Synchronizability of Multilayer Star and Star-Ring Networks

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.


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. e 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 . 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. erefore, in recent years, the focus of complex network research has gradually shifted from the single layer network to the multilayer network. e 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. e paper is structured as follows. Some preliminaries are introduced in Section 2. 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.

Dynamics Model of the Multilayer Network.
e dynamic equation of the i-th node in the multilayer network, with M layers, is [33,34] where _ x K i � f(x K i ) (i � 1, 2, . . ., N; K � 1, 2, . . ., M) describes the isolated dynamics for the i-th node in the K-th layer, f( * ): R n ⟶ R n is a well-defined vector function, H( * ): R n ⟶ R n and a are the inner coupling function and coupling strength for nodes within each layer, respectively, and Γ( * ): R n ⟶ R n and d are the inner coupling function and coupling strength for nodes across layers, respectively. Here, W K � (ω K ij ) ∈ R N×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, ω K ij � 1; otherwise, ω K ij � 0, and there is Let L (K) � − aW K , which is a Laplacian matrix. If the i-th node in the K-th layer is connected with its replica in the L-th It is obvious that D � (d KL i ) ∈ R M×M is also a negative Laplacian matrix.
Let L be the supra-Laplacian matrix of equation (1), L I be the supra-Laplacian matrix representing the interlayer topology, and L L be the supra-Laplacian matrix describing the intralayer topology. en, L can be written as Taking L I to be the Laplacian matrix of the interlayer networks, we have where ⊗ is the Kronecker product, L I � − dD, and I N is the N × N identity matrix. As for L L , it can be represented by the direct sum of the Laplacian matrix L (K) within each layer, namely, e eigenvalues of the supra-Laplacian matrix of the networks are recorded as 0 � λ 1 < λ 2 ≤ λ 3 ≤ ··· ≤ λ max . According to the MSF theory, the synchronizability of network (1) is determined by the minimum nonzero eigenvalue or the ratio R � λ max /λ 2 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 λ 2 , the stronger the synchronizability is; when the synchronous region is bounded, the smaller the R � λ max /λ 2 , the stronger the synchronizability is.

Two Types of Multilayer Networks.
is 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 Figure 1(a), a three-layer star network. For a multilayer star-ring network, each layer is made up of identical starring subnets, as shown in Figure 1(b), a three-layer starring network. e 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 F 1 is connected to its replicas in other layers, and the leaf nodes P i 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 d 0 and between the leaf nodes is d.
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.

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
Considering the star networks composed of M 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 d 0 and between the leaf 3.1. e Eigenvalue Spectrum and Synchronizability of Multilayer Star Networks with d 0 ≠ d. When d 0 ≠ d, the supra-Laplacian matrix of the multilayer star networks is Discrete Dynamics in Nature and Society According to Lemma 1, one obtains the characteristic polynomial of matrix L as Let Discrete Dynamics in Nature and Society and λ I and λ II are the M − 1 multiple roots. rough simplification, one gets erefore, for sufficiently large N, there is λ I ⟶ Md. In a similar way, there is For sufficiently large N, there are To begin with, according to the MSF theory, we study synchronizability of multilayer star networks with different interlayer coupling strength (d 0 ≠ d), as shown in Table 1.

e Eigenvalue Spectrum and Synchronizability of the
(12) e eigenvalues of the networks are 0, Na, a · · · a √√ √√ According to the MSF theory, λ 2 � min Md, a { } and en, we study synchronizability of multilayer star networks with invariant interlayer coupling strength (d 0 � d), as shown in Table 2.
It is worth noting that, when M = 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 d 0 ≠ 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. erefore, the synchronizability of the networks is not affected by the number of nodes N. For bounded synchronous region, the synchronizability depends on N+ (Md 0 /a) or (Na + Md 0 )/Md. When a < Md, the synchronizability is determined by N + (Md 0 /a), and the synchronizability is weakened with the increase of N, M, and d 0 and strengthened with the increase of a. When a > Md, the synchronizability of the networks is determined by Na + Md 0 /Md, and the synchronizability of the network is Discrete Dynamics in Nature and Society strengthened with the increase of M and d and weakened with the increase of a, N, and d 0 ; 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 d 0 = 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.

