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Recently, the consensus problem has attracted extensive attention of researchers from different scientific communities. In this paper, we study the leader–follower network coherence (i.e. the mean steady-state variance of the deviation from the static value of the leader nodes) in the weighted recursive tree networks with assigning the leaders orderly in the initial state. For the weighted recursive tree networks model, we study the characteristic polynomial of the Laplacian submatrix and obtain the relationship for the eigenvalues in two successive generations. The analytical formula of the leader–follower network coherence related by the sum of the reciprocal of all eigenvalues of this submatrix is obtained. The result shows that more number of leaders and the greater weight factor lead to better consensus. [ABSTRACT FROM AUTHOR]

In this paper, a 1D chain network with a reverse weighted edge is introduced. We focus on studying the relationships including the convergence rate and the length, the convergence rate and weight of adding reverse edge relationships. Laplacian characteristic determinant is calculated and subsequently, the sum of the reciprocals of all nonzero Laplacian eigenvalues is obtained. Hence, the analytic expression of global mean weighted first-passage time (GMWFPT) can be deduced. The obtained results show that there exists a linearly positive relationship between GMWFPT and the weight r. [ABSTRACT FROM AUTHOR]

The eigentime identity for random walks on the weighted networks is the expected time for a walker going from a node to another node. Eigentime identity can be studied by the sum of reciprocals of all nonzero Laplacian eigenvalues on the weighted networks. In this paper, we study the weighted (m , n) -flower networks with the weight factor r. We divide the set of the nonzero Laplacian eigenvalues into three subsets according to the obtained characteristic polynomial. Then we obtain the analytic expression of the eigentime identity H t + 1 of the weighted (m , n) -flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take m = 3 , n = 2 as example, and show that the leading term of the eigentime identity on the weighted (3 , 2) -flower networks obey superlinearly, linearly with the network size. [ABSTRACT FROM AUTHOR]

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates q connected branches in each iteration process and assigns the corresponding weight. To reflect the global characteristics of such networks, we study the eigentime identity determined by the reciprocal sum of non-zero eigenvalues of normalized Laplacian matrix. By the recursive relationship of eigenvalues at two successive generations, we find the eigenvalues and their corresponding multiplicities for two cases when q is even or odd. Finally, we obtain the analytical expression of the eigentime identity and the scalings with network size of the weighted scale-free networks. [ABSTRACT FROM AUTHOR]

Complex networks have attracted a great deal of attention from scientific communities, and have been proven as a useful tool to characterize the topologies and dynamics of real and human-made complex systems. Laplacian spectrum of the considered networks plays an essential role in their network properties, which have a wide range of applications in chemistry and others. Firstly, we define one vertex–vertex graph. Then, we deduce the recursive relationship of its eigenvalues at two successive generations of the normalized Laplacian matrix, and we obtain the Laplacian spectrum for vertex–vertex graph. Finally, we show the applications of the Laplacian spectrum, i.e. first-order network coherence, second-order network coherence, Kirchhoff index, spanning tree, and Laplacian-energy-like. [ABSTRACT FROM AUTHOR]

Weighted folded hypercube is an charming variance of the famous hypercube and is superior to the weighted hypercube in many criteria. We mainly study the scaling of network coherence for the weighted folded hypercube that is controlled by a weight factor. Network coherence quantifies the steady-state variance of these fluctuations, and it can be regarded as a measure of robustness of the consensus process to the additive noise. If networks with small steady-state variance have better network coherence, it can be regarded as more robust to noise than networks with low coherence. We firstly calculate the spectra of weighted folded hypercube and obtain the leading terms of network coherence that are quantified as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. Finally, the results show that network coherence depends on iterations and weight factor. Meanwhile, with larger order, the scatings of the first- and second-order network coherence of weighted folded hypercube decrease with the increasing of weight factor. [ABSTRACT FROM AUTHOR]

