# Synchronizability of Multilayer Star and Star-Ring Networks

Authors: Yang Deng; Zhen Jia; Feimei Yang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Synchronization of multilayer complex networks is one of the important frontier issues in network science. In this paper, we strictly derived the analytic expressions of the eigenvalue spectrum of multilayer star and star-ring networks and analyzed the synchronizability of these two networks by using the master stability function (MSF) theory. In particular, we investigated the synchronizability of the networks under different interlayer coupling strength, and the relationship between the synchronizability and structural parameters of the networks (i.e., the number of nodes, intralayer and interlayer coupling strengths, and the number of layers) is discussed. Finally, numerical simulations demonstrated the validity of the theoretical results.


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## Body

## 1. Introduction

Network science is an interdisciplinary subject which abstracts physical, biological, economic and social systems into networks composed of nodes and edges and studies their structural characteristics, dynamic evolution and dynamic characteristics. The network synchronization as an important emerging phenomenon of a population of dynamically interacting units in various fields of science has attracted much attention. Synchronization of complex networks has been widely studied, and much research works have been done over the past few decades and achieved fruitful research results [1–24]. However, most of these works are focusing on isolated networks; in reality, most systems are not isolated but interrelated such as the combination of aviation and railway transportation networks in the transportation system; the interdependence of the server and the terminal system in computer networks; in power infrastructure, the interactive control between the power station and the computer central control system; and in the social network, the compound overlap between the real interpersonal and online interpersonal networks, which constitute more complex networks, called multilayer networks. Therefore, in recent years, the focus of complex network research has gradually shifted from the single layer network to the multilayer network. The research of multilayer network has become an important research direction and attracts tensive attention from scholars.Although the research on multilayer networks is still in its infancy, a series of influential research results have emerged. In 2013, Gómez et al. proposed a diffusion dynamics model based on multilayer networks and the supra-Laplacian matrix of the networks [25]. Granell et al. analyzed the correlation between the two processes of epidemic transmission and epidemic and proposed the information awareness of preventing its infection on the multilayer network [26]. In 2014, Aguirre et al. discussed the eigenvalue spectrum of two completely identical star networks coupled by an edge between layers [27] and considered synchronizability with different coupling modes between layers. In 2016, Xu et al. studied the eigenvalue spectrum and synchronizability of the two-layer star network and theoretically provided the analytic value of the eigenvalue spectrum of the two-layer star network with full connection between layers [28, 29]. In 2017, Li et al. investigated some rules and properties about synchronizability of duplex networks composed of two networks interconnected by two links, for a specific duplex network composed of two star networks, analytical expressions containing the largest and smallest nonzero eigenvalues of the (weighted) Laplacian matrix, and the interlink weight, as well as the network size, which are given for three different interlayer connection patterns [30]. Sun et al. studied the synchronizability of multilayer unidirectional coupling star networks and strictly derived the eigenvalues of such networks in the case of unidirectional coupling between layers [31]. Later, Wei et al. further studied the synchronizability of a double-layer regular network based on the master stability function (MSF) theory by using numerical simulation [32]. In 2019, Tang et al. extended the master stability function to multilayer networks with different intralayer and interlayer coupling functions and derived three master stability equations that determined complete synchronization, intralayer synchronization and interlayer synchronization regions [33]. Deng et al. studied the problem of synchronization of two kinds of multiplex chain networks under different coupling modes between layers and derived the eigenvalue spectrum of the supra-Laplacian matrices of those networks [34]. However, due to the structural complexity of multilayer networks, there is almost no strict theoretical derivation on the eigenvalues of the multilayer network; most of the studies are based on the results of numerical simulation on synchronizability of that. To provide more useful foundations for getting insight into understanding synchronizability of multilayer networks and explore the main influencing factors of synchronizability, in our paper, two kinds of typical multilayer networks (i.e., multilayer star and star-ring networks) are considered on the basis of the literature [28] that studied the synchronizability of the two-layer star network; we not only strictly derived the eigenvalue spectrum of the supra-Laplacian matrices of multilayer star and star-ring networks (not limited to two layers) but also studied the relationships between the synchronizability and structural parameters of that.The paper is structured as follows. Some preliminaries are introduced in Section2. Section 3 studies the eigenvalue spectrum and synchronizability of the multilayer star networks. In Section 4, the eigenvalue spectrum and synchronizability of the multilayer star-ring networks are studied. Section 5 explores the relationship between the synchronizability and structural parameters of two kinds of multilayer networks. Finally, Section 6 gives the conclusion of this paper.


## 2. Preliminaries

### 2.1. Dynamics Model of the Multilayer Network

The dynamic equation of thei-th node in the multilayer network, with M layers, is [33, 34](1)x˙iK=fxiK+a∑j=1NωijKHxjK+d∑L=1MdiKLΓxiL,i=1,2,…,N;K=1,2,…,M,where x˙iK=fxiK (i = 1, 2, …, N; K = 1, 2, …, M) describes the isolated dynamics for the i-th node in the K-th layer, f∗:ℝn⟶ℝn is a well-defined vector function, H∗:ℝn⟶ℝn and a are the inner coupling function and coupling strength for nodes within each layer, respectively, and Γ∗:ℝn⟶ℝn and d are the inner coupling function and coupling strength for nodes across layers, respectively. Here, WK=ωijK∈ℝN×N is the coupling weight configuration matrix of the K-th layer. Explicitly, if the i-th node is connected with the j-th j≠i node within the K-th layer, ωijK=1; otherwise, ωijK=0, and there is(2)ωiiK=−∑j=1j≠iNωijK,i,j=1,2,…,N,K=1,2,…,M.LetL(K) = −aWK, which is a Laplacian matrix. If the i-th node in the K-th layer is connected with its replica in the L-th L≠K layer, diKL=1; otherwise, diKL=0, and there is(3)diKK=−∑L=1L≠KMdiKL,L,K=1,2,…,M.It is obvious thatD=diKL∈ℝM×M is also a negative Laplacian matrix.Letℒ be the supra-Laplacian matrix of equation (1), ℒI be the supra-Laplacian matrix representing the interlayer topology, and ℒL be the supra-Laplacian matrix describing the intralayer topology. Then, ℒ can be written as(4)ℒ=ℒI+ℒL.TakingLI to be the Laplacian matrix of the interlayer networks, we have(5)ℒI=LI⊗IN,where ⊗ is the Kronecker product, LI = −dD, and IN is the N × N identity matrix. As for ℒL, it can be represented by the direct sum of the Laplacian matrix L(K) within each layer, namely,(6)ℒL=L10⋯00L2⋯0⋮⋮⋮00⋯LM=⊕K=1MLK.The eigenvalues of the supra-Laplacian matrix of the networks are recorded as 0 =λ1 < λ2 ≤ λ3 ≤ ··· ≤ λmax. According to the MSF theory, the synchronizability of network (1) is determined by the minimum nonzero eigenvalue or the ratio R = λmax/λ2 of the maximum eigenvalue to the minimum nonzero eigenvalue of the supra-Laplacian matrix ℒ. Generally, when the network synchronous region is unbounded, the greater the λ2, the stronger the synchronizability is; when the synchronous region is bounded, the smaller the R = λmax/λ2, the stronger the synchronizability is.


