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In this paper we study the weighted hierarchical graph which is generated from bipartite graph with N = n 1 + n 2 vertices, in which the weights of edges have been assigned to different values with certain scale. Firstly, we introduce the definition of the modified box dimension. Then for the weighted hierarchical graph we deduce the modified box dimension, dim M B ( { G n } n ∈ N ) = − log r N , depending on the weighted factor r and the number N of copies. Secondly, we mainly study their two average weighted receiving times (AWRTs), 〈 T 〉 I n and 〈 T 〉 II n , of the weighted hierarchical graph on random walk. We discuss two cases. In the case of n 1 n 2 r ≠ n 2 − n 1 , we deduce both AWRTs grow as a power-law function of the network size | V ( G n ) | with the postive exponent, represented by θ = log N ( N n 1 n 2 ) or θ = log N r = 1 − dim M B ( { G n } n ∈ N ) , which means that the bigger the value of the modified box dimension is, the slower the process of receiving information is. In the case of n 1 n 2 r = n 2 − n 1 , both AWRTs tend to constant ( if N < n 1 n 2 ) , the AWRTs grow with increasing order as log N | V ( G n ) | ( if N = n 1 n 2 ) , and both AWRTs grow as a power-law function of the network size | V ( G n ) | with the exponent, represented by θ = log N ( N n 1 n 2 ) > 0 ( if N > n 1 n 2 ) . [ABSTRACT FROM AUTHOR]

Compared with traditional random walk, biased walks have been studied extensively over the past decade especially in the transport and communication networks. In this paper, we first introduce the weighted rose networks. Then, for the weighted rose networks we focus on two biased walks, maximal entropy walk and weight-dependent walk, and obtain the exact expressions of their stationary distributions and mean first-passage times. Finally, we find that the average receiving time for maximal entropy walk is a quadratic function of the weight parameter r while the average receiving time for weighted-dependent walk is a linear function of the weight parameter r. Meanwhile, for the maximal entropy walk, the smaller the value of r is, the more efficient the trapping process is. For the weighted-dependent walk, the larger the value of r (r < r 0 ≈ 2. 6) is, the more efficient for the weighted-dependent walk, the smaller the value of r (r > r 0 ≈ 2. 6) is, the more efficient for the weight-dependent walk. • Weighted rose networks are introduced. • Mean first-passage times for maximal entropy or weight-dependent walk is studied. • Average receiving time and global mean first-passage time are obtained. • Transmission efficiency of maximal entropy walk or weight-dependent walk. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we consider the weighted Dyson hierarchical network, that is a weighted fully-connected network, where the pattern of weights is ruled by a weight factor. Given that the Laplacian operator is intrinsically implied in the analysis of dynamic processes occurring on complex networks, Laplacian spectrum allows addressing analytically a large class of problems. Exploiting the deterministic recursive structure, we are able to derive explicitly all eigenvalues and their corresponding eigenvectors of Laplacian matrix. Further, we derive exact solutions of the trapping time (TT) and average trapping time (ATT) on the weighted Dyson hierarchical network with weight-dependent walk. The obtained results show that TT and ATT are related to the weight factor of the weighted Dyson hierarchical network. Highlights • Model of weighted Dyson hierarchical network is introduced. • Laplacian spectrum of weighted Dyson hierarchical network is obtained. • Eigenvectors of Laplacian matrix are obtained. • The scalings of both TT and ATT can be controlled by the weight factor. [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]

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]

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 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, based on actual road networks, a model of the non-homogeneous double-weighted fractal networks is introduced depending on the number of copies s and two kinds of weight factors w i , r i ( i = 1 , 2 , … , s ) . The double-weights represent the capacity-flowing weights and the cost-traveling weights, respectively. Denote by w i j F the capacity-flowing weight connecting the nodes i and j , and denote by w i j C the cost-traveling weight connecting the nodes i and j . Let w i j F be related to the weight factors w 1 , w 2 , … , w s , and let w i j C be related to the weight factors r 1 , r 2 , … , r s . Assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the capacity-flowing weight of edge linking them. The weighted time for two adjacency nodes is the cost-traveling weight connecting the two nodes. The average weighted receiving time (AWRT) is defined on the non-homogeneous double-weighted fractal networks. AWRT depends on the relationships of the number of copies s and two kinds of weight factors w i , r i ( i = 1 , 2 , … , s ) . The obtained remarkable results display that in the large network, the AWRT grows as a power-law function of the network size N g with the exponent, represented by θ = log s ( w 1 r 1 + w 2 r 2 + ⋯ + w s r s ) < 1 when w 1 r 1 + w 2 r 2 + ⋯ + w s r s ≠ 1 , which means that the smaller the value of w 1 r 1 + w 2 r 2 + ⋯ + w s r s is, the more efficient the process of receiving information is. Especially when w 1 r 1 + w 2 r 2 + ⋯ + w s r s = 1 , AWRT grows with increasing order N g as log N g or ( log N g ) 2 . In the classic fractal networks, the average receiving time (ART) grows with linearly with the network size N g . Thus, the non-homogeneous double-weighted fractal networks are more efficient than classic fractal networks in term of receiving information. [ABSTRACT FROM AUTHOR]

The weighted networks are realistic forms of networks, where a weight is attached to each link. In order to better study searching efficiency of multiple walkers we extend the study on the binary networks to the weighted networks. In this paper, the main aim is to measure the searching efficiency of multiple walkers on the weighted networks. Firstly, we review the theoretical foundations related to our study. Secondly, we introduce the heterogeneous mean-field (HMF) theory and the annealed network approach on the weighted networks. Then, we deduce the analysis formula of mean first parallel passage time (MFPPT). Finally, we study the global mean first parallel passage time (GMFPPT) and compare it to the searching efficiency of a single walker previously studied. The obtained result shows that the GMFPPT follows a uniform power law with the number of walkers. The key of this paper is to apply the heterogeneous mean-field (HMF) theory and the annealed weighted network approach to replace the weighted uncorrelated networks with the weighted fully connected networks and then construct a general probability transfer matrix on the weighted fully connected networks. • Searching efficiency of multiple walkers on the weighted networks. • Analysis formula of mean first parallel passage time is obtained. • The law of global mean first parallel passage time is obtained. • The searching efficiencies for a single walker and multiple walkers are compared. [ABSTRACT FROM AUTHOR]

Recently, corona product graphs have attracted a great deal of attention from scientific communities, such as coding theory, frequency assignment, mathematical chemistry. Meanwhile, weight factor gradually became a research hotpoint because of closing to reality. Hence, we introduce the weighted corona graphs in this paper. The weighted corona product graphs are the in-depth and promotion of the corona product graphs, linking the spectra between the initial graph and the product graph. Then, we deduce the complete information of generalized adjacency and Laplacian eigenvalues of the weighted product graph with different structures. And we generalize the results obtained previously into product corona graph with m (⩾ 1) iterations. • Laplacian spectra of the weighted corona product graphs. • The generalized adjacency eigenvalues of the weighted corona product graphs. • The spectra of the weighted corona product in a finite number of iterations. • Attaching copy graph is a regular graph and a complete bipartite graph. [ABSTRACT FROM AUTHOR]