# Eigenvalue Spectrum and Synchronizability of Two Types of Double-Layer Star-Ring Networks with Hybrid Directional Coupling

Authors: Feimei Yang; Zhen Jia; Yang Deng
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

Complex networks are common in daily life, such as social networks [1], urban transportation networks, and virus spread networks. As the study of complex networks deepens, scholars realized that many networks, both in nature and in society, are highly complex and interdependent or interrelation; for example, in the transportation system, road networks, railway networks, aviation networks, and waterway transport networks are relatively independent and interrelated, which together constitute the superlarge network of the complex transportation system. The congestion of a subway may be related to the simultaneous arrival of trains, buses, and other buses and the instantaneous flow of passengers into the subway station. For another example, in human social activities, a network is formed with each individual as the hub, and each individual interacts with each other online or offline to form a complex social network. Therefore, the research on complex networks cannot be limited to simple regular or random single-layer unidirectional networks. In reality, mostly the networks are small-world, scale-free, or even more complex multilayer or directional coupling networks. At present, abundant research results have been achieved in the study of single-layer complex networks [2–9], while multilayer directional networks have become a new frontier research direction of complex networks.Synchronization is a common set phenomenon in nature, such as the chorus of frogs and the simultaneous twinkling of fireflies. At the same time, people use synchronization to solve many problems in medical, physical, economic, and other fields. In recent years, synchronization and its control of chaos have been deeply studied by scholars [10–14]. Although the research on the synchronization of single-layer complex networks has been relatively mature [15–24], some theories and methods of single-layer networks are difficult to be directly applied to multilayer networks, and multilayer networks have richer and more complex dynamics than single-layer networks. Therefore, the synchronization of multilayer networks is a challenging topic.At present, some preliminary results have been achieved in the research of multilayer complex networks synchronizability [25–28]. In [29], Gómez et al. studied the time scale of diffusion process on multiple networks in 2013 proposed the dynamic model of multilayer networks, and constructed the supra-Laplacian matrix, which provided an idea for later scholars to study multiple complex networks. In [28], Xu et al. studied the eigenvalue spectrum and synchronization of two-layer undirected star networks and compared its conclusion with the numerical simulation results of two-layer BA scale-free networks; it was found that the conclusions on the synchronization ability of these two kinds of networks are very similar. In [30], Sun et al. in 2017, respectively, expanded the star networks from leaf nodes to hub nodes and from hub nodes to leaf nodes from three-layer star networks to multilayer star networks, analyzed the eigenvalue spectrum and synchronization of unidirectional coupled star networks, and gave the analytic expression of eigenvalue. In [31], Deng et al. in 2019 discussed the eigenvalue spectrum and synchronization of multiplex chain networks. The 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. It is of more practical significance to investigate the synchronization of multilayer directed or weighted networks. If a double-layer star-ring network is combined with hybrid directional coupling, how does the supra-Laplacian matrix and the synchronizability change? Generally, for the eigenvalues of the multilayer hybrid directional coupling network, it is difficult to strictly derive the theoretical solution, mostly directly through the numerical simulation to explore the synchronization, and little previous research was conducted on the multilayer hybrid directionally coupling network synchronizability; therefore, a better understanding of the two layers of hybrid directionally coupling network to the topology of the synchronizability is needed. In this article, we studied the double-layer hybrid directionally coupling star-ring networks from leaf nodes to hub nodes and from hub nodes to leaf nodes, respectively. According to the network of Laplacian, its eigenvalue is strictly deduced analytically; then, the stability function method is reused to analyze the network structure parameters and their relationship with the network synchronizability. Finally, the correctness of the theoretical analysis was verified by the numerical simulation.The rest of this article is arranged as follows. Section2 explains some preliminaries. In Section 3, the eigenvalue spectrum and synchronizability of double-layer hybrid directionally coupled star-ring Network I are studied. Section 4 studies the eigenvalue spectrum and synchronizability of double-layer hybrid directionally coupled star-ring Network II. Finally, this article is concluded in Section 5.


## 2. Preliminaries

### 2.1. Dynamics Model of Multilayer Networks

In anM-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, 31–34]:(1)x˙iK=fxiK+aK∑j=1NωijKHxjK+∑L=1MdiKLΓxiL,where 1≤i≤N,1≤K≤M; xiK∈Rn is the state variable of the ith node of the Kth layer. f⋅:Rn⟶Rn is the dynamic equation of a single node; aK is the intralayer coupling strength of the Kth layer. Here, WK=ωijK∈RN×N is the inner coupling matrix of the Kth layer. If there is an edge between the ith node and the jth node (j≠i) in the Kth layer, ωijK=1; otherwise, ωijK=0; there is(2)ωiiK=−∑j=1,j≠iNωijK,i,j=1,2,…,N,K=1,2,…,M.Then,LK=−aKWK is the Laplacian matrix of the Kth layer. Here, H⋅:Rn⟶Rn is the intralayer coupling function. diKL is the interlayer coupling strength of the ith node between the Kth layer and the Lth layer. Accordingly, if there is an edge between the ith node of the Kth layer and the ith node of the Lth layer, diKL=1; otherwise, diKL=0, so there is(3)diKK=−∑L=1,L≠KMdiKL,L,K=1,2,…,M.Then,D=diKL∈RM×M is the interlayer negative Laplacian matrix between the kth layer and the Lth layer. Γ⋅:Rn⟶Rn is the interlayer coupling function.LetℒL be the supra-Laplacian matrix of the intralayer and ℒI be the supra-Laplacian matrix of the interlayer; then, the supra-Laplacian matrix ℒ of model (1) can be written as(4)ℒ=ℒL+ℒI,where ℒL represents the direct sum of the intralayer Laplacian matrix and ℒI represents the Kronecker product between the interlayer Laplacian matrix and the N×N identity matrix, so there is(5)ℒL=L10⋯00L2⋯0⋮⋮⋱⋮00⋯LM=⊕K=1MLK,K=1,2,…,M,ℒI=LI⊗IN×N.In short, the supra-Laplacian matrixℒ of network model (1) is determined by the topology of the network, and the dynamic character will be influenced by many factors, such as the interlayer connections, the intralayer connections, and the different coupling directions and coupling strength between intralayers and interlayers. We focused on the effect of topology on synchronization dynamics in this article.


### 2.2. Judgment Method of Network Synchronizability

The synchronous region of model (1) represents the range of eigenvalues that achieve the network synchronizability. According to the master stability function theory (MSF) [35], there are four kinds of synchronous regions in a network generally.(1)
The synchronous region is unboundedα1,+∞, in which case the network synchronizability is determined by the minimum nonzero eigenvalue λ2 of the corresponding supra-Laplacian matrix L. The larger λ2 is, the stronger the synchronizability is.(2)
The synchronous region is boundedα1,α2, and the network synchronizability is determined by the ratio R between the maximum eigenvalue λmax of the matrix and the minimum nonzero eigenvalue λ2. The smaller R=λmax/λ2 is, the stronger the synchronizability is.(3)
The synchronous region is the union of several unconnected regions. In this case, it is difficult to realize the synchronizability.(4)
When the synchronous region is an empty set, it cannot achieve complete synchronizability no matter how the topology and coupling strength change.Since most of the synchronous regions are in the situation of (1) and (2),λ2 and R=λmax/λ2 are mainly used as indicators to investigate the network synchronizatbility when analyzing the synchronization.In this article, we mainly study two types of double-layer star-ring networks, i.e., the double-layer star-ring network with hybrid directional coupling (Network I) from the leaf node to the hub node and the double-layer star-ring network with hybrid directional coupling (Network II) from hub nodes to leaf nodes. The synchronizability of these two networks, in the case that the synchronous region is unbounded and the synchronous region is bounded, is studied, respectively, andλ2 and R=λmax/λ2 are used as indexes of the synchronizability.


## 2.1. Dynamics Model of Multilayer Networks

In anM-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, 31–34]:(1)x˙iK=fxiK+aK∑j=1NωijKHxjK+∑L=1MdiKLΓxiL,where 1≤i≤N,1≤K≤M; xiK∈Rn is the state variable of the ith node of the Kth layer. f⋅:Rn⟶Rn is the dynamic equation of a single node; aK is the intralayer coupling strength of the Kth layer. Here, WK=ωijK∈RN×N is the inner coupling matrix of the Kth layer. If there is an edge between the ith node and the jth node (j≠i) in the Kth layer, ωijK=1; otherwise, ωijK=0; there is(2)ωiiK=−∑j=1,j≠iNωijK,i,j=1,2,…,N,K=1,2,…,M.Then,LK=−aKWK is the Laplacian matrix of the Kth layer. Here, H⋅:Rn⟶Rn is the intralayer coupling function. diKL is the interlayer coupling strength of the ith node between the Kth layer and the Lth layer. Accordingly, if there is an edge between the ith node of the Kth layer and the ith node of the Lth layer, diKL=1; otherwise, diKL=0, so there is(3)diKK=−∑L=1,L≠KMdiKL,L,K=1,2,…,M.Then,D=diKL∈RM×M is the interlayer negative Laplacian matrix between the kth layer and the Lth layer. Γ⋅:Rn⟶Rn is the interlayer coupling function.LetℒL be the supra-Laplacian matrix of the intralayer and ℒI be the supra-Laplacian matrix of the interlayer; then, the supra-Laplacian matrix ℒ of model (1) can be written as(4)ℒ=ℒL+ℒI,where ℒL represents the direct sum of the intralayer Laplacian matrix and ℒI represents the Kronecker product between the interlayer Laplacian matrix and the N×N identity matrix, so there is(5)ℒL=L10⋯00L2⋯0⋮⋮⋱⋮00⋯LM=⊕K=1MLK,K=1,2,…,M,ℒI=LI⊗IN×N.In short, the supra-Laplacian matrixℒ of network model (1) is determined by the topology of the network, and the dynamic character will be influenced by many factors, such as the interlayer connections, the intralayer connections, and the different coupling directions and coupling strength between intralayers and interlayers. We focused on the effect of topology on synchronization dynamics in this article.