### 2.2. Two Types of Multilayer Networks

This paper focuses on two types of multilayer networks, that is, the multilayer star and star-ring networks. For a multilayer star network, each layer is made up of identical star subnets, as shown in Figure1(a), a three-layer star network. For a multilayer star-ring network, each layer is made up of identical star-ring subnets, as shown in Figure 1(b), a three-layer star-ring network. The nodes in each layers carry the identical dynamics, and each node in one layer is connected to its replicas in other layers (the hub node F1 is connected to its replicas in other layers, and the leaf nodes Pi are connected to their replicas in other layers). Suppose each multilayer network has M layers and N ∗ M nodes (each layer contains N nodes), the intralayer coupling strength of each layer is a, and the interlayer coupling strength between the hub nodes is d0 and between the leaf nodes is d.Figure 1
Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network structure.
(a)
(b)In order to analyze the synchronizability of these two types of multilayer networks, we will separately calculate the eigenvalue spectrum of the networks and analyze the relationship between the synchronizability and structural parameters in the following sections.


## 2.1. Dynamics Model of the Multilayer Network

The dynamic equation of thei-th node in the multilayer network, with M layers, is [33, 34](1)x˙iK=fxiK+a∑j=1NωijKHxjK+d∑L=1MdiKLΓxiL,i=1,2,…,N;K=1,2,…,M,where x˙iK=fxiK (i = 1, 2, …, N; K = 1, 2, …, M) describes the isolated dynamics for the i-th node in the K-th layer, f∗:ℝn⟶ℝn is a well-defined vector function, H∗:ℝn⟶ℝn and a are the inner coupling function and coupling strength for nodes within each layer, respectively, and Γ∗:ℝn⟶ℝn and d are the inner coupling function and coupling strength for nodes across layers, respectively. Here, WK=ωijK∈ℝN×N is the coupling weight configuration matrix of the K-th layer. Explicitly, if the i-th node is connected with the j-th j≠i node within the K-th layer, ωijK=1; otherwise, ωijK=0, and there is(2)ωiiK=−∑j=1j≠iNωijK,i,j=1,2,…,N,K=1,2,…,M.LetL(K) = −aWK, which is a Laplacian matrix. If the i-th node in the K-th layer is connected with its replica in the L-th L≠K layer, diKL=1; otherwise, diKL=0, and there is(3)diKK=−∑L=1L≠KMdiKL,L,K=1,2,…,M.It is obvious thatD=diKL∈ℝM×M is also a negative Laplacian matrix.Letℒ be the supra-Laplacian matrix of equation (1), ℒI be the supra-Laplacian matrix representing the interlayer topology, and ℒL be the supra-Laplacian matrix describing the intralayer topology. Then, ℒ can be written as(4)ℒ=ℒI+ℒL.TakingLI to be the Laplacian matrix of the interlayer networks, we have(5)ℒI=LI⊗IN,where ⊗ is the Kronecker product, LI = −dD, and IN is the N × N identity matrix. As for ℒL, it can be represented by the direct sum of the Laplacian matrix L(K) within each layer, namely,(6)ℒL=L10⋯00L2⋯0⋮⋮⋮00⋯LM=⊕K=1MLK.The eigenvalues of the supra-Laplacian matrix of the networks are recorded as 0 =λ1 < λ2 ≤ λ3 ≤ ··· ≤ λmax. According to the MSF theory, the synchronizability of network (1) is determined by the minimum nonzero eigenvalue or the ratio R = λmax/λ2 of the maximum eigenvalue to the minimum nonzero eigenvalue of the supra-Laplacian matrix ℒ. Generally, when the network synchronous region is unbounded, the greater the λ2, the stronger the synchronizability is; when the synchronous region is bounded, the smaller the R = λmax/λ2, the stronger the synchronizability is.


## 2.2. Two Types of Multilayer Networks

This paper focuses on two types of multilayer networks, that is, the multilayer star and star-ring networks. For a multilayer star network, each layer is made up of identical star subnets, as shown in Figure1(a), a three-layer star network. For a multilayer star-ring network, each layer is made up of identical star-ring subnets, as shown in Figure 1(b), a three-layer star-ring network. The nodes in each layers carry the identical dynamics, and each node in one layer is connected to its replicas in other layers (the hub node F1 is connected to its replicas in other layers, and the leaf nodes Pi are connected to their replicas in other layers). Suppose each multilayer network has M layers and N ∗ M nodes (each layer contains N nodes), the intralayer coupling strength of each layer is a, and the interlayer coupling strength between the hub nodes is d0 and between the leaf nodes is d.Figure 1
Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network structure.
(a)
(b)In order to analyze the synchronizability of these two types of multilayer networks, we will separately calculate the eigenvalue spectrum of the networks and analyze the relationship between the synchronizability and structural parameters in the following sections.


## 3. The Eigenvalue Spectrum and Synchronizability of Multilayer Star Networks

For the derivation of the network eigenvalues, let us introduce the following lemma.Lemma 1.
(see [31]). Let A, B be N × N matrices and M be an integer, then(7)AB⋯BBA⋯B⋮⋮⋱⋮BB⋯AM×M=A+M−1B⋅A−BM−1.
Considering the star networks composed ofM layers (each layer consisting of N nodes), the node dynamics equation of the networks is as shown in equation (1). In the following, we derive the eigenvalue spectrum and synchronizability of the multilayer star networks under two different conditions, namely, the interlayer coupling strength between the hub nodes d0 and between the leaf nodes d is different d0≠d or the same (d0 = d).