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 of d 0 ≠ d and d 0 � d, respectively.

e Eigenvalue Spectrum and Synchronizability of Multilayer Star-Ring Networks with d 0 ≠ d.
For multilayer starring networks, when d 0 ≠ d, one obtains the supra-Laplacian matrix L of the networks: According to Lemma 1, the characteristic polynomial of L is Let By simplifying, Discrete Dynamics in Nature and Society where |H| � .

Numerical Simulation
In this section, we will investigate the synchronizability of multilayer star and star-ring networks under different interlayer coupling strength (d 0 > d, d 0 < d, and d 0 � d) through a large number of numerical simulation experiments.

e Synchronizability of Multilayer Star Networks.
For different interlayer coupling strength (i.e., d 0 > d, d 0 < d, and d 0 � d.), the synchronizability of multilayer star networks varying with the structural parameters is shown in Figures 2-8. e multilayer star-ring networks e case of synchronous region unbounded e case of synchronous region bounded a + 4a sin 2 (π/N − 1) < Md a + 4a sin 2 (π/N − 1) > Md a + 4a sin 2 (π/N − 1) < Md a + 4a sin 2  Discrete Dynamics in Nature and Society 9 When a < Md, it can be seen from the left panel of Figure 2 that λ 2 remains invariant at a specific value λ 2 � a � 1 with increasing N for different d 0 and different values of d 0 have no effect on the synchronizability of the networks, with the unbounded synchronous region. e 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. e smaller d 0 , the stronger the synchronizability, with the bounded synchronous region. When a > Md, from Figure 3(a), λ 2 remains invariant at a specific value λ 2 � Md � 10 with increasing N for d 0 � d; λ 2 increases at first and then almost levels off at an upper bound value λ 2 � Md � 10 for d 0 < d; λ 2 decreases at first and then almost levels off at a lower bound value λ 2 � Md � 10 for d 0 > 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 d 0 have no effect on the synchronizability of the networks. Figure 4 shows the synchronizability of multilayer star networks varying with the intralayer coupling strength a. From Figure 4(a), it is obvious that λ 2 increases linearly from zero and then remains invariant; this means that the synchronizability is strengthened at first and then remains  N, a, d, and M. e multilayer star-ring networks e case of synchronous region unbounded e case of synchronous region bounded a + 4a sin 2 (π/N − 1) < Md a + 4a sin 2 (π/N − 1) > Md a + 4a sin 2 (π/N − 1) < Md a + 4a sin 2   invariant with increasing a. When a < Md, different values of d 0 have no effect on the synchronizability of the networks; when a > Md, the smaller d 0 , 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 d 0 , the stronger the synchronizability, and when a > Md, different values of d 0 have no effect on the synchronizability, with the bounded synchronous region. is 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. Figures 5(a) and 5(b) show the synchronizability is strengthened at first (when a > Md) and then remains invariant (when a < Md) with increasing d. at 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 Figure 6, when a < Md, panel (a) plots that the synchronizability is not affected by the interlayer coupling strength d 0 , with the unbounded synchronous region; panel (b) plots that the synchronizability is weakened with d 0 , with the bounded synchronous region. From Figure 7(a), when a > Md, the synchronizability is strengthened slightly and basically maintained at λ 2 = 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 λ 2 (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.

e Synchronizability of Multilayer Star-Ring Networks.
For different interlayer coupling strength (i.e., d 0 > d, d 0 < d, and d 0 � d), the synchronizability of the multilayer star-ring networks varies with the structural parameters, as shown in Figures 9-15.

12
Discrete Dynamics in Nature and Society From Figure 9, when a < 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.    14 Discrete Dynamics in Nature and Society   is strengthened at first and then almost remains invariant with increasing N for d 0 < d; and the synchronizability is weakened at first and then almost remains invariant with increasing N for d 0 > d, with the unbounded synchronous region. e 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 Figures 11-15, 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.

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. Figure 16 shows λ 2 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), λ 2 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. is 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, λ 2 � a + 4a sin 2 (π/N − 1) for multilayer star-ring networks is larger than λ 2 = a for multilayer star networks; when N is sufficiently large, λ 2 � a + 4a sin 2 (π/N − 1) ⟶ a of multilayer star-ring networks is almost equal to that of multilayer star networks. Figure 17 shows λ 2 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. e 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. e simulation results are consistent with the theoretical results.

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 strength a 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. e 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. erefore, 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. e 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 parameters d, d 0 , 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.

Data Availability
e data used to support the findings of this study are included within the supplementary information file.

Conflicts of Interest
e authors declare that they have no conflicts of interest.