## 2.2. Judgment Method of Network Synchronizability

The synchronous region of model (1) represents the range of eigenvalues that achieve the network synchronizability. According to the master stability function theory (MSF) [35], there are four kinds of synchronous regions in a network generally.(1)
The synchronous region is unboundedα1,+∞, in which case the network synchronizability is determined by the minimum nonzero eigenvalue λ2 of the corresponding supra-Laplacian matrix L. The larger λ2 is, the stronger the synchronizability is.(2)
The synchronous region is boundedα1,α2, and the network synchronizability is determined by the ratio R between the maximum eigenvalue λmax of the matrix and the minimum nonzero eigenvalue λ2. The smaller R=λmax/λ2 is, the stronger the synchronizability is.(3)
The synchronous region is the union of several unconnected regions. In this case, it is difficult to realize the synchronizability.(4)
When the synchronous region is an empty set, it cannot achieve complete synchronizability no matter how the topology and coupling strength change.Since most of the synchronous regions are in the situation of (1) and (2),λ2 and R=λmax/λ2 are mainly used as indicators to investigate the network synchronizatbility when analyzing the synchronization.In this article, we mainly study two types of double-layer star-ring networks, i.e., the double-layer star-ring network with hybrid directional coupling (Network I) from the leaf node to the hub node and the double-layer star-ring network with hybrid directional coupling (Network II) from hub nodes to leaf nodes. The synchronizability of these two networks, in the case that the synchronous region is unbounded and the synchronous region is bounded, is studied, respectively, andλ2 and R=λmax/λ2 are used as indexes of the synchronizability.


## 3. The Eigenvalue Spectrum and Synchronizability of Double-Layer Hybrid Directionally Coupled Star-Ring Network I

### 3.1. The Structure Model of Network I

In this article, the structure of Network I model researched is shown in Figure1. It is assumed that the topological structure of each layer is identical, and the interlayer hub nodes and hub nodes, leaf nodes, and leaf nodes of subnetworks are in one-to-one corresponding coupling. Suppose that the connections of the intralayer nodes are directional; that is, leaf nodes point to hub nodes and the directions between leaf nodes and leaf nodes are consistent, while the connections of the interlayer nodes are undirectional. Let the intralayer coupling strength of leaf nodes be a0, the intralayer coupling strength between leaf nodes and hub nodes be a, the interlayer coupling strength of leaf nodes be d, the interlayer coupling strength of hub nodes be d0, and the number of nodes of subnetwork be N (a total of two layers).Figure 1
Schematic diagram of the structural model of Network I.


### 3.2. The Eigenvalue Spectrum and Synchronizability of Network I

First, according to the structure of the Network I model, its supra-Laplacian matrixℒ1 can be written as(6)ℒ1=d000⋯00−d000⋯00−aa+d+a00⋯0−a00−d0⋯00−a−a0a+d+a0⋯0000−d⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮−a00⋯a+d+a00000⋯−d0−a00⋯−a0a+d+a0000⋯0−d−d000⋯00d000⋯000−d0⋯00−aa+d+a00⋯0−a000−d⋯00−a−a0a+d+a0⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯−d0−a00⋯a+d+a00000⋯0−d−a00⋯−a0a+d+a0.According to the determinant propertyABBA=A+BA−B of the block matrix, the characteristic polynomial of the supra-Laplacian matrix ℒ1 of Network I can be derived as follows:(7)λI−ℒ1∼=λ00⋯00aλ−a−a00⋯0a0aa0λ−a−a0⋯00⋮⋮⋮⋱⋮⋮a00⋯λ−a−a00a00⋯a0λ−a−a0N×Nλ−2d000⋯00aλ−a−2d−a00⋯0a0aa0λ−a−2d−a0⋯00⋮⋮⋮⋱⋮⋮a00⋯λ−a−2d−a00a00⋯a0λ−a−2d−a0N×N=λ−12+Na0N−1+λ−a−a0N−1λ−2d0−12+Na0N−1+λ−a−a0−2dN−1.The eigenvalues of Network I are calculated as follows.When the number of subnetwork nodesN is odd,(8)λ=0,2d0,a,…,a︸N−1/2,a+2a0,…,a+2a0︸N−1/2,a+2d,…,a+2d︸N−1/2,a+2a0+2d,…,a+2a0+2d︸N−1/2.When the number of subnetwork nodesN is even,(9)λ=0,2d0,a,…,a︸N−1,a+2d,…,a+2d︸N−1.According to the odd and even case of the number of subnetwork nodesN, there are also two cases when analyzing the Network I synchronizability.(1)
N is odd, where a+2a0,a+2d,a+2a0+2d andaare N−1/2 multiple roots. λmax is the maximum eigenvalue of ℒ1 and λ2 is the minimum nonzero eigenvalue of ℒ1; then, λ2=min2d0,a,(10)R=λmaxλ2=a+2a0+2dmin2d0,a.For the convenience of doing analysis, the relationship betweenλ2,R=λmax/λ2 and structure parameters is summarized in Table 1.According to the MSF theory, when the synchronous region is unbounded, the synchronizability of Network I is determined byλ2. The larger λ2 is, the stronger the synchronizability is. When the synchronous region is bounded, the synchronizability of Network I is determined by R=a+2a0+2d/2d0. The smaller R is, the stronger the synchronizability is. From the expressions of λ2 and R, it can be seen that whether the synchronous region of Network I is unbounded or bounded, the synchronizability is independent of the number of nodes N of the subnetwork in combination with the changes of λ2 and R with the structure parameters in Table 1. Therefore, the relationship between the synchronizability of Network I and the structure parameters is shown in Table 2.As can be seen from Table2, in the case that N is odd, when the synchronous region is unbounded, the synchronizability of Network I is determined by a,d0, and the synchronizability becomes stronger when a,d0 are enlarged. When the synchronous region is bounded, the synchronizability of Network I is related to a,a0,d and d0 but not to the number of subnetwork nodes. When the intralayer coupling strengths from leaf nodes to hub nodes are large, the increase of d0 enhances the synchronizability, whereas the synchronizability will be weakened by increasing a,a0,d. When the intralayer coupling strengths from leaf nodes to hub nodes are small, only the increase of a can strengthen the synchronizability; increasing a0,d will weaken the synchronizability.(2)
N is even. λ2=min2d0,a,λmax=a+2d,R=λmax/λ2=a+2d/min2d0,a. Then, the relationship between λ2,R=λmax/λ2 and the structural parameters are shown in Table 3.According to the MSF theory, when the synchronous region is unbounded, the synchronizability of Network I is determined byλ2. When the synchronous region is bounded, the synchronizability is determined by R. Therefore, the relationships between the synchronizability and a,d,d0 are obtained as shown in Table 4.Table 1
The change ofλ2andR with the increase of a,a0,d,andd0 of Network I when N is odd.
Increase ofaIncrease ofa0Increase ofdIncrease ofd0λ2=min2d0,aa>2d0UnchangedUnchangedUnchangedIncreaseda<2d0IncreasedUnchangedUnchangedUnchangedR=a+2a0+2d/min2d0,aa>2d0IncreasedIncreasedIncreasedDecreaseda<2d0DecreasedIncreasedIncreasedUnchangedTable 2
The change of synchronizability with the increase ofa,a0,d,andd0 of Network I when N is odd.
Increase ofaIncrease ofa0Increase ofdIncrease ofd0The unbounded synchronous regionλ2=2d0UnchangedUnchangedUnchangedStrengthenedλ2=aStrengthenedUnchangedUnchangedUnchangedThe bounded synchronous regionR=a+2a0+2d/2d0WeakenedWeakenedWeakenedStrengthenedR=a+2a0+2d/aStrengthenedWeakenedWeakenedUnchangedTable 3
The change ofλ2andR with the increase of a,d,andd0 of Network I when N is even.
Increase ofaIncrease ofdIncrease ofd0λ2=min2d0,aa>2d0UnchangedUnchangedIncreaseda<2d0IncreasedUnchangedUnchangedR=a+2d/min2d0,aa>2d0IncreasedIncreasedDecreaseda<2d0DecreasedIncreasedUnchangedTable 4
The change of synchronizability with the increase ofa,d,andd0 of Network I when N is even.
Increase ofaIncrease ofdIncrease ofd0The unbounded synchronous regionλ2=2d0UnchangedUnchangedStrengthenedλ2=aStrengthenedUnchangedUnchangedThe bounded synchronous regionR=a+2d/2d0WeakenedWeakenedStrengthenedR=a+2d/aStrengthenedWeakenedUnchangedAs is shown in Table4, when the synchronous region is unbounded, the synchronizability is only related to a,d0, which indicates that the strong coupling strength between the interlayer coupling strength and the intralayer coupling strength cannot determine the synchronizability. When the synchronous region is bounded, the synchronizability is related to a,d,d0. When the interlayer coupling strength between the hub nodes is small, increasing a,d will weaken the synchronizability of the Network I, but the synchronizability with the increase of d0 will strengthen. When the intralayer coupling strength between the leaf nodes is small, increasing a will enhance the synchronizability, while increasing d will weaken the synchronizability.