### 3.1. The Eigenvalue Spectrum and Synchronizability of Multilayer Star Networks withd0 ≠ d

Whend0 ≠ d, the supra-Laplacian matrix of the multilayer star networks is(8)ℒ=N−1a+M−1d0−a⋯−a−d00⋯0⋯⋯⋯⋯−d00⋯0−aa+M−1d⋯00−d⋯0⋯⋯⋯⋯0−d⋯0⋮⋮⋱⋮⋮⋮⋱⋮⋯⋯⋯⋯⋮⋮⋱⋮−a0⋯a+M−1d00⋯−d⋯⋯⋯⋯00⋯−d−d00⋯0N−1a+M−1d0−a⋯−a⋱⋮⋮⋮⋮0−d⋯0−aa+M−1d⋯0⋱⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋱⋮⋱⋮⋮⋮⋮00⋯−d−a0⋯a+M−1d⋱⋮⋮⋮⋮⋮⋮⋱⋱−d00⋯0⋮⋮⋱⋱0−d⋯0⋮⋮⋱⋱⋮⋮⋱⋮⋮⋮⋱⋱00⋯−d−d00⋯0−d00⋯0N−1a+M−1d0−a⋯−a0−d⋯0⋯⋯⋯⋯0−d⋯0−aa+M−1d⋯0⋮⋮⋱⋮⋯⋯⋯⋯⋮⋮⋱⋮⋮⋮⋱⋮00⋯−d00⋯−d−a0⋯a+M−1d.According to Lemma1, one obtains the characteristic polynomial of matrix ℒ as(9)λI−ℒ=λ−N−1a−Md0aaa⋯aaaλ−a−Md00⋯00a0λ−a−Md0⋯00a00λ−a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−a−Md0a000⋯0λ−a−MdN×NM−1λ−N−1aaaa⋯aaaλ−a00⋯00a0λ−a0⋯00a00λ−a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−a0a000⋯0λ−aN×N,=λλ−Naλ−aN−2λ−a−MdN−2M−1λ−Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1,λ−Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1.LetλI=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2 and λII=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2. Therefore, the eigenvalues of ℒ are 0, Na, a⋯a︸N−2, a+Md⋯a+Md︸N−2M−1, λI⋯λI︸M−1 , λII⋯λII︸M−1 , where a is the N − 2 multiple roots, a + Md is the N−2M−1 multiple roots, and λI and λII are the M − 1 multiple roots. Through simplification, one gets(10)λI=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02=Na+Md+Md0−Na+Md0−Md2+4Mad−d02=12Na1+Md+Md0Na−1+Md0−MdNa2+4Mad−d0Na2,N≫a,d,d0,M≈12Na1+Md+Md0Na−1+Md0−MdNa2,N≫a,d,d0,M=Md.Therefore, for sufficiently largeN, there is λI ⟶ Md. In a similar way, there is(11)λII=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02⟶Na+Md0,N≫a,d,d0,M.For sufficiently largeN, there are λ2=minλI,a⟶minMd,a and R=λmax/λ2=λII/minλI,a⟶Na+Md0/minMd,a. To begin with, according to the MSF theory, we study synchronizability of multilayer star networks with different interlayer coupling strength (d0 ≠ d), as shown in Table 1.Table 1
Whend0 ≠ d, the synchronizability of the multilayer star networks varies with N, a, d, d0, and M.
The multilayer star networks
The case of synchronous region unbounded
The case of synchronous region bounded
a < Md
a > Md
a < Md
a > Md
Synchronizability
With the increase ofN
−
− (N is sufficiently large)
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
−
↑
With the increase ofd0
−
−
↓
↓
−: unchange; ↑: strengthen; ↓: weaken.


### 3.2. The Eigenvalue Spectrum and Synchronizability of the Multilayer Star Networks withd0 = d

Whend0 = d, one gets the characteristic polynomial of the supra-Laplacian matrix ℒ:(12)λI−ℒ=λλ−Naλ−aN−2λ−a−MdN−2M−1λ−Na−MdM−1λ−MdM−1.The eigenvalues of the networks are 0,Na, a⋯a︸N−2, a+Md⋯a+Md︸N−2M−1, Na+Md⋯Na+Md︸M−1, Md⋯Md︸M−1. According to the MSF theory, λ2=minMd,a and R=λmax/λ2=Na+Md/minMd,a. Then, we study synchronizability of multilayer star networks with invariant interlayer coupling strength (d0 = d), as shown in Table 2.Table 2
Whend0 = d, the synchronizability of the multilayer star networks varies with N, a, d, and M.
The multilayer star networks
The case of synchronous region unbounded
The case of synchronous region bounded
a < Md
a > Md
a < Md
a > Md
Synchronizability
With the increase ofN
−
−
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
↓
↑
1−: unchange; ↑: strengthen; ↓: weaken.It is worth noting that, whenM = 2, the results about the synchronizability varying with structural parameters of multilayer star networks are basically consistent with the literature [28], but our results are more general. According to Tables 1 and 2, for sufficiently large N and d0 ≠ d, when the synchronous region is unbounded, the synchronizability of the multilayer star networks depends on the smaller of a and Md, when a > Md, the synchronizability is determined by Md, and when a < Md, the synchronizability is determined by a. Therefore, the synchronizability of the networks is not affected by the number of nodes N. For bounded synchronous region, the synchronizability depends on N+Md0/a or Na+Md0/Md. When a < Md, the synchronizability is determined by N+Md0/a, and the synchronizability is weakened with the increase of N, M, and d0 and strengthened with the increase of a. When a > Md, the synchronizability of the networks is determined by Na+Md0/Md, and the synchronizability of the network is strengthened with the increase of M and d and weakened with the increase of a, N, and d0; hence, d and a have the opposite effect; when N is fixed and d is increased, the synchronizability of the networks can be maintained invariant by increasing a. For d0 = d, the synchronizability varying with structural parameters is basically consistent with that under the different interlayer coupling strength; the difference is that, when a < Md, the synchronizability with invariant interlayer coupling strength is weakened with increasing d, while that with different interlayer coupling strength is invariant with increasing d, with the bounded synchronous region.


## 3.1. The Eigenvalue Spectrum and Synchronizability of Multilayer Star Networks withd0 ≠ d

Whend0 ≠ d, the supra-Laplacian matrix of the multilayer star networks is(8)ℒ=N−1a+M−1d0−a⋯−a−d00⋯0⋯⋯⋯⋯−d00⋯0−aa+M−1d⋯00−d⋯0⋯⋯⋯⋯0−d⋯0⋮⋮⋱⋮⋮⋮⋱⋮⋯⋯⋯⋯⋮⋮⋱⋮−a0⋯a+M−1d00⋯−d⋯⋯⋯⋯00⋯−d−d00⋯0N−1a+M−1d0−a⋯−a⋱⋮⋮⋮⋮0−d⋯0−aa+M−1d⋯0⋱⋮⋮⋮⋮⋮⋮⋱⋮⋮⋮⋱⋮⋱⋮⋮⋮⋮00⋯−d−a0⋯a+M−1d⋱⋮⋮⋮⋮⋮⋮⋱⋱−d00⋯0⋮⋮⋱⋱0−d⋯0⋮⋮⋱⋱⋮⋮⋱⋮⋮⋮⋱⋱00⋯−d−d00⋯0−d00⋯0N−1a+M−1d0−a⋯−a0−d⋯0⋯⋯⋯⋯0−d⋯0−aa+M−1d⋯0⋮⋮⋱⋮⋯⋯⋯⋯⋮⋮⋱⋮⋮⋮⋱⋮00⋯−d00⋯−d−a0⋯a+M−1d.According to Lemma1, one obtains the characteristic polynomial of matrix ℒ as(9)λI−ℒ=λ−N−1a−Md0aaa⋯aaaλ−a−Md00⋯00a0λ−a−Md0⋯00a00λ−a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−a−Md0a000⋯0λ−a−MdN×NM−1λ−N−1aaaa⋯aaaλ−a00⋯00a0λ−a0⋯00a00λ−a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−a0a000⋯0λ−aN×N,=λλ−Naλ−aN−2λ−a−MdN−2M−1λ−Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1,λ−Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1.LetλI=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2 and λII=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2. Therefore, the eigenvalues of ℒ are 0, Na, a⋯a︸N−2, a+Md⋯a+Md︸N−2M−1, λI⋯λI︸M−1 , λII⋯λII︸M−1 , where a is the N − 2 multiple roots, a + Md is the N−2M−1 multiple roots, and λI and λII are the M − 1 multiple roots. Through simplification, one gets(10)λI=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02=Na+Md+Md0−Na+Md0−Md2+4Mad−d02=12Na1+Md+Md0Na−1+Md0−MdNa2+4Mad−d0Na2,N≫a,d,d0,M≈12Na1+Md+Md0Na−1+Md0−MdNa2,N≫a,d,d0,M=Md.Therefore, for sufficiently largeN, there is λI ⟶ Md. In a similar way, there is(11)λII=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02⟶Na+Md0,N≫a,d,d0,M.For sufficiently largeN, there are λ2=minλI,a⟶minMd,a and R=λmax/λ2=λII/minλI,a⟶Na+Md0/minMd,a. To begin with, according to the MSF theory, we study synchronizability of multilayer star networks with different interlayer coupling strength (d0 ≠ d), as shown in Table 1.Table 1
Whend0 ≠ d, the synchronizability of the multilayer star networks varies with N, a, d, d0, and M.
The multilayer star networks
The case of synchronous region unbounded
The case of synchronous region bounded
a < Md
a > Md
a < Md
a > Md
Synchronizability
With the increase ofN
−
− (N is sufficiently large)
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
−
↑
With the increase ofd0
−
−
↓
↓
−: unchange; ↑: strengthen; ↓: weaken.


