Distinct scalings for mean first-passage time of random walks on scale-free networks with the same degree sequence Zhongzhi Zhang, * Wenlei Xie, and Shuigeng Zhou † School of Computer Science, Fudan University, Shanghai 200433, China and Shanghai Key Laboratory of Intelligent Information Processing, Fudan University, Shanghai 200433, China Mo Li Software School, Fudan University, Shanghai 200433, China Jihong Guan ‡ Department of Computer Science and Technology, Tongji University, 4800 Cao’an Road, Shanghai 201804, China Received 5 August 2009; revised manuscript received 25 September 2009; published 8 December 2009 In general, the power-law degree distribution has profound influence on various dynamical processes defined on scale-free networks. In this paper, we will show that power-law degree distribution alone does not suffice to characterize the behavior of trapping problems on scale-free networks, which is an integral major theme of interest for random walks in the presence of an immobile perfect absorber. In order to achieve this goal, we study random walks on a family of one-parameter denoted by q scale-free networks with identical degree sequence for the full range of parameter q, in which a trap is located at a fixed site. We obtain analytically or numerically the mean first-passage time MFPT for the trapping issue. In the limit of large network order number of nodes, for the whole class of networks, the MFPT increases asymptotically as a power-law function of network order with the exponent obviously different for different parameter q, which suggests that power-law degree distribution itself is not sufficient to characterize the scaling behavior of MFPT for random walks at least trapping problem, performed on scale-free networks. DOI: 10.1103/PhysRevE.80.061111 PACS numbers: 05.40.Fb, 89.75.Hc, 05.60.Cd, 89.75.Da I. INTRODUCTION As a fundamental stochastic process, random walks have received considerable attention from the scientific society since they found a wide range of distinct applications in various theoretical and applied fields, such as physics, chem- istry, biology, and computer science, among others 1–3. Among a plethora of interesting issues of random walks, trapping is an integral major one, which plays an important role in an increasing number of disciplines. The so-called trapping issue that was first introduced in 4 is a random- walk problem, where a trap is positioned at a fixed location, absorbing all particles that visit it. The highly desirable quan- tity closely related to the trapping issue is the first-passage time FPT also called trapping time TT. The FPT for a given site node and vertex is the time spent by a walker starting from the site to hit the trap node for the first time. This quantity is very important since it underlies many physical processes 5,6. The average of first-passage times over all starting nodes is referred to as the mean first-passage time MFPT or mean trapping time MTT, which is fre- quently used to measure the efficiency of the trapping prob- lem. One of the most important questions in the research of trapping is determining its efficiency, namely, showing the dependence relation of MFPT on the size of the system where the random walks are performed. Previous studies have provided the answers to the corresponding problems in some particular graphs with simple structure, such as regular lattices 4, Sierpinski fractals 7,8, T-fractal 9, and so forth. However, recent empirical studies 10–12 uncovered that many perhaps most real networks are scale-free char- acterized by a power-law degree distribution Pk k - with the exponent  belonging to interval 2,3, which cannot be described by above simple graphs 13. Thus, it appears quite natural and important to explore the trapping issue on scale- free networks. In recent work 14 –16, we have shown that scale-free property may substantially improve the efficiency of the trapping problem: the MFPT behaves linearly or sub- linearly with the order number of nodes of the scale-free networks, which is in sharp contrast to the superlinear scal- ing obtained for above-mentioned simple graphs 4,7–9. It was speculated that the high efficiency of trapping on scale- free networks is attributed to their power-law property. Al- though scale-free feature can strongly affect the various dy- namics occurring on networks, it was shown that the power- law degree distribution even degree sequence itself does not suffice to characterize some dynamical processes on scale-free networks, e.g., synchronization 17,18, disease spreading 19,20, and the like. Thus far, it is still not known whether degree sequence is sufficient to characterize the be- havior of trapping problem on scale-free networks although it has been shown that the exponent  of power-law degree distribution does not suffice 14,15,21,22. In this paper, we study the trapping problem on a class of scale-free networks with the same degree sequence, which are dominated by a tunable parameter q 23. We determine * zhangzz@fudan.edu.cn † sgzhou@fudan.edu.cn ‡ jhguan@tongji.edu.cn PHYSICAL REVIEW E 80, 061111 2009 1539-3755/2009/806/06111110 ©2009 The American Physical Society 061111-1