### 3.3. Numerical Simulation of Network I Synchronizability

In this section, we verify the correctness of the above theoretical results through a large number of numerical simulations and further explore the value of the structural parameters that enable Network I to achieve optimal synchronizability. Design parameters within the limited scope of the theoretical results after many times of experimental simulation were obtained and a group of more representative data was selected, and as long as the design parameters are within the limited range, the value of the control performance is the same. Furthermore, all the numerical simulations in this article directly use the functions in MATLAB software to calculate the supra-Laplacian eigenvalues of the network, which can ensure the efficiency and stability of the algorithm, which is easy to implement because it does not require high time and space complexity.According to the study in the previous sections, for Network I, when the number of subnetwork nodes is odd, the synchronizability is related to the intralayer coupling strength from leaf nodes to hub nodes, the intralayer coupling strength between the leaf nodes, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. When the number of subnetwork nodes is even, the synchronizability is associated with the intralayer coupling strength from the leaf nodes to the hub nodes, the interlayer coupling strength between the leaf nodes, and the interlayer coupling strength between the hub nodes. Whether the number of subnetwork nodes is odd or even, the number of subnetwork nodes does not affect the synchronizability. In the following numerical simulations, we will takeN = 199 (odd) and N = 200 (even) for numerical experiments to verify the above conclusions. All the numerical simulations in this article directly use the functions in MATLAB software to calculate the supra-Laplacian eigenvalues of the network, which can ensure the efficiency and stability of the algorithm.(1)
N is odd. In the numerical simulations, we take N=199,a0=0.5,d=1,andd0=0.1to investigate the change of network synchronizability with the intralayer coupling strength from the leaf nodes to the hub nodes. The simulation results are shown in Figure 2. For the case of the unbounded synchronous region (Figure 2(a)), when a<2d0, λ2 increases with the increase of a, so the synchronizability strengthens with the increase of a. When a reaches a threshold value 2d0, the synchronizability remains unchanged. Then, 2d0 is the optimal parameter when the synchronizability reaches the maximum. For the case of bounded synchronous region (Figure 2(b)), when a<2d0, R becomes less with the increase of a. When a is equal to the threshold value 2d0, R decreases to the minimum; that is, 2d0 is the optimal structural parameter to make the synchronization reach optimal. When a>2d0, R increases slowly with the increase of a. In other words, the synchronizability strengthens at first and then weakens slowly.TakeN=199,a=1,d=1,andd0=0.1 to investigate how the synchronizability changes with the intralayer coupling strength a0 between the leaf nodes for a>2d0, and take N=199,a=0.1,d=1, and d0=0.1 to investigate how the synchronizability changes with the intralayer coupling strength a0 of the leaf nodes for a<2d0, as shown in Figure 3. For the case that the synchronous region is unbounded (Figure 3(a)), when a>2d0, λ2 is unchanged no matter how a0 changes. When a<2d0, λ2 remains unchanged with the increase of a0, indicating that the synchronizability remains unchanged. For the case where the synchronous region is bounded (Figure 3(b)), when a>2d0, R increases rapidly with the increase of a0. When a<2d0, R increases slowly with the increase of a0, and the influence of a on synchronizability is greater than that of 2d0, but the synchronizability is generally weaker. It can be obtained that the smaller the intralayer coupling strength between the leaf nodes is, the better the synchronizability is.As is shown in Figure4, we take N=199,a=1, a0=0.5,d0=0.1 to examine the change of synchronizability with the increase of the interlayer coupling strength d (when a>2d0) between the leaf nodes, and take N=199,a=0.1,a0=0.5,d0=0.1 to investigate the change of synchronizability with the increase of the interlayer coupling strength d (when a<2d0) of the leaf nodes. Figure 4(a) shows that when a>2d0, λ2 is not influenced by the increase of d. When a<2d0, λ2 remains unchanged with the increase of d. Thus, the synchronizability is not affected by d for the unbounded synchronous region. Figure 4(b) displays that when a>2d0, R increases slowly with the increase of d. When a<2d0, R increases rapidly with the increase of d for the bounded synchronization domain. The interlayer coupling strength from the leaf nodes to the hub nodes a has a great influence on the synchronizability, while the synchronizability is weakened with the increase of d. Therefore, a smaller d is favorable for Network I to achieve optimal synchronizability.As shown in Figure5, we take N=199,a=1, a0=0.5,andd=1 to investigate the change of network synchronizability with the increase of the interlayer coupling strength d0 between the hub nodes. In the case where the synchronous region is unbounded (Figure 5(a)), when a>2d0, λ2 increases with the increase of d0, and when d0 reaches the threshold value a/2, λ2 is not affected by d0. In this case, a/2 is the optimal structural parameter that reaches the optimal synchronizability; that is, the synchronizability strengthens at first and then remains unchanged with the increase of d0. In the case where the synchronous region is bounded (Figure 5(b)), when a>2d0, R decreases, with the increase of d0, and when d0 is about a/2, R starts to remain unchanged; that is, a/2 is the optimal structural parameter that makes the synchronizability reach the optimal. Thus, with the increase of d0, the synchronizability at first strengthens and then remains unchanged.(2)
N is even. As is shown in Figure 6, take N=200,a0=0.5,d=1,d0=0.1 to investigate how the synchronizability changes with the increase of the coupling strength a from the leaf nodes to the hub nodes. In the unbounded synchronous region (Figure 6(a)), when a<2d0, λ2 increases with the increase of a, so the synchronizability strengthens with the increase of a. When a reaches the threshold value of 2d0, the synchronizability is invariant, and then 2d0 is the optimal structural parameter with the optimal synchronizability. In the bounded synchronous region (Figure 6(b)), when a<2d0, as a increases, R is increasingly small. When a is equal to the threshold value 2d0, R decreases to the minimum. Therefore, 2d0 is the optimal structural parameter that achieves the optimal network synchronizability. When a>2d0,R slowly increases along with the augment of a, which indicates that the synchronizability strengthens after the first slowly weaken.Figure 2
WhenN is odd, the synchronizability of Network I changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1) (subgraph: λ2 changes with a (a changes from 0.2 to 1)); (b) R changes with a (a changes from 0.01 to 1) (subgraph: R changes with a (a changes from 0.2 to 1)).
(a)(b)Figure 3
WhenN is odd, the synchronizability of Network I changes with the intralayer coupling strength a0 of the leaf nodes. (a) λ2 changes with a0 (a0 changes from 0.01 to 1); (b) R changes with a0 (a0 changes from 0.01 to 1).
(a)(b)Figure 4
WhenN is odd, the synchronizability of Network I changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 3); (b) R changes with d (d changes from 0.01 to 3).
(a)(b)Figure 5
WhenN is odd, the synchronizability of Network I changes with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)Figure 6
WhenN is even, the synchronizability of Network I changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1) (subgraph: λ2 changes with a (a changes from 0.2 to 1)); (b) R changes with a (a changes from 0.01 to 1) (subgraph: R changes with a (a changes from 0.2 to 1)).
(a)(b)We takeN=200,a=1,a0=0.5,d0=0.1 to examine how the synchronizability changes with the increase of the interlayer coupling strength d between the leaf nodes. Take N=200,a=0.1,a0=0.5,d0=0.1 to investigate how the network synchronizability changes with the increase of the interlayer coupling strength d between the leaf nodes, as shown in Figure 7. Figure 7(a) shows that when a>2d0, λ2 remains unchanged, in the increase of d. When a<2d0, λ2 is not affected by d, indicating that the synchronizability remains unchanged, with the unbounded synchronous region. Notice from Figure 7(b) that when a>2d0, R increases slowly with the increase of d; when a<2d0, R increases rapidly with the increase of d, so the synchronizability weakens with the increase of d, with the bounded synchronous region.Figure 7
WhenN is even, the synchronizability of Network I changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 3); (b) R changes with d (d changes from 0.01 to 3).
(a)(b)TakeN=200,a=1,a0=0.5,d=1 to investigate how the synchronizability changes with the increase of the interlayer coupling strength d0 between the hub nodes (Figure 8). When the synchronous region is unbounded (Figure 8(a)), when a>2d0, λ2 increases with the increase of d0, and when d0 reaches the threshold value a/2, the synchronizability remains unchanged; that is, a/2 is the optimal structural parameter to achieve optimal synchronizability, and the synchronizability at first strengthens and then remains unchanged with the increase of d0. In the case where the synchronous region is bounded (Figure 8(b)), when a>2d0, R decreases with the increase of d0; when d0 increases to a/2, it has no influence on R. Therefore, a/2 is the optimal structural parameter to reach the optimal synchronizability; that is, with the increase of d0, the synchronizability at first strengthens and then remains unchanged.Figure 8
WhenN is even, the synchronizability of Network I changes with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)


## 3.1. The Structure Model of Network I

In this article, the structure of Network I model researched is shown in Figure1. It is assumed that the topological structure of each layer is identical, and the interlayer hub nodes and hub nodes, leaf nodes, and leaf nodes of subnetworks are in one-to-one corresponding coupling. Suppose that the connections of the intralayer nodes are directional; that is, leaf nodes point to hub nodes and the directions between leaf nodes and leaf nodes are consistent, while the connections of the interlayer nodes are undirectional. Let the intralayer coupling strength of leaf nodes be a0, the intralayer coupling strength between leaf nodes and hub nodes be a, the interlayer coupling strength of leaf nodes be d, the interlayer coupling strength of hub nodes be d0, and the number of nodes of subnetwork be N (a total of two layers).Figure 1
Schematic diagram of the structural model of Network I.