## 3.2. The Eigenvalue Spectrum and Synchronizability of the Multilayer Star Networks withd0 = d

Whend0 = d, one gets the characteristic polynomial of the supra-Laplacian matrix ℒ:(12)λI−ℒ=λλ−Naλ−aN−2λ−a−MdN−2M−1λ−Na−MdM−1λ−MdM−1.The eigenvalues of the networks are 0,Na, a⋯a︸N−2, a+Md⋯a+Md︸N−2M−1, Na+Md⋯Na+Md︸M−1, Md⋯Md︸M−1. According to the MSF theory, λ2=minMd,a and R=λmax/λ2=Na+Md/minMd,a. Then, we study synchronizability of multilayer star networks with invariant interlayer coupling strength (d0 = d), as shown in Table 2.Table 2
Whend0 = d, the synchronizability of the multilayer star networks varies with N, a, d, and M.
The multilayer star networks
The case of synchronous region unbounded
The case of synchronous region bounded
a < Md
a > Md
a < Md
a > Md
Synchronizability
With the increase ofN
−
−
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
↓
↑
1−: unchange; ↑: strengthen; ↓: weaken.It is worth noting that, whenM = 2, the results about the synchronizability varying with structural parameters of multilayer star networks are basically consistent with the literature [28], but our results are more general. According to Tables 1 and 2, for sufficiently large N and d0 ≠ d, when the synchronous region is unbounded, the synchronizability of the multilayer star networks depends on the smaller of a and Md, when a > Md, the synchronizability is determined by Md, and when a < Md, the synchronizability is determined by a. Therefore, the synchronizability of the networks is not affected by the number of nodes N. For bounded synchronous region, the synchronizability depends on N+Md0/a or Na+Md0/Md. When a < Md, the synchronizability is determined by N+Md0/a, and the synchronizability is weakened with the increase of N, M, and d0 and strengthened with the increase of a. When a > Md, the synchronizability of the networks is determined by Na+Md0/Md, and the synchronizability of the network is strengthened with the increase of M and d and weakened with the increase of a, N, and d0; hence, d and a have the opposite effect; when N is fixed and d is increased, the synchronizability of the networks can be maintained invariant by increasing a. For d0 = d, the synchronizability varying with structural parameters is basically consistent with that under the different interlayer coupling strength; the difference is that, when a < Md, the synchronizability with invariant interlayer coupling strength is weakened with increasing d, while that with different interlayer coupling strength is invariant with increasing d, with the bounded synchronous region.


## 4. The Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks

In this section, we explore the eigenvalue spectrum and synchronizability of the multilayer star-ring networks in the case ofd0 ≠ d and d0 = d, respectively.


### 4.1. The Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks withd0 ≠ d

For multilayer star-ring networks, whend0 ≠ d, one obtains the supra-Laplacian matrix ℒ of the networks:(13)where ∗≙N−1a+M−1d0 and ⊗≙3a+M−1d. According to Lemma 1, the characteristic polynomial of ℒ is(14)λI−ℒ=λ−N−1a−Md0aaa⋯aaaλ−3a−Mda0⋯0aaaλ−3a−Mda⋯00a0aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3a−Mdaaa00⋯aλ−3a−MdN×NM−1λ−N−1aaaa⋯aaaλ−3aa0⋯0aaaλ−3aa⋯00a0aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3aaaa00⋯aλ−3aN×N.Let(15)CN×NM−1=λ−N−1a−Md0aaa⋯aaaλ−3a−Mda0⋯0aaaλ−3a−Mda⋯00a0aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3a−Mdaaa00⋯aλ−3a−MdN×NM−1,DN×N=λ−N−1aaaa⋯aaaλ−3aa0⋯0aaaλ−3aa⋯00a0aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3aaaa00⋯aλ−3aN×N.By simplifying,(16)CnM−1=λ−Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1⋅λ−Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1HN−1×N−1M−1,where(17)H=1111⋯11aλ−3a−Mda0⋯000aλ−3a−Mda⋯0000aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯λ−3a−Mdaa000⋯aλ−3a−MdN−1×N−1.DN×N=λλ−NaQN−1×N−1=λλ−Na1111⋯11aλ−3aa0⋯000aλ−3aa⋯0000aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯λ−3aaa000⋯aλ−3aN−1×N−1,WhenN is an odd number, the values of λ in Q are(18)5a,a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3.WhenN is an even number, the values of λ in Q are(19)a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2.WhenN is an odd number, the values of λ in |H| are(20)Md+5a,Md+a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3.WhenN is an even number, the values of λ in H are(21)a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2.Thus, the eigenvalues ofℒ are as follows:WhenN is an odd number:(22)0,5a,Na,a+4asin2kπ2N−1︸2,k=2,4,6,…,N−3,Md+5a︸M−1,Md+a+4asin2kπ2N−1︸2M−1,k=2,4,6,…,N−3,Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1,Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1.WhenN is an even number:(23)0,Na,a+4asin2kπ2N−1︸2,k=2,4,6,…,N−2,Md+a+4asin2kπ2N−1︸2M−1,k=2,4,6,…,N−2,Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1,Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1.Let λIII=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2 and λIV=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2. Whether N is an odd or even number, λ max = λIV and λ2=minλIII,a+4asin2π/N−1. Similar to Section 3.1, for sufficiently large N, λIII ⟶ Md and λIV ⟶ Na + Md0; therefore, λ2=minλIII,a+4asin2π/N−1⟶minMd,a+4asin2π/N−1 and R=λmax/λ2λIV/minλIII,a+4asin2π/N−1⟶Na+Md0/minMd,a+4asin2π/N−1. According to the MSF theory, we can summarize the relationship between the synchronizability and structural parameters under the different interlayer coupling strength, as shown in Table 3.Table 3
Whend0 ≠ d, the synchronizability of the multilayer star-ring networks varies with N, a, d, d0, and M.
The multilayer star-ring networks
The case of synchronous region unbounded
The case of synchronous region bounded
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
Synchronizability
With the increase ofN
First ↓, then −
− (N is sufficiently large)
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
−
↑
With the increase ofd0
−
−
↓
↓
1−: unchange; ↑: strengthen; ↓: weaken.


