Anomalous behavior of trapping on a fractal scale-free network

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 〈T 〉 behaves superlinearly as 〈T 〉 ∼ V 2 with an exponent 3 2 much larger than 1, which is in sharp contrast to the scaling 〈T 〉 ∼ V θ with θ 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. Copyright c © EPLA, 2009 Introduction. – In the past decade, there has been a considerable interest in characterizing and understanding the structural properties of networked systems [1]. It has been established that scale-free behavior [2] is one of the most fundamental concepts for a basic understanding of the organization of many real-world systems in nature and society. This scale-free property has a profound effect on almost every aspect on dynamic processes taking place on networks [3], including robustness [4], percolation [5,6], synchronization [7], games [8], epidemic spreading [9], to name just a few. For instance, for a wide range of scale-free networks an epidemic threshold does not exist, and even infections with a low spreading rate will prevail over the entire population in these networks [9]. This is a radical change from the conclusions drawn from classical disease modeling [10]. In addition to the above-mentioned dynamics, some authors have focused their attention on the trapping problem occurring on complex networks [11–17], which is one of the main topic of interest for random walks (a)E-mail: [email protected] (b)E-mail: [email protected] (c)E-mail: [email protected] (diffusion) [18,19]. The classical trapping problem first introduced in [20] is a random-walk issue, where a trap is located at a fixed position, absorbing all particles that visit it. An interesting quantity closely related to the trapping problem is the mean first-passage time (MFPT), which is very important in the study of transport-limited reactions [21,22], and target search [23,24], amongst other physical problems. A result from previous research is that a power-law property can improve the efficiency of transport by diffusion on scale-free networks [11,15–17]: the MFPT, 〈T 〉, scales linearly or sublinearly with the number of network nodes V as 〈T 〉 ∼ V θ with θ= 1 or θ < 1, which shows that the efficiency of trapping processes on scale-free networks is even better than (at least not worse than) that on complete graphs [11], the best possible structure for a fast diffusion (with 〈T 〉 ∼ V ). Although the scale-free topology has a direct effect on other structural characteristics (e.g., average path length [25]) of networks and dynamics running on them, it cannot reflect all the information of the network structure. Recently, it has been discovered that many real-life networks, such as the WWW, metabolic networks, and yeast protein interaction networks have self-similar