Eigentime identity of the weighted (m,n)-flower networks

2020 ◽  
Vol 34 (18) ◽  
pp. 2050159
Author(s):  
Changxi Dai ◽  
Meifeng Dai ◽  
Tingting Ju ◽  
Xiangmei Song ◽  
Yu Sun ◽  
...  
Keyword(s):  
Random Walks ◽  
Characteristic Polynomial ◽  
Network Size ◽  
Weight Factor ◽  
Laplacian Eigenvalues ◽  
Weighted Networks ◽  
Expected Time ◽  
Leading Term ◽  
Analytic Expression ◽  
Markov Spectrum

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 [Formula: see text]-flower networks with the weight factor [Formula: see text]. 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 [Formula: see text] of the weighted [Formula: see text]-flower networks by using the characteristic polynomial of Laplacian and recurrent structure of Markov spectrum. We take [Formula: see text], [Formula: see text] as example, and show that the leading term of the eigentime identity on the weighted [Formula: see text]-flower networks obey superlinearly, linearly with the network size.

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.

EIGENTIME IDENTITIES OF FRACTAL FLOWER NETWORKS

Fractals ◽  
2019 ◽  
Vol 27 (02) ◽  
pp. 1950008 ◽  
Author(s):  
QIANQIAN YE ◽  
JIANGWEN GU ◽  
LIFENG XI
Keyword(s):  
Random Walks ◽  
Characteristic Polynomials ◽  
Expected Time ◽  
Markov Spectrum

The eigentime identity for random walks on networks is the expected time for a walker going from a node to another node. In this paper, our purpose is to calculate the eigentime identities of flower networks by using the characteristic polynomials of normalized Laplacian and recurrent structure of Markov spectrum.

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.

Effect of heterogeneous weights on the average trapping time and two types of random walks in weighted directed networks

2019 ◽  
Vol 33 (05) ◽  
pp. 1950023 ◽  
Author(s):  
Huai Zhang ◽  
Hongjuan Zhang
Keyword(s):  
Random Walks ◽  
Numerical Solutions ◽  
Transition Probability ◽  
Network Size ◽  
Trapping Efficiency ◽  
Directed Networks ◽  
Trapping Time ◽  
Stochastic Parameter ◽  
Weight Parameter ◽  
Trapping Process

In this paper, we define a class of weighted directed networks with the weight of its edge dominated by a weight parameter w. According to the construction of the networks, we study two types of random walks in the weighted directed networks with a trap fixed on the central node, i.e., standard random walks and mixed random walks. For the standard random walks, the trapping process is controlled by a weight parameter w which changes the transition probability of random walks. For the mixed random walks including nonnearest-neighbor hopping, the trapping process is steered by a stochastic parameter [Formula: see text], where [Formula: see text] changes the walking rule. For the above two techniques, we derive both analytically the average trapping time (ATT) as the measure of trapping efficiency, and the obtained analytical expressions are in good agreement with the corresponding numerical solutions for different values of w and [Formula: see text]. The obtained results indicate that ATT scales superlinearly with network size Nn and the weight parameter w changes simultaneously the prefactor and the leading scalings of ATT, while the stochastic parameter [Formula: see text] can only alter the prefactor of ATT and leave the leading scalings of ATT unchanged. This work may help in paving the way for understanding the effects of the link weight and nonnearest-neighbor hopping in real complex systems.

Scalings of first-return time for random walks on generalized and weighted transfractal networks

2019 ◽  
Vol 33 (26) ◽  
pp. 1950306
Author(s):  
Qin Liu ◽  
Weigang Sun ◽  
Suyu Liu
Keyword(s):  
Random Walks ◽  
Large Scale ◽  
Probability Generating Function ◽  
Return Time ◽  
Network Size ◽  
Large Network ◽  
Large Scale Network ◽  
Scale Network ◽  
The Mean ◽  
First Return Time

The first-return time (FRT) is an effective measurement of random walks. Presently, it has attracted considerable attention with a focus on its scalings with regard to network size. In this paper, we propose a family of generalized and weighted transfractal networks and obtain the scalings of the FRT for a prescribed initial hub node. By employing the self-similarity of our networks, we calculate the first and second moments of FRT by the probability generating function and obtain the scalings of the mean and variance of FRT with regard to network size. For a large network, the mean FRT scales with the network size at the sublinear rate. Further, the efficiency of random walks relates strongly with the weight factor. The smaller the weight, the better the efficiency bears. Finally, we show that the variance of FRT decreases with more number of initial nodes, implying that our method is more effective for large-scale network size and the estimation of the mean FRT is more reliable.