### 4.2. The Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks withd0 = d

Similarly, whend0 = d, the eigenvalues of ℒ are as follows:WhenN is an odd number: 0, 5a, Na, a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3, Md︸M−1, Md+5a︸M−1, Md+a+4asin2kπ/2N−1︸2M−1,k=2,4,6,……,N−3, Na+Md︸M−1. When N is an even number: 0, Na, a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2, Md︸M−1, Md+a+4asin2kπ/2N−1︸2M−1,k=2,4,6,…,N−2, Na+Md︸M−1. Hence, λ max = Na + Md and λ2=minMd,a+4asin2π/N−1. According to the MSF theory, we study synchronizability of multilayer star-ring networks with invariant interlayer coupling strength (d0 = d), as shown in Table 4.Table 4
Whend0 = d, the synchronizability of the multilayer star-ring networks varies with N, a, d, and M.
The multilayer star-ring networks
The case of synchronous region unbounded
The case of synchronous region bounded
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
Synchronizability
With the increase ofN
First ↓, then −
−
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
↓
↑
1−: unchange; ↑: strengthen; ↓: weaken.The synchronizability of two kinds of multilayer networks varying with the structural parameters under different (d0 ≠ d) interlayer coupling strength is basically the same as that under the same (d0 = d) interlayer coupling strength.


## 4.1. The Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks withd0 ≠ d

For multilayer star-ring networks, whend0 ≠ d, one obtains the supra-Laplacian matrix ℒ of the networks:(13)where ∗≙N−1a+M−1d0 and ⊗≙3a+M−1d. According to Lemma 1, the characteristic polynomial of ℒ is(14)λI−ℒ=λ−N−1a−Md0aaa⋯aaaλ−3a−Mda0⋯0aaaλ−3a−Mda⋯00a0aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3a−Mdaaa00⋯aλ−3a−MdN×NM−1λ−N−1aaaa⋯aaaλ−3aa0⋯0aaaλ−3aa⋯00a0aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3aaaa00⋯aλ−3aN×N.Let(15)CN×NM−1=λ−N−1a−Md0aaa⋯aaaλ−3a−Mda0⋯0aaaλ−3a−Mda⋯00a0aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3a−Mdaaa00⋯aλ−3a−MdN×NM−1,DN×N=λ−N−1aaaa⋯aaaλ−3aa0⋯0aaaλ−3aa⋯00a0aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮a000⋯λ−3aaaa00⋯aλ−3aN×N.By simplifying,(16)CnM−1=λ−Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1⋅λ−Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02M−1HN−1×N−1M−1,where(17)H=1111⋯11aλ−3a−Mda0⋯000aλ−3a−Mda⋯0000aλ−3a−Md⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯λ−3a−Mdaa000⋯aλ−3a−MdN−1×N−1.DN×N=λλ−NaQN−1×N−1=λλ−Na1111⋯11aλ−3aa0⋯000aλ−3aa⋯0000aλ−3a⋯00⋮⋮⋮⋮⋱⋮⋮0000⋯λ−3aaa000⋯aλ−3aN−1×N−1,WhenN is an odd number, the values of λ in Q are(18)5a,a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3.WhenN is an even number, the values of λ in Q are(19)a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2.WhenN is an odd number, the values of λ in |H| are(20)Md+5a,Md+a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3.WhenN is an even number, the values of λ in H are(21)a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2.Thus, the eigenvalues ofℒ are as follows:WhenN is an odd number:(22)0,5a,Na,a+4asin2kπ2N−1︸2,k=2,4,6,…,N−3,Md+5a︸M−1,Md+a+4asin2kπ2N−1︸2M−1,k=2,4,6,…,N−3,Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1,Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1.WhenN is an even number:(23)0,Na,a+4asin2kπ2N−1︸2,k=2,4,6,…,N−2,Md+a+4asin2kπ2N−1︸2M−1,k=2,4,6,…,N−2,Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1,Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd02︸M−1.Let λIII=Na+Md+Md0−Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2 and λIV=Na+Md+Md0+Na+Md+Md02−4N−1aMd+aMd0+M2dd0/2. Whether N is an odd or even number, λ max = λIV and λ2=minλIII,a+4asin2π/N−1. Similar to Section 3.1, for sufficiently large N, λIII ⟶ Md and λIV ⟶ Na + Md0; therefore, λ2=minλIII,a+4asin2π/N−1⟶minMd,a+4asin2π/N−1 and R=λmax/λ2λIV/minλIII,a+4asin2π/N−1⟶Na+Md0/minMd,a+4asin2π/N−1. According to the MSF theory, we can summarize the relationship between the synchronizability and structural parameters under the different interlayer coupling strength, as shown in Table 3.Table 3
Whend0 ≠ d, the synchronizability of the multilayer star-ring networks varies with N, a, d, d0, and M.
The multilayer star-ring networks
The case of synchronous region unbounded
The case of synchronous region bounded
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
Synchronizability
With the increase ofN
First ↓, then −
− (N is sufficiently large)
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
−
↑
With the increase ofd0
−
−
↓
↓
1−: unchange; ↑: strengthen; ↓: weaken.


## 4.2. The Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks withd0 = d

Similarly, whend0 = d, the eigenvalues of ℒ are as follows:WhenN is an odd number: 0, 5a, Na, a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−3, Md︸M−1, Md+5a︸M−1, Md+a+4asin2kπ/2N−1︸2M−1,k=2,4,6,……,N−3, Na+Md︸M−1. When N is an even number: 0, Na, a+4asin2kπ/2N−1︸2,k=2,4,6,…,N−2, Md︸M−1, Md+a+4asin2kπ/2N−1︸2M−1,k=2,4,6,…,N−2, Na+Md︸M−1. Hence, λ max = Na + Md and λ2=minMd,a+4asin2π/N−1. According to the MSF theory, we study synchronizability of multilayer star-ring networks with invariant interlayer coupling strength (d0 = d), as shown in Table 4.Table 4
Whend0 = d, the synchronizability of the multilayer star-ring networks varies with N, a, d, and M.
The multilayer star-ring networks
The case of synchronous region unbounded
The case of synchronous region bounded
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
a
+
4
a
sin
2
π
/
N
−
1
<
M
d
a
+
4
a
sin
2
π
/
N
−
1
>
M
d
Synchronizability
With the increase ofN
First ↓, then −
−
↓
↓
With the increase ofM
−
↑
↓
↑
With the increase ofa
↑
−
↑
↓
With the increase ofd
−
↑
↓
↑
1−: unchange; ↑: strengthen; ↓: weaken.The synchronizability of two kinds of multilayer networks varying with the structural parameters under different (d0 ≠ d) interlayer coupling strength is basically the same as that under the same (d0 = d) interlayer coupling strength.