## 3.2. The Eigenvalue Spectrum and Synchronizability of Network I

First, according to the structure of the Network I model, its supra-Laplacian matrixℒ1 can be written as(6)ℒ1=d000⋯00−d000⋯00−aa+d+a00⋯0−a00−d0⋯00−a−a0a+d+a0⋯0000−d⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮−a00⋯a+d+a00000⋯−d0−a00⋯−a0a+d+a0000⋯0−d−d000⋯00d000⋯000−d0⋯00−aa+d+a00⋯0−a000−d⋯00−a−a0a+d+a0⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯−d0−a00⋯a+d+a00000⋯0−d−a00⋯−a0a+d+a0.According to the determinant propertyABBA=A+BA−B of the block matrix, the characteristic polynomial of the supra-Laplacian matrix ℒ1 of Network I can be derived as follows:(7)λI−ℒ1∼=λ00⋯00aλ−a−a00⋯0a0aa0λ−a−a0⋯00⋮⋮⋮⋱⋮⋮a00⋯λ−a−a00a00⋯a0λ−a−a0N×Nλ−2d000⋯00aλ−a−2d−a00⋯0a0aa0λ−a−2d−a0⋯00⋮⋮⋮⋱⋮⋮a00⋯λ−a−2d−a00a00⋯a0λ−a−2d−a0N×N=λ−12+Na0N−1+λ−a−a0N−1λ−2d0−12+Na0N−1+λ−a−a0−2dN−1.The eigenvalues of Network I are calculated as follows.When the number of subnetwork nodesN is odd,(8)λ=0,2d0,a,…,a︸N−1/2,a+2a0,…,a+2a0︸N−1/2,a+2d,…,a+2d︸N−1/2,a+2a0+2d,…,a+2a0+2d︸N−1/2.When the number of subnetwork nodesN is even,(9)λ=0,2d0,a,…,a︸N−1,a+2d,…,a+2d︸N−1.According to the odd and even case of the number of subnetwork nodesN, there are also two cases when analyzing the Network I synchronizability.(1)
N is odd, where a+2a0,a+2d,a+2a0+2d andaare N−1/2 multiple roots. λmax is the maximum eigenvalue of ℒ1 and λ2 is the minimum nonzero eigenvalue of ℒ1; then, λ2=min2d0,a,(10)R=λmaxλ2=a+2a0+2dmin2d0,a.For the convenience of doing analysis, the relationship betweenλ2,R=λmax/λ2 and structure parameters is summarized in Table 1.According to the MSF theory, when the synchronous region is unbounded, the synchronizability of Network I is determined byλ2. The larger λ2 is, the stronger the synchronizability is. When the synchronous region is bounded, the synchronizability of Network I is determined by R=a+2a0+2d/2d0. The smaller R is, the stronger the synchronizability is. From the expressions of λ2 and R, it can be seen that whether the synchronous region of Network I is unbounded or bounded, the synchronizability is independent of the number of nodes N of the subnetwork in combination with the changes of λ2 and R with the structure parameters in Table 1. Therefore, the relationship between the synchronizability of Network I and the structure parameters is shown in Table 2.As can be seen from Table2, in the case that N is odd, when the synchronous region is unbounded, the synchronizability of Network I is determined by a,d0, and the synchronizability becomes stronger when a,d0 are enlarged. When the synchronous region is bounded, the synchronizability of Network I is related to a,a0,d and d0 but not to the number of subnetwork nodes. When the intralayer coupling strengths from leaf nodes to hub nodes are large, the increase of d0 enhances the synchronizability, whereas the synchronizability will be weakened by increasing a,a0,d. When the intralayer coupling strengths from leaf nodes to hub nodes are small, only the increase of a can strengthen the synchronizability; increasing a0,d will weaken the synchronizability.(2)
N is even. λ2=min2d0,a,λmax=a+2d,R=λmax/λ2=a+2d/min2d0,a. Then, the relationship between λ2,R=λmax/λ2 and the structural parameters are shown in Table 3.According to the MSF theory, when the synchronous region is unbounded, the synchronizability of Network I is determined byλ2. When the synchronous region is bounded, the synchronizability is determined by R. Therefore, the relationships between the synchronizability and a,d,d0 are obtained as shown in Table 4.Table 1
The change ofλ2andR with the increase of a,a0,d,andd0 of Network I when N is odd.
Increase ofaIncrease ofa0Increase ofdIncrease ofd0λ2=min2d0,aa>2d0UnchangedUnchangedUnchangedIncreaseda<2d0IncreasedUnchangedUnchangedUnchangedR=a+2a0+2d/min2d0,aa>2d0IncreasedIncreasedIncreasedDecreaseda<2d0DecreasedIncreasedIncreasedUnchangedTable 2
The change of synchronizability with the increase ofa,a0,d,andd0 of Network I when N is odd.
Increase ofaIncrease ofa0Increase ofdIncrease ofd0The unbounded synchronous regionλ2=2d0UnchangedUnchangedUnchangedStrengthenedλ2=aStrengthenedUnchangedUnchangedUnchangedThe bounded synchronous regionR=a+2a0+2d/2d0WeakenedWeakenedWeakenedStrengthenedR=a+2a0+2d/aStrengthenedWeakenedWeakenedUnchangedTable 3
The change ofλ2andR with the increase of a,d,andd0 of Network I when N is even.
Increase ofaIncrease ofdIncrease ofd0λ2=min2d0,aa>2d0UnchangedUnchangedIncreaseda<2d0IncreasedUnchangedUnchangedR=a+2d/min2d0,aa>2d0IncreasedIncreasedDecreaseda<2d0DecreasedIncreasedUnchangedTable 4
The change of synchronizability with the increase ofa,d,andd0 of Network I when N is even.
Increase ofaIncrease ofdIncrease ofd0The unbounded synchronous regionλ2=2d0UnchangedUnchangedStrengthenedλ2=aStrengthenedUnchangedUnchangedThe bounded synchronous regionR=a+2d/2d0WeakenedWeakenedStrengthenedR=a+2d/aStrengthenedWeakenedUnchangedAs is shown in Table4, when the synchronous region is unbounded, the synchronizability is only related to a,d0, which indicates that the strong coupling strength between the interlayer coupling strength and the intralayer coupling strength cannot determine the synchronizability. When the synchronous region is bounded, the synchronizability is related to a,d,d0. When the interlayer coupling strength between the hub nodes is small, increasing a,d will weaken the synchronizability of the Network I, but the synchronizability with the increase of d0 will strengthen. When the intralayer coupling strength between the leaf nodes is small, increasing a will enhance the synchronizability, while increasing d will weaken the synchronizability.


