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Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network structure.
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Synchronization of multilayer complex networks is one of the important frontier issues in network science. In this paper, we strictly derived the analytic expressions of the eigenvalue spectrum of multilayer star and star-ring networks and analyzed the synchronizability of these two networks by using the master stability function (MSF) theory. In p...
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Citations
... Later, people studied the influence of the number and mode of interlayer connections on the synchronizability [16][17][18]. On the basis of single-layer networks research, the study of network synchronizability is also extended to M-layer and more complex network structures [19][20][21][22]. But the above researches all consider the same structure for different layers. ...
... The dynamics of node i in a two-layer network satisfies the following equation [15,19]: ...
... By observing the eigenvalues of multi-layer networks [15][16][17][19][20][21] and repeating calculations, we found the eigenvalues of supra-Laplacian matrix having the form of , where belong to complex number field. Therefore, the eigenvalues of supra-Laplacian matrix for the generalized two-layer network can be given as . ...
This paper focuses on a generalized two-layer network and its synchronizability, which randomly generate different topologies at each layer. This kind of network can better describe some irregular networks in reality. From the master stability function method of network synchronization analysis, we estimate the largest eigenvalue and the lowest nonzero eigenvalue of the supra-Laplacian matrix. Then, the influence of node coupling strength on the synchronizability of generalized two-layer networks is analyzed. We obtain that the enhancement of node coupling strength can promote network synchronization in bounded and unbounded synchronization regions. In the end, we perform numerical simulations based on theoretical analysis. The numerical results also show that the more nodes, the stronger the synchronizability under the unbounded synchronization region, and the opposite is true for the bounded synchronization region. The results have a certain guiding significance for the synchronous application of general network in reality.
... Li [27] proposed a two-layer dumbbell network model to analyze the synchronizability and verified the synchronization through a simple numerical simulation. Deng et al. [28] researched the synchronizability of multilayer chain networks. However, in real-world complex networks, whether they are single-layer or multilayer, studying the synchronizability of networks with weighted, directed and variable coupling strength is more meaningful. ...
We investigate the synchronizability of multilayer star-ring networks. Two types of multilayer networks, including aggregated coupling and divergent coupling, are established based on the connections between the hub node and the leaf nodes in the subnetwork. The eigenvalue spectrum of the two types of multilayer networks is strictly derived, and the correlation between topological parameters and synchronizability is analyzed by the master stability function framework. Moreover, the variable coupling strength has been investigated, revealing that it is significantly related to the synchronizability of the aggregated coupling while having no influence on the divergent coupling. Furthermore, the validity of the synchronizability analysis is obtained by implementing adaptive control on the multilayer star-ring networks previously mentioned. Calculations and comparisons show that the differences caused by the sizes of multilayer networks and interlayer coupling strength are not negligible. Finally, numerical examples are also provided to validate the effectiveness of the theoretical analysis.
... Zhang et al. studied the synchronizability of multi-layer K-nearest-neighbor networks and analyzed the impacts of some parameters (such as the network size, the number of layers) on network synchronizability [28]. Deng et al. compared the synchronizability of single-center three-layer star-ring networks and discussed the relationships among the parameters in the case of the unbounded and bounded synchronous regions [29]. Inspired by the above literature, the main contributions of this paper are as follows: 1) We defined two kinds of multi-layer dual-center star networks. ...
... Compared with the numbers of 2N nodes in this paper, the synchronizability of multi-layer dual-center coupled star networks is weaker than that of single-center coupled star networks. When N is large enough, we can calculate λ 2 and r of single-center star-ring networks with 2N nodes as follows [29]: ...
In the research on complex networks, synchronizability is a significant measurement of network nature. Several research studies center around the synchronizability of single-layer complex networks and few studies on the synchronizability of multi-layer networks. Firstly, this paper calculates the Laplacian spectrum of multi-layer dual-center coupled star networks and multi-layer dual-center coupled star–ring networks according to the master stability function (MSF) and obtains important indicators reflecting the synchronizability of the above two network structures. Secondly, it discusses the relationships among synchronizability and various parameters, and numerical simulations are given to illustrate the effectiveness of the theoretical results. Finally, it is found that the two sorts of networks studied in this paper are of the same synchronizability, and compared with that of a single-center network structure, the synchronizability of two dual-center structures is relatively weaker.