## 5. Numerical Simulation

In this section, we will investigate the synchronizability of multilayer star and star-ring networks under different interlayer coupling strength (d0 > d, d0 < d, and d0 = d) through a large number of numerical simulation experiments.


### 5.1. The Synchronizability of Multilayer Star Networks

For different interlayer coupling strength (i.e.,d0 > d, d0 < d, and d0 = d.), the synchronizability of multilayer star networks varying with the structural parameters is shown in Figures 2–8.Figure 2
The synchronizability of multilayer star networksvs. varying the number of nodes N (a = 1, M = 10, d = 1, and a < Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 3
The synchronizability of multilayer star networksvs. varying the number of nodes N (a = 20, M = 10, d = 1, and a > Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 4
The synchronizability of multilayer star networksvs. varying the intralayer coupling strength a (N = 300, M = 10, and d = 1). (a) λ2 with respect to varying a for different d0. (b) R with respect to varying a for different d0.
(a)
(b)Figure 5
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d (N = 300, M = 10, a = 20, and d0 = 2). (a) λ2 with respect to varying d. (b) R with respect to varying d.
(a)
(b)Figure 6
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 1, d = 1, and a < Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 7
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 20, d = 1, and a > Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 8
The synchronizability of multilayer star networksvs. varying the number of the layers M (N = 100, a = 10, and d = 1). (a) λ2 with respect to varying M for different d0. (b) R with respect to varying M for different d0.
(a)
(b)Whena < Md, it can be seen from the left panel of Figure 2 that λ2 remains invariant at a specific value λ2 = a = 1 with increasing N for different d0 and different values of d0 have no effect on the synchronizability of the networks, with the unbounded synchronous region. The right panel reveals that R increases with increasing N, and the observation reveals that the synchronizability is weakened with increasing the number of nodes in each layer. The smaller d0, the stronger the synchronizability, with the bounded synchronous region. When a > Md, from Figure 3(a), λ2 remains invariant at a specific value λ2 = Md = 10 with increasing N for d0 = d; λ2 increases at first and then almost levels off at an upper bound value λ2 = Md = 10 for d0 < d; λ2 decreases at first and then almost levels off at a lower bound value λ2 = Md = 10 for d0 > d, with the unbounded synchronous region. From Figure 3(b), R increases with increasing N, the synchronizability is weakened with increasing N, and different values of d0 have no effect on the synchronizability of the networks.Figure4 shows the synchronizability of multilayer star networks varying with the intralayer coupling strength a. From Figure 4(a), it is obvious that λ2 increases linearly from zero and then remains invariant; this means that the synchronizability is strengthened at first and then remains invariant with increasing a. When a < Md, different values of d0 have no effect on the synchronizability of the networks; when a > Md, the smaller d0, the worse the synchronizability, with the unbounded synchronous region. From Figure 4(b), one can observe that, with increasing a, R first decreases (when a < Md) and then increases sharply (when a > Md), and the synchronizability is strengthened at first (a < Md) and then weakened with increasing intralayer coupling strength (a > Md). When a < Md, the smaller d0, the stronger the synchronizability, and when a > Md, different values of d0 have no effect on the synchronizability, with the bounded synchronous region. This is because when the intralayer coupling strength far exceeds the interlayer coupling strength, the structure of the multiplex network is similar to the community structure of multiple clusters; therefore, the synchronizability is weakened.Figures5(a) and 5(b) show the synchronizability is strengthened at first (when a > Md) and then remains invariant (when a < Md) with increasing d. That is to say, whether the synchronous region is unbounded or bounded, when d<a/M, the synchronizability of the networks is strengthened, and when d>a/M, the synchronizability of the networks remains basically invariant; therefore, there exists an optimal value of d=a/M for interlayer coupling strength to maximizing synchronizability of multilayer star networks. When d is too large, it not conducive to synchronization.As is shown in Figure6, when a < Md, panel (a) plots that the synchronizability is not affected by the interlayer coupling strength d0, with the unbounded synchronous region; panel (b) plots that the synchronizability is weakened with d0, with the bounded synchronous region. From Figure 7(a), when a > Md, the synchronizability is strengthened slightly and basically maintained at λ2 = Md = 10, with the unbounded synchronous region; from Figure 7(b), the synchronizability is also strengthened slightly, with the bounded synchronous region. Figure 8(a) shows that increasing the number of layers M increases λ2 (increases gradually at first and then approaches Md), and therefore, the synchronizability is strengthened at first and then basically maintained invariant, with the unbounded synchronous region. Figure 8(b) shows that the synchronizability is strengthened at first and then weakened slightly with increasing the number of layers, with the bounded synchronous region.


### 5.2. The Synchronizability of Multilayer Star-Ring Networks

For different interlayer coupling strength (i.e.,d0 > d, d0 < d, and d0 = d), the synchronizability of the multilayer star-ring networks varies with the structural parameters, as shown in Figures 9–15.Figure 9
The synchronizability of multilayer star-ring networksvs. varying the number of nodes N (a = 1, M = 10, d = 1, and a < Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 10
The synchronizability of multilayer star-ring networksvs. varying the number of nodes N (a = 20, M = 10, d = 1, and a > Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 11
The synchronizability of multilayer star-ring networksvs. varying the intralayer coupling strength a (N = 300, M = 10, and d = 1). (a) λ2 with respect to varying a for different d0. (b) R with respect to varying a for different d0.
(a)
(b)Figure 12
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d (N = 300, M = 10, a = 20, and d0 = 2). (a) λ2 with respect to varying d. (b) R with respect to varying d.
(a)
(b)Figure 13
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 1, d = 1, and a < Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 14
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 20, d = 1, and a > Md) (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 15
The synchronizability of multilayer star-ring networksvs. varying the number of the layers M (N = 100, a = 10, and d = 1). (a) λ2 with respect to varying M for different d0. (b) R with respect to varying M for different d0.
(a)
(b)From Figure 9, whena < Md, the synchronizability of multilayer star-ring networks is weakened at first and then remains invariant with increasing N, with the unbounded synchronous region. When the synchronous region is bounded, the synchronizability is weakened with increasing the number of nodes in each layer N.Figure10 displays that the synchronizability remains invariant with increasing N for d0 = d; the synchronizability is strengthened at first and then almost remains invariant with increasing N for d0 < d; and the synchronizability is weakened at first and then almost remains invariant with increasing N for d0 > d, with the unbounded synchronous region. The influences of the number of nodes on synchronizability for the multilayer star-ring networks are similar to those for the multilayer star networks. When the synchronous region is bounded, the synchronizability is also weakened with increasing the number of nodes in each layer N.As can be obtained from Figures11–15, the influences of the structural parameters (i.e., the number of layers, the interlayer coupling strength, and the intralayer coupling strength) on synchronizability for the multilayer star-ring networks are basically similar to those for the multilayer star networks; a few different points between the two will be introduced in Section 5.3.


