Abstract
In this paper, we present the weighted scale-free treelike networks controlled by the weight factor r and the parameter m. Based on the network structure, we study two types of weight-dependent walks with a highest-degree trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. Although some properties have been revealed in weighted networks, studies on mixed weight-dependent walks are still less and remain a challenge. For the weighted scale-free treelike network, we derive exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process. The obtained results show that ATT is related to weight factor r, parameter m and spectral dimension of the weighted network. We find that in different range of the weight factor r, the leading term of ATT grows differently, i.e., superlinearly, linearly and sublinearly with the network size. Furthermore, the obtained results show that changing the walking rule has no effect on the leading scaling of the trapping efficiency. All results in this paper can help us get deeper understanding about the effect of link weight, network structure and the walking rule on the properties and functions of complex networks.
Highlights
The past decade has witnessed a great deal of activity devoted to complex networks by the scientific community, since a number of real-life systems in nature and society can be described by complex networks
In existing studies of complex networks, an important issue is to uncover the influence of topological characteristics on the dynamical processes occurring on networks[4]
In order to discover what topological characteristics of network have important effect on diffusion, a lot of work focus on the average trapping time (ATT) which is used to characterize the efficiency of diffusion in different networks[8,9,10]
Summary
The past decade has witnessed a great deal of activity devoted to complex networks by the scientific community, since a number of real-life systems in nature and society can be described by complex networks. Computer networks can range from a local network area to a wide regional network To study these real networks, we develop a mathematical framework and construct the weighted scale-free treelike networks in this paper. It should be mentioned that we only studied the trapping problem in a theoretical model of weighted scale-free treelike networks, whether the conclusion holds for random networks, which needs further investigations in their future research. In contrast to nearest-neighbor weight-dependent walks, there are few theoretical studies on the influence of mixed weight-dependent walks on the network, especially on the trapping problem. It is quite important and significant to further explore how weight distribution and walking rule affect diffusion on weighted networks
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