... e star-ring network is a regular network derived from the star network. In [32], Deng et al. researched the relationship between the eigenvalue spectrum, structure parameters, and synchronizability of multilayer star-ring networks with no weight and no direction, especially analyzing the change of synchronizability under different interlayer coupling strengths. However, in our real complex network, whether it is a single-layer complex network or a multilayer complex network, edges between nodes have directions or weights. ...
... In an M-layer network, if there are N nodes in each layer of the subnetwork, then the dynamic equation of the ith node can be written as follows [28,[31][32][33][34]: ...
... Different from the results of other similar networks, the parity of the number of star-ring subnetwork nodes in the two kinds of networks studied in this article has a certain impact on the synchronization, and the synchronization with odd or even of the subnetwork nodes has a different dependence on the structure parameters. Deng et al. [32] studied multilayer undirected star-ring and star networks. By comparing the simulation results of the interlayer coupling strength in this article with the case that the interlayer coupling strength is not equal in [32], it is found that the influence of the interlayer coupling strength on the network synchronization ability is the same only in the case of Network I and the odd number of subnetwork nodes. ...
In this study, we studied the eigenvalue spectrum and synchronizability of two types of double-layer hybrid directionally coupled star-ring networks, namely, the double-layer star-ring networks with the leaf node pointing to the hub node (Network I) and the double-layer star-ring networks with the hub node pointing to the leaf node (Network II). We strictly derived the eigenvalue spectrum of the supra-Laplacian matrix of these two types of networks and analyzed the relationship between the synchronizability and the structural parameters of networks based on the master stability function theory. Furthermore, the correctness of the theoretical results was verified through numerical simulations, and the optimum structural parameters were obtained to achieve the optimal synchronizability.
... To better understand the relationships between topological parameters and synchronizability, it is necessary to give a more rigorous theoretical analysis. Recently, the analytical expressions for the eigenvalues of multilayer fully-connected networks, star networks, chain networks, and star-ring networks were derived to analyze the synchronizability [26][27][28][29][30]. To the best of our knowledge, very little work has been devoted to studying the synchronizability of multilayer K-nearest-neighbor networks. ...
In this paper, the synchronizability of multilayer K-nearest-neighbor networks is studied by using the master stability function method. The analytical expressions for the eigenvalues of the supra-Laplacian matrix are given for two-layer and multilayer K-nearest-neighbor networks. In addition, the impacts of various topological parameters (such as the network size, the node degree, the number of layers, the intra-layer and the inter-layer coupling strengths) on the network synchronizability are discussed. Finally, the theoretical results are verified through numerical simulation.
This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.
The system model on synchronizability problem of complex networks with multi-layer structure is closer to the real network than the usual single-layer case. Based on the master stability equation (MSF), this paper studies the eigenvalue spectrum of two k-layer variable coupling windmill-type networks. In the case of bounded and unbounded synchronization domain, the relationships between the synchronizability of the layered windmill-type networks and network parameters, such as the numbers of nodes and layers, inter-layers coupling strength, are studied. The simulation of the synchronizability of the layered windmill-type networks are given, and they verify the theoretical results well. Finally, the optimization schemes of the synchronizability are given from the perspective of single-layer and multi-layer networks, and it was found that the synchronizability of the layered windmill-type networks can be improved by changing the parameters appropriately.
Synchronization of multiplex networks has been a topical issue in network science. Dumbbell networks are very typical structures in complex networks which are distinguished from both regular star networks and general community structures, whereas the synchronous dynamics of a double-layer dumbbell network relies on the interlink patterns between layers. In this paper, two kinds of double-layer dumbbell networks are defined according to different interlayer coupling patterns: one with the single-link coupling pattern between layers and the other with the two-link coupling pattern between layers. Furthermore, the largest and smallest nonzero eigenvalues of the Laplacian matrix are calculated analytically and numerically for the single-link coupling pattern and also obtained numerically for the two-link coupling pattern so as to characterize the synchronizability of double-layer dumbbell networks. It is shown that interlayer coupling patterns have a significant impact on the synchronizability of multiplex systems. Finally, a numerical example is provided to verify the effectiveness of theoretical analysis. Our findings can facilitate company managers to select optimal interlayer coupling patterns and to assign proper parameters in terms of improving the efficiency and reducing losses of the whole team.