### 5.3. Comparison of Two Types of Network Synchronizability

In this section, we will discuss the difference of synchronizability between multilayer star and star-ring networks under the same structural parameters.Figure16 shows λ2 and eigenratio R numerically calculated from supra-Laplacian matrices of multilayer star and star-ring networks with varying scale size N in the case of a < Md. When N is small (approximately N < 50), as can be observed from Figure 16(a), λ2 for multilayer star-ring networks is larger than that for multilayer star networks; Figure 16(b) shows R for multilayer star networks is larger than that for multilayer star-ring networks. This implies that, when N is small, the multilayer star-ring network has better synchronizability as that of the multilayer star network. When N is sufficiently large, the multilayer star-ring network has the same synchronizability as that of the multilayer star network, and the synchronizability of the two types of networks is nearly invariant with increasing N, implying that a larger size has little effect on the synchronizability of the two networks, for the unbounded synchronous region; while leads to linearly weakened synchronizability for two types of networks, for the bounded synchronous region. It is obvious that the simulation results are consistent with the theoretical results, for the case of a < Md; when N is small enough, λ2=a+4asin2π/N−1 for multilayer star-ring networks is larger than λ2 = a for multilayer star networks; when N is sufficiently large, λ2=a+4asin2π/N−1⟶a of multilayer star-ring networks is almost equal to that of multilayer star networks.Figure 16
The synchronizability of multilayer star and star-ring networks, wherea = 1, d = 1, M = 10, and a < Md. (a) λ2 with respect to varying N for multilayer star and star-ring networks. (b) R with respect to varying N for multilayer star and star-ring networks.
(a)
(b)Figure17 shows λ2 and eigenratio R numerically calculated from supra-Laplacian matrices of multilayer star and star-ring networks with varying scale size N in the case of a > Md. When a > Md, the multilayer star-ring network has the same synchronizability as that of the multilayer star network under taking the same structural parameter values. The synchronizability of the two networks is determined by the number of the layers M and the interlayer coupling strength d between leaf nodes and invariant with increasing N, for the unbounded synchronous region; the synchronizability of the two networks is weakened with increasing N, for the bounded synchronous region. The simulation results are consistent with the theoretical results.Figure 17
The synchronizability of multilayer star and star-ring networks, where a = 20,d = 1, M = 10, and a > Md. (a) λ2 with respect to varying N for multilayer star and star-ring networks. (b) R with respect to varying N for multilayer star and star-ring networks.
(a)
(b)


## 5.1. The Synchronizability of Multilayer Star Networks

For different interlayer coupling strength (i.e.,d0 > d, d0 < d, and d0 = d.), the synchronizability of multilayer star networks varying with the structural parameters is shown in Figures 2–8.Figure 2
The synchronizability of multilayer star networksvs. varying the number of nodes N (a = 1, M = 10, d = 1, and a < Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 3
The synchronizability of multilayer star networksvs. varying the number of nodes N (a = 20, M = 10, d = 1, and a > Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 4
The synchronizability of multilayer star networksvs. varying the intralayer coupling strength a (N = 300, M = 10, and d = 1). (a) λ2 with respect to varying a for different d0. (b) R with respect to varying a for different d0.
(a)
(b)Figure 5
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d (N = 300, M = 10, a = 20, and d0 = 2). (a) λ2 with respect to varying d. (b) R with respect to varying d.
(a)
(b)Figure 6
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 1, d = 1, and a < Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 7
The synchronizability of multilayer star networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 20, d = 1, and a > Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 8
The synchronizability of multilayer star networksvs. varying the number of the layers M (N = 100, a = 10, and d = 1). (a) λ2 with respect to varying M for different d0. (b) R with respect to varying M for different d0.
(a)
(b)Whena < Md, it can be seen from the left panel of Figure 2 that λ2 remains invariant at a specific value λ2 = a = 1 with increasing N for different d0 and different values of d0 have no effect on the synchronizability of the networks, with the unbounded synchronous region. The right panel reveals that R increases with increasing N, and the observation reveals that the synchronizability is weakened with increasing the number of nodes in each layer. The smaller d0, the stronger the synchronizability, with the bounded synchronous region. When a > Md, from Figure 3(a), λ2 remains invariant at a specific value λ2 = Md = 10 with increasing N for d0 = d; λ2 increases at first and then almost levels off at an upper bound value λ2 = Md = 10 for d0 < d; λ2 decreases at first and then almost levels off at a lower bound value λ2 = Md = 10 for d0 > d, with the unbounded synchronous region. From Figure 3(b), R increases with increasing N, the synchronizability is weakened with increasing N, and different values of d0 have no effect on the synchronizability of the networks.Figure4 shows the synchronizability of multilayer star networks varying with the intralayer coupling strength a. From Figure 4(a), it is obvious that λ2 increases linearly from zero and then remains invariant; this means that the synchronizability is strengthened at first and then remains invariant with increasing a. When a < Md, different values of d0 have no effect on the synchronizability of the networks; when a > Md, the smaller d0, the worse the synchronizability, with the unbounded synchronous region. From Figure 4(b), one can observe that, with increasing a, R first decreases (when a < Md) and then increases sharply (when a > Md), and the synchronizability is strengthened at first (a < Md) and then weakened with increasing intralayer coupling strength (a > Md). When a < Md, the smaller d0, the stronger the synchronizability, and when a > Md, different values of d0 have no effect on the synchronizability, with the bounded synchronous region. This is because when the intralayer coupling strength far exceeds the interlayer coupling strength, the structure of the multiplex network is similar to the community structure of multiple clusters; therefore, the synchronizability is weakened.Figures5(a) and 5(b) show the synchronizability is strengthened at first (when a > Md) and then remains invariant (when a < Md) with increasing d. That is to say, whether the synchronous region is unbounded or bounded, when d<a/M, the synchronizability of the networks is strengthened, and when d>a/M, the synchronizability of the networks remains basically invariant; therefore, there exists an optimal value of d=a/M for interlayer coupling strength to maximizing synchronizability of multilayer star networks. When d is too large, it not conducive to synchronization.As is shown in Figure6, when a < Md, panel (a) plots that the synchronizability is not affected by the interlayer coupling strength d0, with the unbounded synchronous region; panel (b) plots that the synchronizability is weakened with d0, with the bounded synchronous region. From Figure 7(a), when a > Md, the synchronizability is strengthened slightly and basically maintained at λ2 = Md = 10, with the unbounded synchronous region; from Figure 7(b), the synchronizability is also strengthened slightly, with the bounded synchronous region. Figure 8(a) shows that increasing the number of layers M increases λ2 (increases gradually at first and then approaches Md), and therefore, the synchronizability is strengthened at first and then basically maintained invariant, with the unbounded synchronous region. Figure 8(b) shows that the synchronizability is strengthened at first and then weakened slightly with increasing the number of layers, with the bounded synchronous region.


