Coherence analysis of a class of weighted tree-like polymer networks

2018 ◽  
Vol 32 (05) ◽  
pp. 1850064 ◽  
Author(s):  
Jiaojiao He ◽  
Meifeng Dai ◽  
Yue Zong ◽  
Jiahui Zou ◽  
Yu Sun ◽  
...  
Keyword(s):  
Polymer Networks ◽  
Laplacian Matrix ◽  
Network Size ◽  
Second Order ◽  
Weight Factor ◽  
Stochastic Disturbances ◽  
Weighted Tree ◽  
Consensus Dynamics ◽  
Successive Generations ◽  
Relationship Of

Complex networks have elicited considerable attention from scientific communities. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. Firstly, we introduce a class of weighted tree-like polymer networks with the weight factor. Then, we deduce the recursive relationship of the eigenvalues of Laplacian matrix at two successive generations. Finally, we calculate the first- and second-order network coherence quantifying as the sum and square sum of reciprocals of all nonzero Laplacian eigenvalues. 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 and the scalings of second-order coherence with network size obey five laws along with the range of the weight factor.

Consensus dynamics on a family of weighted recursive trees

2019 ◽  
Vol 33 (02) ◽  
pp. 1950003 ◽  
Author(s):  
Meifeng Dai ◽  
Yue Zong ◽  
Jiaojiao He ◽  
Tingting Ju ◽  
Yu Sun ◽  
...  
Keyword(s):  
Weight Distribution ◽  
Close Relation ◽  
Network Size ◽  
Second Order ◽  
Weight Factor ◽  
Consensus Algorithms ◽  
The Family ◽  
Consensus Dynamics ◽  
Research Goal ◽  
Recursive Trees

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 [Formula: see text]. 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.

EIGENTIME IDENTITY FOR WEIGHT-DEPENDENT WALK ON A CLASS OF WEIGHTED FRACTAL SCALE-FREE HIERARCHICAL-LATTICE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (08) ◽  
pp. 1950138
Author(s):  
BO WU ◽  
ZHIZHUO ZHANG ◽  
YINGYING CHEN ◽  
TINGTING JU ◽  
MEIFENG DAI ◽  
...  
Keyword(s):  
Iteration Process ◽  
Laplacian Matrix ◽  
Network Size ◽  
Hierarchical Lattice ◽  
Analytical Expression ◽  
Scale Free ◽  
Scale Free Networks ◽  
Reciprocal Sum ◽  
Successive Generations ◽  
Relationship Of

In this paper, we construct a class of weighted fractal scale-free hierarchical-lattice networks. Each edge in the network generates [Formula: see text] 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 [Formula: see text] 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.

Coherence analysis of a family of weighted star-composed networks

2019 ◽  
Vol 33 (23) ◽  
pp. 1950264
Author(s):  
Meifeng Dai ◽  
Tingting Ju ◽  
Yongbo Hou ◽  
Jianwei Chang ◽  
Yu Sun ◽  
...  
Keyword(s):  
Kronecker Product ◽  
Future Research ◽  
Weight Factor ◽  
Node Number ◽  
Laplacian Eigenvalues ◽  
First Order ◽  
Successive Generations ◽  
Network Consistency ◽  
Relationship Of ◽  
The Relationship

Recently, the study of many kinds of weighted networks has received the attention of researchers in the scientific community. In this paper, first, a class of weighted star-composed networks with a weight factor is introduced. We focus on the network consistency in linear dynamical system for a class of weighted star-composed networks. The network consistency can be characterized as network coherence by using the sum of reciprocals of all nonzero Laplacian eigenvalues, which can be obtained by using the relationship of Laplacian eigenvalues at two successive generations. Remarkably, the Laplacian matrix of the class of weighted star-composed networks can be represented by the Kronecker product, then the properties of the Kronecker product can be used to obtain conveniently the corresponding characteristic roots. In the process of finding the sum of reciprocals of all nonzero Laplacian eigenvalues, the key step is to obtain the relationship of Laplacian eigenvalues at two successive generations. Finally, we obtain the main results of the first- and second-order network coherences. The obtained results show that if the weight factor is 1 then the obtained results in this paper coincide with the previous results on binary networks, otherwise the scalings of the first-order network coherence are related to the node number of attaching copy graph, the weight factor and generation number. Surprisingly, the scalings of the first-order network coherence are independent of the node number of initial graph. Consequently, it will open up new perspectives for future research.