The consensus dynamics with additive stochastic disturbances are characterized by the network coherence, which is the robustness of consensus algorithms when the nodes are subject to external perturbations. In this paper, the research goal is to obtain the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. One innovation point of this paper is the structure of a family of the weighted recursive trees with weight factor. We mainly obtain the exact expressions and scalings of network coherence on the family of weighted recursive trees. The scalings of first-order network coherence with network size obey three laws along with the range of the weight factor, while the scalings of second-order network coherence obey four laws along with the range of the weight factor. In addition, the scalings of first- and second-order network coherence on our studied networks are smaller than those performed on other studied networks when 1 m + 1 < r ≤ 1. The obtained results indicate that the efficiency of network coherence on the weighted network has close relation to the weight distribution, and we can design a better weight distribution to make the coherence of network more efficient. [ABSTRACT FROM AUTHOR]

Abstract The eigenvalues of the normalized Laplacian matrix of a network provide information on its structural properties and some relevant dynamical aspects, in particular for weight-dependent walk. In order to get the eigentime identity for weight-dependent walk, we need to obtain the eigenvalues and their multiplicities of the Laplacian matrix. Firstly, the model of the weighted scale-free triangulation networks is constructed. Then, the eigenvalues and their multiplicities of transition weight matrix are presented, after the recursive relationship of those eigenvalues at two successive generations are given. Consequently, the Laplacian spectrum is obtained. Finally, the analytical expression of the eigentime identity, indicating that the eigentime identity grows sublinearly with the network order, is deduced. Highlights • Eigentime identity of the weighted scale-free triangulation networks. • Transition weight matrix for weight-dependent walk. • Recursive relationship of those eigenvalues of transition weight matrix. • Eigentime identity grows sublinearly with the network order. [ABSTRACT FROM AUTHOR]

Laplacian spectrum gives a lot of useful information about complex structural properties and relevant dynamical aspects, which has attracted the attention of mathematicians. We introduced the weighted scale-free network inspired by the binary scale-free network. First, the weighted scale-free network with a weight factor is constructed by an iterative way. In the next step, we use the definition of eigenvalue and eigenvector to obtain the recursive relationship of its eigenvalues and multiplicities at two successive generations. Through analysis of eigenvalues of transition weight matrix we find that multiplicities of eigenvalues 0 of transition matrix are different for the binary scale-free network and the weighted scale-free network. Then, we obtain the eigenvalues for the normalized Laplacian matrix of the weighted scale-free network by using the obtained eigenvalues of transition weight matrix. Finally, we show some applications of the Laplacian spectrum in calculating eigentime identity and Kirchhoff index. The leading term of these indexes are completely different for the binary and the weighted scale-free network. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For a network, knowledge of its Laplacian eigenvalues is central to understand its structure and dynamics. In this paper, we study the Laplacian spectra and their applications for a family of treelike networks. Firstly, in order to obtain the Laplacian spectra, we calculate the constant term and monomial coefficient of characteristic polynomial of the Laplacian matrix for a family of treelike networks. By using the Vieta theorem, we then obtain the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix. Finally, we determine some interesting quantities that are related to the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix, such as Kirchhoff index, global mean-first passage time (GMFPT). [ABSTRACT FROM AUTHOR]

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 1 d < r ≤ 1. [ABSTRACT FROM AUTHOR]

The eigenvalues of the normalized Laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects, in particular those related to weight-dependent walk. In this paper, we first present a study on the transition weight matrix of a weighted network. To get the eigentime identity for weight-dependent walk and weighted counting of spanning trees, we need to obtain all the eigenvalues and their multiplicities of the transition weight matrix. Then we obtain the recursive relationship of its eigenvalues at two successive generations of transition weight matrix. By substituting, we can obtain the relationship of normalized Laplacian matrix's eigenvalues at two successive generations. Using the relationship and Vietas formulas, we obtain the scalings of the eigentime identity. Afterwards, we classify normalized Laplacian matrix's eigenvalues and compute the product of all nonzero normalized Laplacian eigenvalues by the product recursive relationship. The product is used to obtain weighted counting of spanning trees. Finally, by weighted counting of spanning trees, we validate the obtained eigenvalues and their multiplicities. The obtained results show that the weight factor has a strong effect on the behavior of weight-dependent walks. [ABSTRACT FROM AUTHOR]