## 5.2. The Synchronizability of Multilayer Star-Ring Networks

For different interlayer coupling strength (i.e.,d0 > d, d0 < d, and d0 = d), the synchronizability of the multilayer star-ring networks varies with the structural parameters, as shown in Figures 9–15.Figure 9
The synchronizability of multilayer star-ring networksvs. varying the number of nodes N (a = 1, M = 10, d = 1, and a < Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 10
The synchronizability of multilayer star-ring networksvs. varying the number of nodes N (a = 20, M = 10, d = 1, and a > Md). (a) λ2 with respect to varying N for different d0. (b) R with respect to varying N for different d0.
(a)
(b)Figure 11
The synchronizability of multilayer star-ring networksvs. varying the intralayer coupling strength a (N = 300, M = 10, and d = 1). (a) λ2 with respect to varying a for different d0. (b) R with respect to varying a for different d0.
(a)
(b)Figure 12
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d (N = 300, M = 10, a = 20, and d0 = 2). (a) λ2 with respect to varying d. (b) R with respect to varying d.
(a)
(b)Figure 13
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 1, d = 1, and a < Md). (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 14
The synchronizability of multilayer star-ring networksvs. varying the interlayer coupling strength d0 (N = 300, M = 10, a = 20, d = 1, and a > Md) (a) λ2 with respect to varying d0. (b) R with respect to varying d0.
(a)
(b)Figure 15
The synchronizability of multilayer star-ring networksvs. varying the number of the layers M (N = 100, a = 10, and d = 1). (a) λ2 with respect to varying M for different d0. (b) R with respect to varying M for different d0.
(a)
(b)From Figure 9, whena < Md, the synchronizability of multilayer star-ring networks is weakened at first and then remains invariant with increasing N, with the unbounded synchronous region. When the synchronous region is bounded, the synchronizability is weakened with increasing the number of nodes in each layer N.Figure10 displays that the synchronizability remains invariant with increasing N for d0 = d; the synchronizability is strengthened at first and then almost remains invariant with increasing N for d0 < d; and the synchronizability is weakened at first and then almost remains invariant with increasing N for d0 > d, with the unbounded synchronous region. The influences of the number of nodes on synchronizability for the multilayer star-ring networks are similar to those for the multilayer star networks. When the synchronous region is bounded, the synchronizability is also weakened with increasing the number of nodes in each layer N.As can be obtained from Figures11–15, the influences of the structural parameters (i.e., the number of layers, the interlayer coupling strength, and the intralayer coupling strength) on synchronizability for the multilayer star-ring networks are basically similar to those for the multilayer star networks; a few different points between the two will be introduced in Section 5.3.


## 5.3. Comparison of Two Types of Network Synchronizability

In this section, we will discuss the difference of synchronizability between multilayer star and star-ring networks under the same structural parameters.Figure16 shows λ2 and eigenratio R numerically calculated from supra-Laplacian matrices of multilayer star and star-ring networks with varying scale size N in the case of a < Md. When N is small (approximately N < 50), as can be observed from Figure 16(a), λ2 for multilayer star-ring networks is larger than that for multilayer star networks; Figure 16(b) shows R for multilayer star networks is larger than that for multilayer star-ring networks. This implies that, when N is small, the multilayer star-ring network has better synchronizability as that of the multilayer star network. When N is sufficiently large, the multilayer star-ring network has the same synchronizability as that of the multilayer star network, and the synchronizability of the two types of networks is nearly invariant with increasing N, implying that a larger size has little effect on the synchronizability of the two networks, for the unbounded synchronous region; while leads to linearly weakened synchronizability for two types of networks, for the bounded synchronous region. It is obvious that the simulation results are consistent with the theoretical results, for the case of a < Md; when N is small enough, λ2=a+4asin2π/N−1 for multilayer star-ring networks is larger than λ2 = a for multilayer star networks; when N is sufficiently large, λ2=a+4asin2π/N−1⟶a of multilayer star-ring networks is almost equal to that of multilayer star networks.Figure 16
The synchronizability of multilayer star and star-ring networks, wherea = 1, d = 1, M = 10, and a < Md. (a) λ2 with respect to varying N for multilayer star and star-ring networks. (b) R with respect to varying N for multilayer star and star-ring networks.
(a)
(b)Figure17 shows λ2 and eigenratio R numerically calculated from supra-Laplacian matrices of multilayer star and star-ring networks with varying scale size N in the case of a > Md. When a > Md, the multilayer star-ring network has the same synchronizability as that of the multilayer star network under taking the same structural parameter values. The synchronizability of the two networks is determined by the number of the layers M and the interlayer coupling strength d between leaf nodes and invariant with increasing N, for the unbounded synchronous region; the synchronizability of the two networks is weakened with increasing N, for the bounded synchronous region. The simulation results are consistent with the theoretical results.Figure 17
The synchronizability of multilayer star and star-ring networks, where a = 20,d = 1, M = 10, and a > Md. (a) λ2 with respect to varying N for multilayer star and star-ring networks. (b) R with respect to varying N for multilayer star and star-ring networks.
(a)
(b)


## 6. Discussion and Conclusion

In this paper, the eigenvalue spectrum of the supra-Laplacian matrices of multilayer star and star-ring networks is strictly derived theoretically, and the synchronizability of two kinds of multilayer networks varying with the structural parameters is analyzed. In the case of unbounded synchronous region, the synchronizability of two types of networks is related to the intralayer coupling strength and interlayer coupling strength (the network scale is fixed) under different interlayer coupling strength. When the intralayer coupling strengtha is relatively weak, the synchronizability of the networks only depends on the intralayer coupling strength a. When the interlayer coupling strength d between the leaf nodes is weak, the synchronizability only depends on the interlayer coupling strength d, that is, the synchronizability is determined by the weaker of the two. In the case of the bounded synchronous region, when the intralayer coupling strength is weak, the increase of the intralayer coupling strength will enhance the synchronizability of the networks, while the increase of the interlayer coupling strength between the leaf nodes will weaken the synchronizability. When the interlayer coupling strength between the leaf nodes d is weak, the increase of the interlayer coupling strength between the leaf nodes will enhance the synchronizability of the networks, while the increases of the intralayer coupling strength will weaken the synchronizability. The same change of the intralayer coupling strength and the interlayer coupling strength between the leaf nodes has an opposite effect on the synchronizability of the two types of networks. Therefore, when the scale of the networks is fixed (M and N are unchanged), the interlayer coupling strength d increases, we can increase the intralayer coupling strength a to maintain the synchronizability. The multilayer star-ring networks have better synchronizability than that of multilayer star networks, for sufficiently small N.So far, there are many unresolved issues in the research studies on the multilayer network. For example, for the multilayer star network and star-ring network proposed in this paper, how to adjust or control the relationship between parametersd, d0, and a to maintain or improve the synchronizability of the networks? How to optimize the structural parameters of the networks to achieve the best synchronizability? All these need further study. In a word, many theoretical and practical problems have to be challenged.



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*Source: 9143917-2020-04-01.xml*