Applications of Laplacian spectrum for the vertex–vertex graph

2019 ◽  
Vol 33 (17) ◽  
pp. 1950184 ◽  
Author(s):  
Tingting Ju ◽  
Meifeng Dai ◽  
Changxi Dai ◽  
Yu Sun ◽  
Xiangmei Song ◽  
...  
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
First Order ◽  
Laplacian Energy ◽  
Wide Range ◽  
Vertex Graph ◽  
Network Properties ◽  
Successive Generations ◽  
Relationship Of

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.

CHARACTERISTIC POLYNOMIAL OF ADJACENCY OR LAPLACIAN MATRIX FOR WEIGHTED TREELIKE NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950074 ◽  
Author(s):  
MEIFENG DAI ◽  
YONGBO HOU ◽  
CHANGXI DAI ◽  
TINGTING JU ◽  
YU SUN ◽  
...  
Keyword(s):  
Characteristic Polynomial ◽  
Future Study ◽  
Laplacian Matrix ◽  
Block Matrix ◽  
Weight Factor ◽  
Laplacian Spectrum ◽  
Weighted Networks ◽  
Adjacency Spectrum ◽  
Analytic Expression ◽  
Successive Generations

In recent years, weighted networks have been extensively studied in various fields. This paper studies characteristic polynomial of adjacency or Laplacian matrix for weighted treelike networks. First, a class of weighted treelike networks with a weight factor is introduced. Then, the relationships of adjacency or the Laplacian matrix at two successive generations are obtained. Finally, according to the operation of the block matrix, we obtain the analytic expression of the characteristic polynomial of the adjacency or the Laplacian matrix. The obtained results lay the foundation for the future study of adjacency spectrum or Laplacian spectrum.

The average weighted receiving time with weight-dependent walk on a family of double-weighted polymer networks

2019 ◽  
Vol 30 (08) ◽  
pp. 1950063
Author(s):  
Fei Zhang ◽  
Dandan Ye ◽  
Changling Han ◽  
Wei Chen ◽  
Yingze Zhang
Keyword(s):  
Polymer Networks ◽  
Network Size ◽  
Trapping Efficiency ◽  
Large Network ◽  
Weight Factor ◽  
Current Node ◽  
Weight Factors ◽  
Edge Linking

In this paper, a family of the double-weighted polymer networks is introduced depending on the number of copies [Formula: see text] and two weight factors [Formula: see text]. The double-weights represent the selected weights and the consumed weights, respectively. Denote by [Formula: see text] the selected weight connecting the nodes [Formula: see text] and [Formula: see text], and denote by [Formula: see text] the consumed weight connecting the nodes [Formula: see text] and [Formula: see text]. Let [Formula: see text] be related to the weight factor [Formula: see text], and let [Formula: see text] be related to the weight factors [Formula: see text]. Assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the selected weight of edge linking them. The weighted time for two adjacency nodes is the consumed weight connecting the two nodes. The average weighted receiving time (AWRT) is defined on the double-weighted polymer networks. Our results show that in large network, the leading behaviors of AWRT for the double-weighted polymer networks follow distinct scalings, with the trapping efficiency associated with the network size [Formula: see text], the number of copies [Formula: see text], and two weight factors [Formula: see text]. We also found that the scalings of the AWRT with weight-dependent walk in double-weighted polymer networks is due to the use of the weight-dependent walk and the weighted time. The dominant reason is the range of each weight factor. To investigate the reason of the scalings, the AWRT for four cases are discussed.

Second-Order Nonlinear Optical Interpenetrating Polymer Networks

1995 ◽  
pp. 198-204 ◽  
Author(s):  
S. Marturunkakul ◽  
J. I. Chen ◽  
L. Li ◽  
X. L. Jiang ◽  
R. J. Jeng ◽  
...  
Keyword(s):  
Nonlinear Optical ◽  
Polymer Networks ◽  
Second Order ◽  
Interpenetrating Polymer Networks

Determining entire mean first-passage time for Cayley networks

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009 ◽  
Author(s):  
Xiaoqian Wang ◽  
Meifeng Dai ◽  
Yufei Chen ◽  
Yue Zong ◽  
Yu Sun ◽  
...  
Keyword(s):  
First Passage Time ◽  
Polymer Networks ◽  
Laplacian Matrix ◽  
Constant Term ◽  
Passage Time ◽  
Mean First Passage Time ◽  
First Passage ◽  
Laplacian Spectra ◽  
Self Similar ◽  
The Relationship

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.

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