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Shen Jie, Tang Longkun. Intra-layer synchronization in duplex networks. Chinese Physics B, 2018, 27(10): 100503  
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Intra-layer synchronization in duplex networks
Shen Jie1, Tang Longkun1, 2, †
Fujian Province University Key Laboratory of Computation Science, School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China
Department of Mathematics & Statistics, Georgia State University, Atlanta 30303, USA

 

† Corresponding author. E-mail: tomlk@hqu.edu.cn

Project supported in part by the National Natural Science Foundation of China (Grant Nos. 61573004 and 11501221), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (Grant No. ZQN-YX301), the Program for New Century Excellent Talents in Fujian Province University in 2016, and the Project of Education and Scientific Research for Middle and Young Teachers in Fujian Province, China (Grant Nos. JAT170027 and JA15030).

Abstract

This paper explores the intra-layer synchronization in duplex networks with different topologies within layers and different inner coupling patterns between, within, and across layers. Based on the Lyapunov stability method, we prove theoretically that the duplex network can achieve intra-layer synchronization under some appropriate conditions, and give the thresholds of coupling strength within layers for different types of inner coupling matrices across layers. Interestingly, for a certain class of coupling matrices across layers, it needs larger coupling strength within layers to ensure the intra-layer synchronization when the coupling strength across layers become larger, intuitively opposing the fact that the intra-layer synchronization is seemly independent of the coupling strength across layers. Finally, numerical simulations further verify the theoretical results.

PACS: 05.45.Xt;05.45.-a
Keyword:multi-layer network;intra-layer synchronization;Lyapunov stability method
1. Introduction

Networks are ubiquitous in the real world, such as the Internet, power grid, biological and ecological networks, social networks, and so on. It is a very significant thing to unveil the mechanism behind network collective behaviors, which is still a research focus in network science fields. Synchronization, as one of the collective behaviors, has attracted a great deal of attention,[1–14] since a great deal of work[15] has been published, involving in phase synchronization, complete synchronization, cluster synchronization, generalized synchronization, and even chimera state.[16] Most of the works on synchronization investigated the relationship between synchronization and topological structures, and how to implement synchronization through the controllers or coupling strategies.

Very recently in the last few years, there has been increasing interest in the multi-layer (or multiplex) networks or hypernetworks[22–26] where there are different connectivity patterns between layers, since one kind of networks is always interconnected with other kinds of networks, such as power networks and computer networks, transport networks and communication networks, and the single composite (single connectivity pattern) network model does not well describe the feature of multi-layer networks. Naturally, synchronization in multi-layer networks becomes a hot research topic, and the related works[27–31] appear. Among them, He et al.[31] investigated the complete synchronization of multi-agents in multi-layer networks through the Lyapunov stability method, and state that complete synchronization depends on the joint synchronization region and the eigenvalues of each layer. Based on the master stability function method, with the hypothesis that the inner coupling function in multiplex networks is the same in each layer and across layers, some works[29,30] explored the complete synchronization and synchronizability from the view point of network topological structures. Our recent work[32] extended the master stability framework to multiplex networks with different inner coupling functions within and across layers. With it, we studied simultaneously complete, intra-layer, and inter-layer synchronizations, and showed that the inner coupling function across layers has an important impact on the intra-layer synchornization, but it is still completely unclear.

Intra-layer synchronization can be considered to some extent as the mesoscale level between the microscopic level (such as individual behaviors) and macroscopic large-scale level (such as complete synchronization), which has received increasing attention. Gambuzza et al.[33] analyzed the intra-layer synchronization of a population of oscillators indirectly coupled through an inhomogeneous medium. Shortly afterwards, Jalan and Singh[34] investigated nodes interactions in one layer affecting the cluster synchronizability of the other layer, and showed that at weak couplings, the multiplexing enhances the cluster synchronizability, while at strong couplings this enhancement depends on the architecture as well as the connection density of the other layer. Rakshit et al.[35] explored the intra-layer and inter-layer synchronizations in multiplex networks where the intra-layer coupling interactions are switched stochastically, and showed that the higher switching frequency enhances both intra-layer and inter-layer synchronizations. Tang et al.[36] studied the impact of the inter-layer coupling function and the intra-layer coupling delay on the intra-layer synchronization, and indicatd that the inter-layer coupling strength neither improves nor weakens the intra-layer synchronizability for certain inter-layer coupling functions. However, these investigations are based on the master stability function method, which is an approach for local stabilization.