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.

SCALING OF THE AVERAGE RECEIVING TIME ON A FAMILY OF WEIGHTED HIERARCHICAL NETWORKS

Fractals ◽  
2016 ◽  
Vol 24 (03) ◽  
pp. 1650038 ◽  
Author(s):  
YU SUN ◽  
MEIFENG DAI ◽  
YANQIU SUN ◽  
SHUXIANG SHAO
Keyword(s):  
Bipartite Graph ◽  
Power Law ◽  
Complete Bipartite Graph ◽  
Network Size ◽  
Large Network ◽  
Weight Factor ◽  
Hierarchical Networks ◽  
Current Node ◽  
Edge Linking ◽  
Power Law Function

In this paper, based on the un-weight hierarchical networks, a family of weighted hierarchical networks are introduced, the weight factor is denoted by [Formula: see text]. The weighted hierarchical networks depend on the number of nodes in complete bipartite graph, denoted by [Formula: see text], [Formula: see text] and [Formula: see text]. 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 [Formula: see text], the ART grows as a power-law function of the network size [Formula: see text] with the exponent, represented by [Formula: see text], [Formula: see text]. This means that the smaller the value of [Formula: see text], the more efficient the process of receiving information. When [Formula: see text], the ART grows with increasing order [Formula: see text] as [Formula: see text] or [Formula: see text].

Spectra analysis and network coherence for weighted folded hypercube

2019 ◽  
Vol 33 (11) ◽  
pp. 1950094 ◽  
Author(s):  
Meifeng Dai ◽  
Jiaojiao He ◽  
Huiling Wu ◽  
Xianbin Wu
Keyword(s):  
Steady State ◽  
Additive Noise ◽  
Second Order ◽  
Weight Factor ◽  
Consensus Process ◽  
Spectra Analysis ◽  
Laplacian Eigenvalues ◽  
Low Coherence ◽  
Folded Hypercube ◽  
Robust To Noise

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.

Mean first return time for random walks on weighted networks

2015 ◽  
Vol 26 (06) ◽  
pp. 1550068 ◽  
Author(s):  
Xing-Li Jing ◽  
Xiang Ling ◽  
Jiancheng Long ◽  
Qing Shi ◽  
Mao-Bin Hu
Keyword(s):  
Random Walk ◽  
Complex Networks ◽  
Random Walks ◽  
Stationary Distribution ◽  
Real World ◽  
Return Time ◽  
Edge Weight ◽  
Weighted Networks ◽  
Strength Based ◽  
First Return Time

Random walks on complex networks are of great importance to understand various types of phenomena in real world. In this paper, two types of biased random walks on nonassortative weighted networks are studied: edge-weight-based random walks and node-strength-based random walks, both of which are extended from the normal random walk model. Exact expressions for stationary distribution and mean first return time (MFRT) are derived and examined by simulation. The results will be helpful for understanding the influences of weights on the behavior of random walks.

FIRST-ORDER NETWORK COHERENCE AND EIGENTIME IDENTITY ON THE WEIGHTED CAYLEY NETWORKS

Fractals ◽  
2017 ◽  
Vol 25 (05) ◽  
pp. 1750049 ◽  
Author(s):  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YUE ZONG ◽  
JIAHUI ZOU ◽  
YUFEI CHEN ◽  
...  
Keyword(s):  
First Passage Time ◽  
Analytical Formula ◽  
Passage Time ◽  
Small Degree ◽  
Network Size ◽  
Mean First Passage Time ◽  
Weight Factor ◽  
First Passage ◽  
First Order ◽  
Definition Of

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.

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