## 3.3. Numerical Simulation of Network I Synchronizability

In this section, we verify the correctness of the above theoretical results through a large number of numerical simulations and further explore the value of the structural parameters that enable Network I to achieve optimal synchronizability. Design parameters within the limited scope of the theoretical results after many times of experimental simulation were obtained and a group of more representative data was selected, and as long as the design parameters are within the limited range, the value of the control performance is the same. Furthermore, all the numerical simulations in this article directly use the functions in MATLAB software to calculate the supra-Laplacian eigenvalues of the network, which can ensure the efficiency and stability of the algorithm, which is easy to implement because it does not require high time and space complexity.According to the study in the previous sections, for Network I, when the number of subnetwork nodes is odd, the synchronizability is related to the intralayer coupling strength from leaf nodes to hub nodes, the intralayer coupling strength between the leaf nodes, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. When the number of subnetwork nodes is even, the synchronizability is associated with the intralayer coupling strength from the leaf nodes to the hub nodes, the interlayer coupling strength between the leaf nodes, and the interlayer coupling strength between the hub nodes. Whether the number of subnetwork nodes is odd or even, the number of subnetwork nodes does not affect the synchronizability. In the following numerical simulations, we will takeN = 199 (odd) and N = 200 (even) for numerical experiments to verify the above conclusions. All the numerical simulations in this article directly use the functions in MATLAB software to calculate the supra-Laplacian eigenvalues of the network, which can ensure the efficiency and stability of the algorithm.(1)
N is odd. In the numerical simulations, we take N=199,a0=0.5,d=1,andd0=0.1to investigate the change of network synchronizability with the intralayer coupling strength from the leaf nodes to the hub nodes. The simulation results are shown in Figure 2. For the case of the unbounded synchronous region (Figure 2(a)), when a<2d0, λ2 increases with the increase of a, so the synchronizability strengthens with the increase of a. When a reaches a threshold value 2d0, the synchronizability remains unchanged. Then, 2d0 is the optimal parameter when the synchronizability reaches the maximum. For the case of bounded synchronous region (Figure 2(b)), when a<2d0, R becomes less with the increase of a. When a is equal to the threshold value 2d0, R decreases to the minimum; that is, 2d0 is the optimal structural parameter to make the synchronization reach optimal. When a>2d0, R increases slowly with the increase of a. In other words, the synchronizability strengthens at first and then weakens slowly.TakeN=199,a=1,d=1,andd0=0.1 to investigate how the synchronizability changes with the intralayer coupling strength a0 between the leaf nodes for a>2d0, and take N=199,a=0.1,d=1, and d0=0.1 to investigate how the synchronizability changes with the intralayer coupling strength a0 of the leaf nodes for a<2d0, as shown in Figure 3. For the case that the synchronous region is unbounded (Figure 3(a)), when a>2d0, λ2 is unchanged no matter how a0 changes. When a<2d0, λ2 remains unchanged with the increase of a0, indicating that the synchronizability remains unchanged. For the case where the synchronous region is bounded (Figure 3(b)), when a>2d0, R increases rapidly with the increase of a0. When a<2d0, R increases slowly with the increase of a0, and the influence of a on synchronizability is greater than that of 2d0, but the synchronizability is generally weaker. It can be obtained that the smaller the intralayer coupling strength between the leaf nodes is, the better the synchronizability is.As is shown in Figure4, we take N=199,a=1, a0=0.5,d0=0.1 to examine the change of synchronizability with the increase of the interlayer coupling strength d (when a>2d0) between the leaf nodes, and take N=199,a=0.1,a0=0.5,d0=0.1 to investigate the change of synchronizability with the increase of the interlayer coupling strength d (when a<2d0) of the leaf nodes. Figure 4(a) shows that when a>2d0, λ2 is not influenced by the increase of d. When a<2d0, λ2 remains unchanged with the increase of d. Thus, the synchronizability is not affected by d for the unbounded synchronous region. Figure 4(b) displays that when a>2d0, R increases slowly with the increase of d. When a<2d0, R increases rapidly with the increase of d for the bounded synchronization domain. The interlayer coupling strength from the leaf nodes to the hub nodes a has a great influence on the synchronizability, while the synchronizability is weakened with the increase of d. Therefore, a smaller d is favorable for Network I to achieve optimal synchronizability.As shown in Figure5, we take N=199,a=1, a0=0.5,andd=1 to investigate the change of network synchronizability with the increase of the interlayer coupling strength d0 between the hub nodes. In the case where the synchronous region is unbounded (Figure 5(a)), when a>2d0, λ2 increases with the increase of d0, and when d0 reaches the threshold value a/2, λ2 is not affected by d0. In this case, a/2 is the optimal structural parameter that reaches the optimal synchronizability; that is, the synchronizability strengthens at first and then remains unchanged with the increase of d0. In the case where the synchronous region is bounded (Figure 5(b)), when a>2d0, R decreases, with the increase of d0, and when d0 is about a/2, R starts to remain unchanged; that is, a/2 is the optimal structural parameter that makes the synchronizability reach the optimal. Thus, with the increase of d0, the synchronizability at first strengthens and then remains unchanged.(2)
N is even. As is shown in Figure 6, take N=200,a0=0.5,d=1,d0=0.1 to investigate how the synchronizability changes with the increase of the coupling strength a from the leaf nodes to the hub nodes. In the unbounded synchronous region (Figure 6(a)), when a<2d0, λ2 increases with the increase of a, so the synchronizability strengthens with the increase of a. When a reaches the threshold value of 2d0, the synchronizability is invariant, and then 2d0 is the optimal structural parameter with the optimal synchronizability. In the bounded synchronous region (Figure 6(b)), when a<2d0, as a increases, R is increasingly small. When a is equal to the threshold value 2d0, R decreases to the minimum. Therefore, 2d0 is the optimal structural parameter that achieves the optimal network synchronizability. When a>2d0,R slowly increases along with the augment of a, which indicates that the synchronizability strengthens after the first slowly weaken.Figure 2
WhenN is odd, the synchronizability of Network I changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1) (subgraph: λ2 changes with a (a changes from 0.2 to 1)); (b) R changes with a (a changes from 0.01 to 1) (subgraph: R changes with a (a changes from 0.2 to 1)).
(a)(b)Figure 3
WhenN is odd, the synchronizability of Network I changes with the intralayer coupling strength a0 of the leaf nodes. (a) λ2 changes with a0 (a0 changes from 0.01 to 1); (b) R changes with a0 (a0 changes from 0.01 to 1).
(a)(b)Figure 4
WhenN is odd, the synchronizability of Network I changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 3); (b) R changes with d (d changes from 0.01 to 3).
(a)(b)Figure 5
WhenN is odd, the synchronizability of Network I changes with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)Figure 6
WhenN is even, the synchronizability of Network I changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1) (subgraph: λ2 changes with a (a changes from 0.2 to 1)); (b) R changes with a (a changes from 0.01 to 1) (subgraph: R changes with a (a changes from 0.2 to 1)).
(a)(b)We takeN=200,a=1,a0=0.5,d0=0.1 to examine how the synchronizability changes with the increase of the interlayer coupling strength d between the leaf nodes. Take N=200,a=0.1,a0=0.5,d0=0.1 to investigate how the network synchronizability changes with the increase of the interlayer coupling strength d between the leaf nodes, as shown in Figure 7. Figure 7(a) shows that when a>2d0, λ2 remains unchanged, in the increase of d. When a<2d0, λ2 is not affected by d, indicating that the synchronizability remains unchanged, with the unbounded synchronous region. Notice from Figure 7(b) that when a>2d0, R increases slowly with the increase of d; when a<2d0, R increases rapidly with the increase of d, so the synchronizability weakens with the increase of d, with the bounded synchronous region.Figure 7
WhenN is even, the synchronizability of Network I changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 3); (b) R changes with d (d changes from 0.01 to 3).
(a)(b)TakeN=200,a=1,a0=0.5,d=1 to investigate how the synchronizability changes with the increase of the interlayer coupling strength d0 between the hub nodes (Figure 8). When the synchronous region is unbounded (Figure 8(a)), when a>2d0, λ2 increases with the increase of d0, and when d0 reaches the threshold value a/2, the synchronizability remains unchanged; that is, a/2 is the optimal structural parameter to achieve optimal synchronizability, and the synchronizability at first strengthens and then remains unchanged with the increase of d0. In the case where the synchronous region is bounded (Figure 8(b)), when a>2d0, R decreases with the increase of d0; when d0 increases to a/2, it has no influence on R. Therefore, a/2 is the optimal structural parameter to reach the optimal synchronizability; that is, with the increase of d0, the synchronizability at first strengthens and then remains unchanged.Figure 8
WhenN is even, the synchronizability of Network I changes with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)


## 4. The Eigenvalue Spectrum and Synchronizability of Double-Layer Hybrid Directionally Coupled Star-Ring Network II

### 4.1. The Structure Model of Network II

The structure model of Network II studied in this section is shown in Figure9. It is assumed that the topological structure of each layer of the double-layer hybrid directional coupled star-ring network is identical, and the interlayer hub nodes and hub nodes, the leaf nodes, and the leaf nodes of the double-layer subnetwork are in one-to-one coupling. Provided that the connections between the intralayer nodes are directional; that is, from the hub nodes to the leaf nodes, the intralayer direction between the leaf nodes and the leaf nodes is consistent, while the interlayer connection is undirected. Let the intralayer coupling strength between the leaf nodes bea0, the intralayer coupling strength from the hub nodes to the leaf nodes be a, the interlayer coupling strength between the leaf nodes be d, the interlayer coupling strength between the hub nodes be d0, and the number of nodes in each layer subnetwork be N (a total of two layers of the Network II).Figure 9
Schematic diagram of the structural model of Network II.


