It has been recently reported that trapping problem can characterize various dynamical processes taking place on complex networks. However, most works focused on the case of binary networks, and dynamical processes on weighted networks are poorly understood. In this paper, we study two kinds of biased walks including standard weight-dependent walk and mixed weight-dependent walk on the weighted scale-free treelike networks with a trap at the central node. Mixed weight-dependent walk including non-nearest neighbor jump appears in many real situations, but related studies are much less. By the construction of studied networks in this paper, we determine all the eigenvalues of the fundamental matrix for two kinds of biased walks and show that the largest eigenvalue has an identical dominant scaling as that of the average trapping time (ATT). Thus, we can obtain the leading scaling of ATT by a more convenient method and avoid the tedious calculation. The obtained results show that the weight factor has a significant effect on the ATT, and the smaller the value of the weight factor, the more efficient the trapping process is. Comparing the standard weight-dependent walk with mixed weight-dependent walk, although next-nearest-neighbor jumps have no main effect on the trapping process, they can modify the coefficient of the dominant term for the ATT.

Complex networks are omnipresent in science and in our real life, and have been the focus of intense interest. It is vital to research the impact of their characters on the dynamic progress occurring on complex networks for weight-dependent walk. In this paper, we first consider the weight-dependent walk on one kind of transfractal (or fractal) which is named the weighted transfractal [Formula: see text]-flowers. And we pay attention to the first return time (FRT). We mainly calculate the mean and variance of FRT for a prescribed hub (i.e. the most concerned nodes) in virtue of exact probability generating function and its properties. Then, we obtain the mean and the secondary moment of the first return time. Finally, using the relationship among the variance, mean and the secondary moment, we obtain the variance of FRT and the scaling properties of the mean and variance of FRT on weighted transfractals [Formula: see text]-flowers.

In this paper, we first study the first-order network coherence, characterized by the entire mean first-passage time (EMFPT) for weight-dependent walk, on the weighted Cayley networks with the weight factor. The analytical formula of the EMFPT is obtained by the definition of the EMFPT. The obtained results show that the scalings of first-order coherence with network size obey four laws along with the range of the weight factor. Then, we study eigentime identity quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on the weighted Cayley networks with the weight factor. We show that all their eigenvalues can be obtained by calculating the roots of several small-degree polynomials defined recursively. The obtained results show that the scalings of the eigentime identity on the weighted Cayley networks obey two laws along with the range of the weight factor.

This paper investigates consensus dynamics in a dynamical system with additive stochastic disturbances that is characterized as network coherence by using the Laplacian spectrum. We introduce a class of weighted networks based on a complete graph and investigate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. First, the recursive relationship of its eigenvalues at two successive generations of Laplacian matrix is deduced. Then, we compute the sum and square sum of reciprocal of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first- and second-order coherence with network size obey four and five laws, respectively, along with the range of the weight factor. Finally, it indicates that the scalings of our studied networks are smaller than other studied networks when 1d<r≤1.

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.