These above researches motivate us to focus on the duplex networks where the intra- and inter-layers coupling functions are different, and employ the Lyapunov stability method (a global method) to investigate the intra-layer synchronization. The main contribution in this paper is to give the intra-layer synchronization criterion and the threshold of intra-layer strengths for three classes of inter-layer coupling matrices, and analyze and discuss their influences on the intra-layer synchronization.

The rest of this paper is organized as follows. A duplex network with different inner coupling function within and across layers is introduced in Section 2. In Section 3, the stability analysis based on the Lyapunov stability method is performed, and the corresponding synchronization criteria and thresholds are presented for different types of inner coupling functions across layers. In Section 4, some numerical examples are provided for further verifying the theoretical results obtained in Section 3. Finally, some conclusions and discussions are given in Section 5.

2. A duplex network model

A duplex network consisting of two layers whose one-to-one connections between layers can be described as

where and dominate the i-th node of the x-layer and y-layer, respectively. and are adjacent matrices of x-layer and y-layer, respectively, representing the topological structures within layers. If node i is connected with node j in the same layer, then , otherwise. c and d are the coupling strengths within and across layers, respectively.

Function H(·) : Rn → Rn is the inner coupling one within layers, describing the communication pattern between state variables of nodes. Γ(·) : Rn → Rn is the inner coupling function between the node in one layer and its corresponding node in the other layer. They can be the linear or nonlinear functions. For simplicity, H(·) and Γ(·) are set as the different linear functions here, namely, H(x) = Hx and Γ(x) = Γx. Hereafter, we also call H and Γ the inner coupling matrices for convenience. The diagram of duplex networks consisting of x-layer and y-layer is shown in Fig. 1.

Fig. 1. (color online) A diagram of the duplex network with 6 nodes in each layer.
3. Intra-layer synchronization stability analysis

The so-called intra-layer synchronization means that all the nodes of each layer synchronize to the common state called the intra-layer synchronous state eventually, and the synchronous states are different from each other. The duplex network achieves complete synchronization if the synchronous states are the same. Here, we focus on the nonidentical intra-layer synchronous state in each layer, and theoretically analyze the intra-layer synchronization stability via the Lyapunov stability method.

To begin with, Laplacian matrix Lx(y) = (lij)m × m associated with the adjacent matrix Ax(y) is described as

which satisfies the dissipative property that . Taking an example for further illustration, the adjacent matrices within layers of the duplex network shown in Fig. 1.

and their Laplacian matrices are

With the Laplacian matrices, network (1) can be rewritten as

When network (2) obtains the intra-layer synchronization, then the x-layer and the y-layer tend to the different synchronous states, respectively. Here, suppose is the synchronous solution of the x-layer, and is the one of the y-layer. It follows that

and similarly,

Due to and , one can get

Next, some assumptions and lemmas are addressed before giving the main theoretical results.

Denote M = max{a1, a2}, , , , , , , , and , it yields

Since L is a symmetric and zero row sum matrix due to the fact that Lx and Ly are symmetric and zero row sum matrices, there exists an orthogonal matrix such that

where Λ is a diagonal matrix whose diagonal elements being , and and are eigenvalues of Lx and Ly, respectively.

Let α = (UT ⊗ In)δ and denote where αi is an n-dimensional vector, and take c = c0 + 1 where

we can get

where λ2(Λ ⊗ H) is the smallest nonzero eigenvalue of Λ ⊗ H, and λmax((UTQU) ⊗ Γ) is the largest eigenvalue of (UTQU) ⊗ Γ.

According to LaSalle’s invariance principle,[39] the network (4) or network (2) achieves synchronization, and the proof is completed.

