In this paper, based on the un-weight hierarchical networks, a family of weighted hierarchical networks are introduced, the weight factor is denoted by r. The weighted hierarchical networks depend on the number of nodes in complete bipartite graph, denoted by n1, n2 and n = n1 + n2. Assume that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the weight of edge linking them. We deduce the analytical expression of the average receiving time (ART). The obtained remarkable results display two conditions. In the large network, when nr > n1n2, the ART grows as a power-law function of the network size |V (Gk)| with the exponent, represented by θ =logn( nr n1n2 ), 0 < θ < 1. This means that the smaller the value of θ, the more efficient the process of receiving information. When nr ≤ n1n2, the ART grows with increasing order |V (Gk)| as logn|V (Gk)| or (logn|V (Gk)|)2.

Scaling of the average receiving time on a family of weighted hierarchical networks

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 first study on the spectra of the normalized Laplacian of weighted iterated triangulations of graphs. We analytically obtain all the eigenvalues, as well as their multiplicities from two successive generations. As an example of application of these results, we then derive closed-form expressions for their multiplicative Kirchhoff index, Kemeny’s constant and number of weighted spanning trees.

Spectral Analysis for Weighted Iterated Triangulations of Graphs

With the rapid spread of the Internet, the transmission of rumors via the electronic media has become ubiquitous. In the present paper, a double-identity rumor spreading model is established by considering the rumor spreading identities and real-world identities of the rumor creator, the rumor controller and the normal user. The diversity of these real-world identities encompasses five different attributes and the transition process between different rumor spreading identities. A new forgetting mechanism is proposed through a consideration of the influence of the interaction behavior on the new forgetting mechanism. Then, based on the simulation experiments of the proposed model, the impacts of the forgetting mechanism and the time lag between rumor creator and rumor controller on the spreading process of rumor are discussed. The results show that both elements play important roles in the dynamics of rumor propagation.

A double-identity rumor spreading model

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 ) .

The modified box dimension and average weighted receiving time of the weighted hierarchical graph

In this paper a family of weighted fractal networks, in which the weights of edges have been assigned to different values with certain scale, are studied. For the case of the weighted fractal networks the definition of modified box dimension is introduced, and a rigorous proof for its existence is given. Then, the modified box dimension depending on the weighted factor and the number of copies is deduced. Assuming that the walker, at each step, starting from its current node, moves uniformly to any of its nearest neighbors. The weighted time for two adjacency nodes is the weight connecting the two nodes. Then the average weighted receiving time (AWRT) is a corresponding definition. The obtained remarkable result displays that in the large network, when the weight factor is larger than the number of copies, the AWRT grows as a power law function of the network order with the exponent, being the reciprocal of modified box dimension. This result shows that the efficiency of the trapping process depends on the modified box dimension: the larger the value of modified box dimension, the more efficient the trapping process is.

/pdf/modified-box-dimension-and-average-weighted-receiving-time-34xa9md9p8.pdf

Modified box dimension and average weighted receiving time on the weighted fractal networks.

In this paper we introduce a new kind of scale-free and small-world networks with multiple hubs based on classic networks and present the trapping problem on them. We show that in the multiple-hub networks, the growth of MFPT with the traps located on the hub or peripheral nodes displays quite a difference from the single-hub networks. Our conclusions are directed to possible applications in rumor or epidemic spreading phenomena.

Trapping on modular scale-free and small-world networks with multiple hubs

In this paper, given a time series generated by a certain dynamical system, we construct a new class of scale-free networks with fractal structure based on the subshift of finite type and base graphs. To simplify our model, we suppose the base graphs are bipartite graphs and the subshift has the special form. When embedding our growing network into the plane, we find its image is a graph-directed self-affine fractal, whose Hausdorff dimension is related to the power law exponent of cumulative degree distribution. It is known that a large spectral gap in terms of normalized Laplacian is usually associated with small mixing time, which makes facilitated synchronization and rapid convergence possible. Through an elaborate analysis of our network, we can estimate its Cheeger constant, which controls the spectral gap by Cheeger inequality. As a result of this estimation, when the bipartite base graph is complete, we give a sharp condition to ensure that our networks are well-connected with rapid mixing property.

Zhixiong Wen

Papers

Trapping on modular scale-free and small-world networks with multiple hubs

A class of scale-free networks with fractal structure based on subshift of finite type