### 4.2. The Eigenvalue Spectrum and Synchronizability of Network II

According to the structural model of Network II, the supra-Laplacian matrixℒ2 of its network is as follows:(11)ℒ2=d0+N−1a−a−a⋯−a−a−d000⋯000d+a00⋯0−a00−d0⋯000−a0d+a0⋯0000−d⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯d+a00000⋯−d0000⋯−a0d+a0000⋯0−d−d000⋯00d0+N−1a−a−a⋯−a−a0−d0⋯000d+a00⋯0−a000−d⋯000−a0d+a0⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯−d0000⋯d+a00000⋯0−d000⋯−a0d+a0.It can also be deduced that the characteristic polynomial of Network II is(12)λI−ℒ2∼=λ−N−1aaa⋯aa0λ−a00⋯0a00a0λ−a0⋯00⋮⋮⋮⋱⋮⋮000⋯λ−a00000⋯a0λ−a0N×Nλ−2d0−N−1aaa⋯aa0λ−2d−a00⋯0a00a0λ−2d−a0⋯00⋮⋮⋮⋱⋮⋮000⋯λ−2d−a00000⋯a0λ−2d−a0N×N=λ−N−1a−1Na0N−1+λ−a0N−1λ−2d0−N−1a−1Na0N−1+λ−2d−a0N−1.Therefore, whenN is odd, the eigenvalues λ of ℒ2 are 0,…,0︸N−1/2,2a0,…,2a0︸N−1/2,N−1a,2d+2a0,…,2d+2a0︸N−1/22d,…,2d︸N−1/2,2d0+N−1a. When N is even, the eigenvalues λ of ℒ2 are 0,…,0︸N−1,2d,…,2d︸N−1,N−1a,2d0+N−1a.The eigenvalues of the supra-Laplacian matrixℒ2 of Network II are different because of the odd and even number of subnetwork nodes. Here, we analyze the synchronizability of Network II according to the odd and even number of subnetwork nodes.(1)
N is odd. λ2=min2a0,2d, R=λmax/λ2=2d0+N−1a/min2a0,2d. Variations ofλ2,R of Network II are summarized in Table 5.According to the MSF theory, when the synchronous region is unbounded, the synchronizability is determined byλ2=min2a0,2d. The larger λ2 is, the stronger the synchronizability is. When the synchronous region is bounded, the synchronizability of Network II is determined by R=2d0+N−1a/min2a0,2d. The smaller R is, the stronger the synchronizability is. The relationship between the synchronizability and the structural parameters is shown in Table 6.It can be seen from Table6 that when the synchronous region is unbounded, the synchronizability is related to a0,d, and with the increase of a0,d, the synchronizability will strengthen. When the synchronous region is bounded, the synchronizability is related to a0. When the intralayer coupling strength between the leaf node is small, the increase of a0 enhances the synchronizability, while the increase of N,a,d0 weakens the synchronizability. When the interlayer coupling strength between the leaf node is weak, the synchronizability strengthens with the increase of d but weakens with the increase of N,a,d0.(2)
N is even. λ2=2d,λmax=2d0+N−1a, R=λmax/λ2=2d0+N−1a/2d.The variation ofλ2,R of Network II is concluded in Table 7.According to the MSF theory, when the network synchronous region is unbounded, the synchronizability is determined byλ2=2d. When the network synchronous region is bounded, the network synchronizability of Network II is determined by R=2d0+N−1a/2d. Whether the synchronous region is unbounded or bounded is independent of the intralayer coupling strength between leaf nodes a0.Table 5
The change ofλ2andR with the increase of N,a,a0,d,andd0 of Network II when N is odd.
Increase ofNIncrease ofaIncrease ofa0Increase ofdIncrease ofd0λ2=min2d,2a0d>a0UnchangedUnchangedIncreasedUnchangedUnchangedd<a0UnchangedUnchangedUnchangedIncreasedUnchangedR=2d0+N−1a/min2d,2a0d>a0IncreasedIncreasedDecreasedUnchangedIncreasedd<a0IncreasedIncreasedUnchangedDecreasedIncreasedTable 6
The change of synchronizability with the increase ofN,a,a0,d,andd0 of Network II when N is odd.
Increase ofNIncrease ofaIncrease ofa0Increase ofdIncrease ofd0The unbounded synchronous regionλ2=2a0UnchangedUnchangedStrengthenedUnchangedUnchangedλ2=2dUnchangedUnchangedUnchangedStrengthenedUnchangedThe bounded synchronous regionR=2d0+N−1a/2a0WeakenedWeakenedStrengthenedUnchangedWeakenedR=2d0+N−1a/2dWeakenedWeakenedUnchangedStrengthenedWeakenedTable 7
The change ofλ2andR with N,a,d,andd0 of Network II when N is even.
Increase ofNIncrease ofaIncrease ofdIncrease ofd0λ2=2dUnchangedUnchangedIncreasedUnchangedR=2d0+N−1a/2dIncreaseIncreasedDecreasedIncreasedTherefore, the synchronizability is enhanced with the increase ofd in the unbounded synchronous region. The increase of d enhances the synchronizability of Network II, but the increase of N,a,d0 weakens the synchronizability in the bounded synchronous region.


### 4.3. Numerical Simulation of Network II Synchronizability

The section on the theoretical results shows that, in Network II, when the number of nodes in each layer of subnetwork is odd, the synchronizability is related to the number of subnetwork nodes, the intralayer coupling strength from the hub node to the leaf node, and the intralayer coupling strength between the leaf nodes, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. When the number of subnetwork nodes is even, the synchronizability is related to the number of subnetwork nodes, the intralayer coupling strength between the hub node and the leaf node, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. In the following numerical simulations, we will takeN = 199 (odd) and N = 200 (even) separately to carry out numerical experiments to verify the above theoretical results.(1)
N is odd. Take a=1,a0=0.5,d=1,d0=0.1 to examine the synchronizability of Network II with the change of the number of subnetwork nodes N for a0>d, and take a=1,a0=0.5,d=0.3,d0=0.1 to investigate the synchronizability as the number of subnetwork nodes N changes for a0<d, as shown in Figure 10. It can be seen from Figure 10 that for the case of the unbounded synchronous region (Figure 10(a)), when a0>d, λ2 is unchanged with the increase of N. When a0<d, λ2 remains unchanged with the increase of N; then, the synchronizability is invariant with the increase of N. For the case of the bounded synchronous region (Figure 10(b)), R increases with the increase of N in both a0>d and a0<d, indicating that the greater the number of subnetwork nodes, the more unfavorable the synchronizability.TakeN=199,a0=0.5,d=1,d0=0.1 to investigate the synchronizability with the change of the intralayer coupling strength a from the hub nodes to the leaf nodes, and takeN=199,a0=0.5,d=0.3,d0=0.1 to investigate the change in synchronizability with the intralayer coupling strength a from the hub nodes to the leaf nodes, as shown in Figure 11. Figure 11(a) shows that neither a0>d nor a0<d, λ2 is not affected by a, indicating that a has no influence on the synchronizability, in the unbounded synchronous region. Figure 11(b) displays that when the synchronous region is bounded, R increases in both a0>d and a0<d with the increase of a. Moreover, the interlayer coupling strength between the leaf nodes d has a great influence on the synchronizability, which makes the synchronizability weaken faster. The synchronizability weakens with the increase of a, and the larger a is, the more unfavorable the synchronizability is.Notice from Figure12 (N=199,a=1,d=1, d0=0.1) how the synchronizability changes with the intralayer coupling strength a0 between the leaf nodes. For the case of the unbounded synchronous region (Figure 12(a)), when a0=d, λ2 increases with the increase of a0, and when a0 increases to the threshold value d, λ2 remains unchanged with the increase of a0. In this case, d is the optimal structural parameter that enables Network II to reach the optimal synchronizability. The synchronizability at first strengthens and then remains unchanged with the increase of a0. For the case of the bounded synchronous region (Figure 12(b)), when a0=d, R rapidly decreases with the increase of a0; when a0 increases to the threshold value d, R starting along with the increase of a0 remains invariant; thus, the threshold value d is to achieve the optimal synchronizability when the structure parameters are optimal. Therefore, with the increase of a0, the synchronizability strengthens before remaining invariant.As shown in Figure13, take N=199,a=1,a0=0.5,d0=0.1 to examine the synchronizability with the change of the interlayer coupling strength d between the leaf nodes. For the case of the unbounded synchronous region (Figure 13(a)), when a0>d, λ2 increases with the increase of d. When d increases to the threshold value a0, λ2 remains unchanged with the increase of d; at this time, d is the optimal structural parameter to reach the optimal synchronizability. It shows that the synchronizability strengthens at first and then remains unchanged with the increase of d. For the case of the bounded synchronous region (Figure 13(b)), when a0>d, R decreases rapidly with the increase of d. When d increases to the threshold value a0, R does not change with the increase of d. The threshold value a0 is the optimal structural parameter for optimal synchronizability. It is concluded that, with the increase of d, the synchronizability will strengthen at first and then remains unchanged.The simulation results are shown in Figure14. Take N=199,a=1,a0=0.5,d=1 to investigate how the synchronizability changes with the interlayer coupling strength d0 (when a0>d) between the hub nodes. Take N=199,a=1,a0=0.5,d=0.3 to investigate how the synchronizability changes with the interlayer coupling strength d0 (when a0<d) between the hub nodes. In the unbounded synchronous region (Figure 14(a)), either a0>d or a0<d, and λ2 are basically unchanged with the increase of d0, indicating that the synchronizability is unchanged with the increase of d0. In the bounded synchronous region (Figure 14(b)), R increases slowly with the increase of d0 in both a0>d and a0<d. Therefore, the synchronizability weakens with the increase of d0.(2)
N is even. For Network II with the even of the subnetwork nodes, it is found that the synchronizability is influenced by the number of subnetwork nodes, the intralayer coupling strength from the hub node to the leaf node, the interlayer coupling strength between the leaf nodes, and the interlayer coupling strength between the hub nodes, which are the same as those when the number of the subnetwork nodes is odd for a0>d, namely, Figure 10 (the red line) and Figure 15, Figure 11 (the red line) and Figure 16, Figure 13 (when d<a0) and Figure 17, and Figure 14 (the red line) and Figure 18. The synchronizability strengthens with the increase of d in the unbounded synchronous region. The synchronizability is enhanced with the increase of d but weakens with the increase of N,a,d0, in the bounded synchronous region. Moreover, the intralayer coupling strength a0 between the leaf nodes has no effect on the synchronizability when N is even. At this time, the edges between the intralayer leaf nodes are removed, which is the two-layer unidirectional coupled star network from the hub nodes to the leaf node. The above conclusions are the same as the simulation results of the star network with unidirectional coupling between the three-layer hub nodes and the leaf node in [30].Figure 10
WhenN is odd, the synchronizability of Network II changes with the number of the subnetwork nodes N. (a) λ2 changes with N (N changes from 9 to 599); (b) R changes with N (N changes from 9 to 599).
(a)(b)Figure 11
WhenN is odd, the synchronizability of Network II changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1); (b) R changes with a (a changes from 0.01 to 1).
(a)(b)Figure 12
WhenN is odd, the synchronizability of Network II changes with the intralayer coupling strength a0 of the leaf nodes. (a) λ2 changes with a0 (a0 changes from 0.01 to 2); (b) R changes with a0 (a0 changes from 0.01 to 2).
(a)(b)Figure 13
WhenN is odd, the synchronizability of Network II changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 1); (b) R changes with d (d changes from 0.01 to 1).
(a)(b)Figure 14
WhenN is odd, the synchronizability of Network II varies with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)Figure 15
WhenN is even, the synchronizability of Network II changes with the number of subnetwork nodes N. (a) λ2 changes with N (N changes from 10 to 600); (b) R changes with N (N changes from 10 to 600).
(a)(b)Figure 16
WhenN is even, the synchronizability of Network II changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1); (b) R changes with a (a changes from 0.01 to 1).
(a)(b)Figure 17
WhenN is even, the synchronizability of Network II changes with the interlayer coupling strength d between the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 1); (b) R changes with d (d changes from 0.01 to 1).
(a)(b)Figure 18
WhenN is even, the synchronizability of Network II varies with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)


## 4.1. The Structure Model of Network II

The structure model of Network II studied in this section is shown in Figure9. It is assumed that the topological structure of each layer of the double-layer hybrid directional coupled star-ring network is identical, and the interlayer hub nodes and hub nodes, the leaf nodes, and the leaf nodes of the double-layer subnetwork are in one-to-one coupling. Provided that the connections between the intralayer nodes are directional; that is, from the hub nodes to the leaf nodes, the intralayer direction between the leaf nodes and the leaf nodes is consistent, while the interlayer connection is undirected. Let the intralayer coupling strength between the leaf nodes bea0, the intralayer coupling strength from the hub nodes to the leaf nodes be a, the interlayer coupling strength between the leaf nodes be d, the interlayer coupling strength between the hub nodes be d0, and the number of nodes in each layer subnetwork be N (a total of two layers of the Network II).Figure 9
Schematic diagram of the structural model of Network II.