4. Numerical simulations

To show the effectiveness of the theoretical results, we take the Lorenz chaotic system as network nodal dynamics in both layers:

where σ = 10, β = 28, and γ = 8/3. Since the Lorenz chaotic system is global Lipschtiz continuous and bounded, one can get the positive constant M = 50 by estimating the ultimate bound of the Lorenz chaotic system.

The duplex network is composed of x-layer and y-layer, and each layer has 100 Lorenz chaotic oscillators, respectively. The x-layer is a BA scale-free network generated from the random graph with the initial condition of m0 = 4. The y-layer is an NW small-world network generated from the ring network with the connection probability of p = 0.1. After fixing the intra-layer topologies, the smallest eigenvalues of Laplacian matrices are easily obtained. Here, , , and the intra-layer inner coupling matrix H = diag{1, 1, 1}.

To measure network synchronization, we define the intra-layer synchronization (sync.) error of x-layer and y-layer:

and the complete synchronization error:

where , and

For further verification of the theoretical results, we select different inter-layer inner coupling matrices Γ for numerical simulations. Specifically, we take d = 0.1, and c = 88 (which satisfies ) for the cases of Γ = diag(1, 1, 0) and Γ = diag(0, 0, 1); for the cases of Γ = diag(−1, −1, −1) and Γ = diag(−1, 1, 1). Figures 2–5 show that each layer in the duplex network converges to the respective synchronous state, which verifies our theoretical results.

Fig. 2. (color online) The intra-layer synchronization error (a) and x variables (b) of time, here Γ = diag{1, 1, 0}, c = 88, and d = 0.1. The small picture (b2) inserted in panel (a) is the magnification of one segment in panel (b1).
Fig. 3. (color online) The intra-layer synchronization error (a) and x variables (b) of time, here Γ = diag{0, 0, 1}, c = 88, and d = 0.1. The small picture (b2) inserted in panel (a) is the magnification of one segment in (b1).
Fig. 4. (color online) The intra-layer synchronization error (a) and x variables (b) of time, here Γ = diag{−1, 1, 1}, c = 91, and d = 0.1. The small picture (b2) inserted in panel (a) is the magnification of one segment in panel (b1).
Fig. 5. (color online) The intra-layer synchronization error (a) and x variables (b) of time, here Γ = diag{−1, −1, −1}, c = 91, and d = 0.1. The small picture (b2) inserted in panel (a) is the magnification of one segment in panel (b1).
Fig. 6. (color online) Synchronization error of time for the duplex network with 50 nodes in each layer, Γ = diag{−1, −1, −1} and different combinations of c and d. EXY, EX, and EY represent complete synchronization, synchronization of x-layer and y-layer, respectively. It indicates that a larger c is necessary for realizing intra-layer synchronization when d increases from 0.1 to 1.
5. Conclusions and discussions

In summary, focusing on the duplex network where there are different topological structures in each layer and different inner coupling functions between, within, and across layers, we have theoretically studied the intra-layer synchronization based on the Lyapunov stability method, and give some synchronization criteria as well as the threshold of coupling strengths within layers. The threshold depends not only on the intra-layer topologies and nodal dynamics, but also on the inter-layer coupling strength. When fixing nodal dynamics and taking a positive-definite matrix as the inner coupling one across layers, the threshold of intra-layer coupling strengths is determined by the intra-layer topologies, i.e., the smaller of nonzero smallest eigenvalues of the intra-layer Laplacians. More interesting, for negative-definite (even indefinite) coupling matrices across layers, the inter-layer coupling strength has an influence on the intra-layer synchronization. In particular, a large inter-layer coupling strength needs a large intra-layer coupling strength to drive the oscillators in each layer into their respective synchronous states. Our results also show that when the nodal dynamics between two layers is the same, the negative-definite (even indefinite) coupling matrix across layers may ensure no inter-layer but mere intra-layer synchronization, coinciding with the result in Ref. [32] analyzed by the master stability function method. Further theoretical investigation on why some coupling matrices across layers can guarantee mere intra-layer synchronization but others cannot is interesting. The results also can work for multiplex (more than two-layer) networks.

However, what happens if the delay or directed topology is taken account into, and what is the influence of delays and directed connections on the intra-layer synchronization? As these problems remain unsolved, it is worth further study.

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