an integral major theme of random walks (diffusion) [1–3], which is relevant to a wide range of applications and has led to a growing number of theoretical and practical investigation over the past decades [4–9]. The trapping problem, first introduced in [10], is in fact a random-walk issue, where a trap is positioned at a given location, which absorbs all particles visiting it. The primarily interesting quantity closely related to trapping problem is the average trapping time, also referred to as the mean first-passage time (MFPT), which is useful in the study of transport-limited reactions [11, 12], target search [13, 14] and other physical problems. An important question in the study of trapping is how the MFPT scales with the size of the system. There are some well-known results providing answers to the corresponding questions in the cases of some graphs with simple topology, including regular lattices [10], Sierpinski fractals [15, 16], T-fractal [17], and so on. However, these graphs are not suitable to describe real systems [18] encountered in everyday experience, most of which are scalefree [19] and small-world [20] that have been shown to influence profoundly various dynamical processes running on networks [21, 22]. Thus, it is natural and interesting (a)[email protected] (b)[email protected] (c)[email protected] to explore the trapping problem on networks with general structure embedded in real life. Although a lot of activities have been devoted to studying random walks on complex networks [23–27], work about trapping problem on scale-free small-world graphs is much less [28]. In the paper, we investigate the trapping problem on the Apolloian network [29, 30] with scale-free and smallworld properties. We focus on a specific aspect of random walks in the presence of a single trap situated at a given node with the largest degree (hub node). We obtain an exact analytical solution for the MFPT and the dependence of this primary quantity on the system size. We show that the Apolloian network is a preferred architecture that minimizes the increase of MFPT with network size, compared with regular networks and fractals. Introduction to the Apollonian network. – We first introduce the Apollonian packing [31], from which the Apollonian network is derived. There are two commonly used Apollonian packings that differ mainly in initial configurations. The first packing is constructed by starting with three mutually touching disks, the interstice of which is a curvilinear triangle. In the first generation a disk is inscribed, touching all the sides of this curvilinear triangle. For subsequent generations we indefinitely repeat the packing process for all the new curvilinear triangles. In