ArticlePDF Available

Synchronizability of Multilayer Star and Star-Ring Networks

Discrete Dynamics in Nature and Society

April 2020
2020(1):1-20

DOI:10.1155/2020/9143917

License
CC BY 4.0

Authors:

Yang Deng

Harbin Institute of Technology Shenzhen

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.

Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network structure.

…

Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network structure.

…

The synchronizability of multilayer star networks vs. 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.

…

+29

The synchronizability of multilayer star networks vs. 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.

…

Figures - available from: Discrete Dynamics in Nature and Society

This content is subject to copyright. Terms and conditions apply.

Access to this full-text is provided by Wiley.

Learn more

Content available from Discrete Dynamics in Nature and Society

This content is subject to copyright. Terms and conditions apply.

Research Article

Synchronizability of Multilayer Star and Star-Ring Networks

Yang Deng, Zhen Jia , and Feimei Yang

College of Science, Guilin University of Technology, Guilin 541004, China

Correspondence should be addressed to Zhen Jia; jjjzzz0@163.com

Received 25 November 2019; Accepted 17 February 2020; Published 1 April 2020

Academic Editor: Nikos I. Karachalios

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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. 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 [1–24]. However, most

of these works are focusing on isolated networks; in re-

ality, most systems are not isolated but interrelated such as

the combination of aviation and railway transportation

networks in the transportation system; the interdepen-

dence 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 inter-

personal networks, which constitute more complex net-

works, called multilayer networks. erefore, in recent

years, the focus of complex network research has grad-

ually 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, G´

omez et al. proposed a diﬀusion dy-

namics 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 synchro-

nizability with diﬀerent coupling modes between layers. In

2016, Xu et al. studied the eigenvalue spectrum and syn-

chronizability 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

Hindawi

Discrete Dynamics in Nature and Society

Volume 2020, Article ID 9143917, 20 pages

https://doi.org/10.1155/2020/9143917

given for three diﬀerent interlayer connection patterns [30].

Sun et al. studied the synchronizability of multilayer uni-

directional 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 dif-

ferent intralayer and interlayer coupling functions and de-

rived 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 syn-

chronizability 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 net-

works. In Section 4, the eigenvalue spectrum and synchro-

nizability of the multilayer star-ring networks are studied.

Section 5 explores the relationship between the synchroniz-

ability 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. e dynamic

equation of the i-th node in the multilayer network, with M

layers, is [33, 34]

i�f xK

􏼐 􏼑+a􏽘

j�1

ωK

ij H xK

􏼐 􏼑

+d􏽘

L�1

dKL

iΓxL

􏼐 􏼑, i �1,2,. . . , N;K�1,2,..., M,

(1)

where _

i�f(xK

i)(i�1, 2, . . .,N;K�1, 2, ...,M) describes

the isolated dynamics for the i-th node in the K-th layer,

f(∗):Rn⟶Rnis a well-deﬁned vector function,

H(∗):Rn⟶Rnand aare the inner coupling function and

coupling strength for nodes within each layer, respectively,

and Γ(∗):Rn⟶Rnand dare the inner coupling function

and coupling strength for nodes across layers, respectively.

Here, WK� (ωK

ij )∈RN×Nis the coupling weight conﬁgu-

ration 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,

ωK

ij �1; otherwise, ωK

ij �0, and there is

ωK

ii � − 􏽘

j�1

j≠i

ωK

ij , i, j �1,2,..., N, K �1,2,. . . , M.

(2)

Let L

(K)

� − aW

, 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, dKL

i�1; otherwise, dKL

i�0, and there is

dKK

i� − 􏽘

L�1

L≠K

dKL

i, L, K �1,2,..., M.

(3)

It is obvious that D� (dKL

i)∈RM×Mis also a negative

Laplacian matrix.