## 4.2. The Eigenvalue Spectrum and Synchronizability of Network II

According to the structural model of Network II, the supra-Laplacian matrixℒ2 of its network is as follows:(11)ℒ2=d0+N−1a−a−a⋯−a−a−d000⋯000d+a00⋯0−a00−d0⋯000−a0d+a0⋯0000−d⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯d+a00000⋯−d0000⋯−a0d+a0000⋯0−d−d000⋯00d0+N−1a−a−a⋯−a−a0−d0⋯000d+a00⋯0−a000−d⋯000−a0d+a0⋯00⋮⋮⋮⋱⋮⋮⋮⋮⋮⋱⋮⋮000⋯−d0000⋯d+a00000⋯0−d000⋯−a0d+a0.It can also be deduced that the characteristic polynomial of Network II is(12)λI−ℒ2∼=λ−N−1aaa⋯aa0λ−a00⋯0a00a0λ−a0⋯00⋮⋮⋮⋱⋮⋮000⋯λ−a00000⋯a0λ−a0N×Nλ−2d0−N−1aaa⋯aa0λ−2d−a00⋯0a00a0λ−2d−a0⋯00⋮⋮⋮⋱⋮⋮000⋯λ−2d−a00000⋯a0λ−2d−a0N×N=λ−N−1a−1Na0N−1+λ−a0N−1λ−2d0−N−1a−1Na0N−1+λ−2d−a0N−1.Therefore, whenN is odd, the eigenvalues λ of ℒ2 are 0,…,0︸N−1/2,2a0,…,2a0︸N−1/2,N−1a,2d+2a0,…,2d+2a0︸N−1/22d,…,2d︸N−1/2,2d0+N−1a. When N is even, the eigenvalues λ of ℒ2 are 0,…,0︸N−1,2d,…,2d︸N−1,N−1a,2d0+N−1a.The eigenvalues of the supra-Laplacian matrixℒ2 of Network II are different because of the odd and even number of subnetwork nodes. Here, we analyze the synchronizability of Network II according to the odd and even number of subnetwork nodes.(1)
N is odd. λ2=min2a0,2d, R=λmax/λ2=2d0+N−1a/min2a0,2d. Variations ofλ2,R of Network II are summarized in Table 5.According to the MSF theory, when the synchronous region is unbounded, the synchronizability is determined byλ2=min2a0,2d. The larger λ2 is, the stronger the synchronizability is. When the synchronous region is bounded, the synchronizability of Network II is determined by R=2d0+N−1a/min2a0,2d. The smaller R is, the stronger the synchronizability is. The relationship between the synchronizability and the structural parameters is shown in Table 6.It can be seen from Table6 that when the synchronous region is unbounded, the synchronizability is related to a0,d, and with the increase of a0,d, the synchronizability will strengthen. When the synchronous region is bounded, the synchronizability is related to a0. When the intralayer coupling strength between the leaf node is small, the increase of a0 enhances the synchronizability, while the increase of N,a,d0 weakens the synchronizability. When the interlayer coupling strength between the leaf node is weak, the synchronizability strengthens with the increase of d but weakens with the increase of N,a,d0.(2)
N is even. λ2=2d,λmax=2d0+N−1a, R=λmax/λ2=2d0+N−1a/2d.The variation ofλ2,R of Network II is concluded in Table 7.According to the MSF theory, when the network synchronous region is unbounded, the synchronizability is determined byλ2=2d. When the network synchronous region is bounded, the network synchronizability of Network II is determined by R=2d0+N−1a/2d. Whether the synchronous region is unbounded or bounded is independent of the intralayer coupling strength between leaf nodes a0.Table 5
The change ofλ2andR with the increase of N,a,a0,d,andd0 of Network II when N is odd.
Increase ofNIncrease ofaIncrease ofa0Increase ofdIncrease ofd0λ2=min2d,2a0d>a0UnchangedUnchangedIncreasedUnchangedUnchangedd<a0UnchangedUnchangedUnchangedIncreasedUnchangedR=2d0+N−1a/min2d,2a0d>a0IncreasedIncreasedDecreasedUnchangedIncreasedd<a0IncreasedIncreasedUnchangedDecreasedIncreasedTable 6
The change of synchronizability with the increase ofN,a,a0,d,andd0 of Network II when N is odd.
Increase ofNIncrease ofaIncrease ofa0Increase ofdIncrease ofd0The unbounded synchronous regionλ2=2a0UnchangedUnchangedStrengthenedUnchangedUnchangedλ2=2dUnchangedUnchangedUnchangedStrengthenedUnchangedThe bounded synchronous regionR=2d0+N−1a/2a0WeakenedWeakenedStrengthenedUnchangedWeakenedR=2d0+N−1a/2dWeakenedWeakenedUnchangedStrengthenedWeakenedTable 7
The change ofλ2andR with N,a,d,andd0 of Network II when N is even.
Increase ofNIncrease ofaIncrease ofdIncrease ofd0λ2=2dUnchangedUnchangedIncreasedUnchangedR=2d0+N−1a/2dIncreaseIncreasedDecreasedIncreasedTherefore, the synchronizability is enhanced with the increase ofd in the unbounded synchronous region. The increase of d enhances the synchronizability of Network II, but the increase of N,a,d0 weakens the synchronizability in the bounded synchronous region.