Much information about the structural properties and dynamical aspects of a network is measured by the eigenvalues of its normalized Laplacian matrix. In this paper, we aim to present a study on the spectra of the normalized Laplacian of weighted tree-like fractals. We analytically obtain the relationship between the eigenvalues and their multiplicities for two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index and Kemeny’s constant. [ABSTRACT FROM AUTHOR]

In this paper, we first analytically calculate the eigenvalues of the transition matrix of a structure with very complex architecture and their multiplicities. We call this structure polymer network. Based on the eigenvalues obtained in the iterative manner, we then calculate the eigentime identity. We highlight two scaling behaviors (logarithmic and linear) for this quantity, strongly depending on the value of the weight factor. Finally, by making use of the obtained eigenvalues, we determine the weighted counting of spanning trees. [ABSTRACT FROM AUTHOR]

In this paper, we consider the entire mean first-passage time (EMFPT) with random walks for Cayley networks. We use Laplacian spectra to calculate the EMFPT. Firstly, we calculate the constant term and monomial coefficient of characteristic polynomial. By using the Vieta theorem, we then obtain the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix. Finally, we obtain the scaling of the EMFPT for Cayley networks by using the relationship between the sum of reciprocals of all nonzero eigenvalues of Laplacian matrix and the EMFPT. We expect that our method can be adapted to other types of self-similar networks, such as vicsek networks, polymer networks. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the eigentime identities quantifying as the sum of reciprocals of all nonzero normalized Laplacian eigenvalues on a family of treelike networks and the polymer networks. Firstly, for a family of treelike networks, it is shown that all their eigenvalues can be obtained by computing the roots of several small-degree polynomials defined recursively. We obtain the scalings of the eigentime identity on a family of treelike with network size N n is N n ln N n . Then, for the polymer networks, we apply the spectral decimation approach to determine analytically all the eigenvalues and their corresponding multiplicities. Using the relationship between the generation and the next generation of eigenvalues we obtain the scalings of the eigentime identity on the polymer networks with network size N n is N n ln N n . By comparing the eigentime identities on these two kinds of networks, their scalings with network size N n are all N n ln N n . [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose the entire mean weighted first-passage time (EMWFPT) for the first time in the literature. The EMWFPT is obtained by the sum of the reciprocals of all nonzero Laplacian eigenvalues on weighted networks. Simplified calculation of EMWFPT is the key quantity in the study of infinite families of weighted tree networks, since the weighted complex systems have become a fundamental mechanism for diverse dynamic processes. We base on the relationships between characteristic polynomials at different generations of their Laplacian matrix and Laplacian eigenvalues to compute EMWFPT. This technique of simplified calculation of EMWFPT is significant both in theory and practice. In this paper, firstly, we introduce infinite families of weighted tree networks with recursive properties. Then, we use the sum of the reciprocals of all nonzero Laplacian eigenvalues to calculate EMWFPT, which is equal to the average of MWFPTs over all pairs of nodes on infinite families of weighted networks. In order to compute EMWFPT, we try to obtain the analytical expressions for the sum of the reciprocals of all nonzero Laplacian eigenvalues. The key step here is to calculate the constant terms and the coefficients of first-order terms of characteristic polynomials. Finally, we obtain analytically the closed-form solutions to EMWFPT on the weighted tree networks and show that the leading term of EMWFPT grows superlinearly with the network size. [ABSTRACT FROM AUTHOR]

We consider first-order network coherence in 5-rose graphs with stochastic disturbances. Our analysis for first-order systems is facilitated by connections between first-order network coherence and the sum of reciprocals of all nonzero Laplacian eigenvalues. Our analysis reveals that 5-rose networks with small size have high network coherence and can be considered to be more robust to noise than networks with low coherence. The method used in this paper is practical and easy to handle. • The first-order network coherence in the 5-rose graphs is studied. • The 5-rose graphs is introduced. • Computing the roots of several small-degree polynomials. [ABSTRACT FROM AUTHOR]