## 4.3. Numerical Simulation of Network II Synchronizability

The section on the theoretical results shows that, in Network II, when the number of nodes in each layer of subnetwork is odd, the synchronizability is related to the number of subnetwork nodes, the intralayer coupling strength from the hub node to the leaf node, and the intralayer coupling strength between the leaf nodes, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. When the number of subnetwork nodes is even, the synchronizability is related to the number of subnetwork nodes, the intralayer coupling strength between the hub node and the leaf node, the interlayer coupling strength between the hub nodes, and the interlayer coupling strength between the leaf nodes. In the following numerical simulations, we will takeN = 199 (odd) and N = 200 (even) separately to carry out numerical experiments to verify the above theoretical results.(1)
N is odd. Take a=1,a0=0.5,d=1,d0=0.1 to examine the synchronizability of Network II with the change of the number of subnetwork nodes N for a0>d, and take a=1,a0=0.5,d=0.3,d0=0.1 to investigate the synchronizability as the number of subnetwork nodes N changes for a0<d, as shown in Figure 10. It can be seen from Figure 10 that for the case of the unbounded synchronous region (Figure 10(a)), when a0>d, λ2 is unchanged with the increase of N. When a0<d, λ2 remains unchanged with the increase of N; then, the synchronizability is invariant with the increase of N. For the case of the bounded synchronous region (Figure 10(b)), R increases with the increase of N in both a0>d and a0<d, indicating that the greater the number of subnetwork nodes, the more unfavorable the synchronizability.TakeN=199,a0=0.5,d=1,d0=0.1 to investigate the synchronizability with the change of the intralayer coupling strength a from the hub nodes to the leaf nodes, and takeN=199,a0=0.5,d=0.3,d0=0.1 to investigate the change in synchronizability with the intralayer coupling strength a from the hub nodes to the leaf nodes, as shown in Figure 11. Figure 11(a) shows that neither a0>d nor a0<d, λ2 is not affected by a, indicating that a has no influence on the synchronizability, in the unbounded synchronous region. Figure 11(b) displays that when the synchronous region is bounded, R increases in both a0>d and a0<d with the increase of a. Moreover, the interlayer coupling strength between the leaf nodes d has a great influence on the synchronizability, which makes the synchronizability weaken faster. The synchronizability weakens with the increase of a, and the larger a is, the more unfavorable the synchronizability is.Notice from Figure12 (N=199,a=1,d=1, d0=0.1) how the synchronizability changes with the intralayer coupling strength a0 between the leaf nodes. For the case of the unbounded synchronous region (Figure 12(a)), when a0=d, λ2 increases with the increase of a0, and when a0 increases to the threshold value d, λ2 remains unchanged with the increase of a0. In this case, d is the optimal structural parameter that enables Network II to reach the optimal synchronizability. The synchronizability at first strengthens and then remains unchanged with the increase of a0. For the case of the bounded synchronous region (Figure 12(b)), when a0=d, R rapidly decreases with the increase of a0; when a0 increases to the threshold value d, R starting along with the increase of a0 remains invariant; thus, the threshold value d is to achieve the optimal synchronizability when the structure parameters are optimal. Therefore, with the increase of a0, the synchronizability strengthens before remaining invariant.As shown in Figure13, take N=199,a=1,a0=0.5,d0=0.1 to examine the synchronizability with the change of the interlayer coupling strength d between the leaf nodes. For the case of the unbounded synchronous region (Figure 13(a)), when a0>d, λ2 increases with the increase of d. When d increases to the threshold value a0, λ2 remains unchanged with the increase of d; at this time, d is the optimal structural parameter to reach the optimal synchronizability. It shows that the synchronizability strengthens at first and then remains unchanged with the increase of d. For the case of the bounded synchronous region (Figure 13(b)), when a0>d, R decreases rapidly with the increase of d. When d increases to the threshold value a0, R does not change with the increase of d. The threshold value a0 is the optimal structural parameter for optimal synchronizability. It is concluded that, with the increase of d, the synchronizability will strengthen at first and then remains unchanged.The simulation results are shown in Figure14. Take N=199,a=1,a0=0.5,d=1 to investigate how the synchronizability changes with the interlayer coupling strength d0 (when a0>d) between the hub nodes. Take N=199,a=1,a0=0.5,d=0.3 to investigate how the synchronizability changes with the interlayer coupling strength d0 (when a0<d) between the hub nodes. In the unbounded synchronous region (Figure 14(a)), either a0>d or a0<d, and λ2 are basically unchanged with the increase of d0, indicating that the synchronizability is unchanged with the increase of d0. In the bounded synchronous region (Figure 14(b)), R increases slowly with the increase of d0 in both a0>d and a0<d. Therefore, the synchronizability weakens with the increase of d0.(2)
N is even. For Network II with the even of the subnetwork nodes, it is found that the synchronizability is influenced by the number of subnetwork nodes, the intralayer coupling strength from the hub node to the leaf node, the interlayer coupling strength between the leaf nodes, and the interlayer coupling strength between the hub nodes, which are the same as those when the number of the subnetwork nodes is odd for a0>d, namely, Figure 10 (the red line) and Figure 15, Figure 11 (the red line) and Figure 16, Figure 13 (when d<a0) and Figure 17, and Figure 14 (the red line) and Figure 18. The synchronizability strengthens with the increase of d in the unbounded synchronous region. The synchronizability is enhanced with the increase of d but weakens with the increase of N,a,d0, in the bounded synchronous region. Moreover, the intralayer coupling strength a0 between the leaf nodes has no effect on the synchronizability when N is even. At this time, the edges between the intralayer leaf nodes are removed, which is the two-layer unidirectional coupled star network from the hub nodes to the leaf node. The above conclusions are the same as the simulation results of the star network with unidirectional coupling between the three-layer hub nodes and the leaf node in [30].Figure 10
WhenN is odd, the synchronizability of Network II changes with the number of the subnetwork nodes N. (a) λ2 changes with N (N changes from 9 to 599); (b) R changes with N (N changes from 9 to 599).
(a)(b)Figure 11
WhenN is odd, the synchronizability of Network II changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1); (b) R changes with a (a changes from 0.01 to 1).
(a)(b)Figure 12
WhenN is odd, the synchronizability of Network II changes with the intralayer coupling strength a0 of the leaf nodes. (a) λ2 changes with a0 (a0 changes from 0.01 to 2); (b) R changes with a0 (a0 changes from 0.01 to 2).
(a)(b)Figure 13
WhenN is odd, the synchronizability of Network II changes with the interlayer coupling strength d of the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 1); (b) R changes with d (d changes from 0.01 to 1).
(a)(b)Figure 14
WhenN is odd, the synchronizability of Network II varies with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)Figure 15
WhenN is even, the synchronizability of Network II changes with the number of subnetwork nodes N. (a) λ2 changes with N (N changes from 10 to 600); (b) R changes with N (N changes from 10 to 600).
(a)(b)Figure 16
WhenN is even, the synchronizability of Network II changes with the intralayer coupling strength a from the leaf nodes to the hub nodes. (a) λ2 changes with a (a changes from 0.01 to 1); (b) R changes with a (a changes from 0.01 to 1).
(a)(b)Figure 17
WhenN is even, the synchronizability of Network II changes with the interlayer coupling strength d between the leaf nodes. (a) λ2 changes with d (d changes from 0.01 to 1); (b) R changes with d (d changes from 0.01 to 1).
(a)(b)Figure 18
WhenN is even, the synchronizability of Network II varies with the interlayer coupling strength d0 of the leaf nodes. (a) λ2 changes with d0 (d0 changes from 0.01 to 1); (b) R changes with d0 (d0 changes from 0.01 to 1).
(a)(b)


## 5. Conclusion and Discussion

In this article, we study the eigenvalue spectrum and synchronizability of two types of double-layer hybrid directional coupling star-ring network. One is the double-layer hybrid directional coupled star-ring networks from the leaf nodes to the hub nodes (Network I); the other is the double-layer hybrid directional coupling star-ring network (Network II) from the hub node to the leaf node. The eigenvalue spectrum of the two types of double-layer star-ring networks is derived strictly in theory, and the relationship between the synchronization and the structural parameters is analyzed. 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. The rest of the cases are that the influence of synchronizability is the same between the interlayer coupling strength of the leaf node in this article and multilayer undirected star and star-ring networks of the interlayer coupling strength of the hub node, and the interlayer coupling strength between hub nodes in our article has the same effect on the synchronization as the interlayer coupling strength between leaf nodes of the multilayer undirected star and star-ring networks.For Network I, in the case that the synchronous region is unbounded, no matter whether the number of subnetwork nodes is odd or even, the synchronizability is enhanced with the increase of the intralayer coupling strength from the leaf nodes to the hub nodes and the interlayer coupling strength between the hub nodes. In the case that the synchronous region is bounded, when the intralayer coupling strength from the leaf nodes to the hub nodes is weaker, the synchronizability strengthens with the increase of the intralayer coupling strength from the leaf nodes to the hub nodes but weakens with the increase of the interlayer coupling strength between the leaf nodes. When the interlayer coupling strength between the hub nodes is weaker, without the parity of the number of subnetwork nodes, the synchronizability is enhanced with the increase of the interlayer coupling strength between the hub nodes but weakens with the increase of the intralayer coupling strength between the leaf nodes to the hub nodes and the interlayer coupling strength between the leaf nodes. When the number of subnetwork nodes is odd, the synchronizability weakens with the increase of the intralayer coupling strength between the leaf nodes. When the number of subnetwork nodes is even, the intralayer coupling strength between the leaf nodes has no effect on network synchronizability; in this case, studying network synchronizability can remove the edge between the leaf nodes of the ring, to reduce the physical memory and to improve the work efficiency, and the results and three layers of the leaf nodes pointing to the hub nodes of unidirectionally coupling conclusion are consistent with the star networks in [30].For Network II, the intralayer coupling strength between the leaf nodes has an influence on the synchronizability when the number of subnetwork nodes is odd but not when the number of subnetwork nodes is even. When the number of subnetwork nodes is even, the conclusion is the same as that of the three-layer star networks, unidirectionally coupling from the hub nodes to the leaf nodes in [30]. Therefore, when the number of subnetwork nodes is even, synchronizability research can reduce physical memory and improve the working efficiency by deleting the edges between the leaf nodes in the ring. In the case that the synchronous region is unbounded, when the number of subnetwork nodes is odd, the synchronizability is enhanced with the increase of the intralayer coupling strength between the leaf nodes and the interlayer coupling strength between the leaf nodes. When the number of nodes in the subnetwork is even, the synchronizability is enhanced with the increase of the interlayer coupling strength between the leaf nodes. For the bounded synchronous region, when the intralayer coupling strength between the leaf nodes is weaker, in the case that the number of subnetwork nodes is odd, the synchronizability is enhanced with the increase of the intralayer coupling strengthened between the leaf nodes, but with the increase of the number of subnetwork nodes, the intralayer coupling strength between the hub nodes to the leaf nodes and the interlayer coupling strength between the hub nodes are weakened. When the interlayer coupling strength between the leaf nodes is weaker, whether the number of nodes in the subnetwork is odd or even, the synchronizability strengthens with the increase of the interlayer coupling strength between the leaf nodes but weakens with the increase of the number of nodes in the subnetwork, the intralayer coupling strength between the hub nodes to the leaf nodes, and the interlayer coupling strength between the hub nodes.


## 6. Prospect

Due to the large scale and complex structure of multilayer networks, its network dynamics properties are also very abundant, but it is very difficult to research, especially to obtain the results of the theoretical derivation. The theoretical research of the network in this article can be used to study the dynamics of more general multilayer hybrid directional networks. The simulation results can help better to understand the design and control of real complex systems. Here, we obtain the theoretical results of the eigenvalue spectrum of two kinds of two-layer hybrid directionally coupled star-ring networks. However, if we generalize the model to any multilayer network, it will be more challenging to solve its eigenvalue spectrum. Secondly, what is the eigenvalue spectrum and synchronizability of the two-layer network model that we studied, when there is a direction between the hub nodes of the interlayer or between the leaf nodes of the interlayer? If the intralayer coupling strength between the leaf nodes is unchanged, how do other parameters change so that the synchronizability remains unchanged? We will continue to devote ourselves to the in-depth study of these issues. There are various multilayer directional networks in the real world. At present, the theoretical model of multilayer directional networks is still weak in practical application, mainly because of the gap between the theoretical model and the actual data. With the further investigation of the theory of multilayer hybrid directional network, the theoretical model will be more and more close to the real system, and the practical application will be increasingly common.



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