Let Lbe the supra-Laplacian matrix of equation (1), LI

be the supra-Laplacian matrix representing the interlayer

topology, and LLbe the supra-Laplacian matrix describing

the intralayer topology. en, Lcan be written as

L�LI+LL.(4)

Taking L

to be the Laplacian matrix of the interlayer

networks, we have

LI�LI⊗IN,(5)

where ⊗is the Kronecker product, L

� − dD, and I

is the N

×Nidentity matrix. As for LL, it can be represented by the

direct sum of the Laplacian matrix L

(K)

within each layer,

namely,

LL�

L(1)0· · · 0

0L(2)· · · 0

⋮ ⋮ ⋮

0 0 · · · L(M)

⎛

⎜

⎝⎞

⎟

⎠�⊕

K�1L(K).(6)

e eigenvalues of the supra-Laplacian matrix of the

networks are recorded as 0 �λ

<λ

≤λ

≤··· ≤λ

max

According to the MSF theory, the synchronizability of net-

work (1) is determined by the minimum nonzero eigenvalue

or the ratio R�λ

max

/λ

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

, the stronger the synchro-

nizability is; when the synchronous region is bounded, the

smaller the R�λ

max

/λ

, the stronger the synchronizability is.

2.2. 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

2Discrete Dynamics in Nature and Society

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. 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

is connected to its

replicas in other layers, and the leaf nodes P

are con-

nected to their replicas in other layers). Suppose each

multilayer network has Mlayers and N∗Mnodes (each

layer contains Nnodes), the intralayer coupling strength

of each layer is a, and the interlayer coupling strength

between the hub nodes is d

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 rela-

tionship between the synchronizability and structural pa-

rameters in the following sections.

3. The Eigenvalue Spectrum and

Synchronizability of Multilayer

Star Networks

For the derivation of the network eigenvalues, let us in-

troduce the following lemma.

Lemma 1 (see [31]). Let A, B be N ×N matrices and M be an

integer, then

A B · · · B

B A · · · B

⋮ ⋮ ⋱ ⋮

B B · · · A

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌M×M

� |A+(M−1)B| · |A−B|(M−1).

(7)

Considering the star networks composed of Mlayers

(each layer consisting of Nnodes), the node dynamics

equation of the networks is as shown in equation (1). In the

following, we derive the eigenvalue spectrum and syn-

chronizability of the multilayer star networks under two

diﬀerent conditions, namely, the interlayer coupling

strength between the hub nodes d

and between the leaf

nodes dis diﬀerent (d0≠d)or the same (d

�d).

3.1. e Eigenvalue Spectrum and Synchronizability of Mul-

tilayer Star Networks with d

≠d. When d

≠d, the supra-

Laplacian matrix of the multilayer star networks is

(a)

(b)

Figure 1: Schematic diagram of the network structure. (a) A three-layer star network structure. (b) A three-layer star-ring network

structure.

Discrete Dynamics in Nature and Society 3

L�

(N−1)a+ (M−1)d0−a· · · − a−d00· · · 0· · · · · · · · · · · · − d00· · · 0

−a a + (M−1)d· · · 0 0 −d· · · 0· · · · · · · · · · · · 0−d· · · 0

⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ · · · · · · · · · · · · ⋮ ⋮ ⋱ ⋮

−a0· · · a+ (M−1)d0 0 · · · − d· · · · · · · · · · · · 0 0 · · · − d

−d00· · · 0(N−1)a+ (M−1)d0−a· · · − a⋱ ⋮ ⋮ ⋮ ⋮

0−d· · · 0−a a + (M−1)d· · · 0⋱ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮ ⋱ ⋮ ⋮ ⋮ ⋮

0 0 · · · − d−a0· · · a+ (M−1)d⋱ ⋮ ⋮ ⋮ ⋮

⋮ ⋮ ⋱ ⋱ −d00· · · 0

⋮ ⋮ ⋱ ⋱ 0−d· · · 0

⋮ ⋮ ⋱ ⋱ ⋮ ⋮ ⋱ ⋮

⋮ ⋮ ⋱ ⋱ 0 0 · · · − d

−d00· · · 0−d00· · · 0(N−1)a+ (M−1)d0−a· · · − a

0−d· · · 0· · · ·· · · · · · · · 0−d· · · 0−a a + (M−1)d· · · 0

⋮ ⋮ ⋱ ⋮ · · · ·· · · · · · · · ⋮ ⋮ ⋱ ⋮ ⋮ ⋮ ⋱ ⋮

0 0 · · · − d0 0 ·· · − d−a0· · · a+ (M−1)d

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(8)

According to Lemma 1, one obtains the characteristic

polynomial of matrix Las

|λI−L| �

λ− (N−1)a−Md0a a a · · · a a

aλ−a−M d 0 0 · · · 0 0

a0λ−a−M d 0· · · 0 0

a0 0 λ−a−M d · · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a0 0 0 · · · λ−a−M d 0

a0 0 0 · · · 0λ−a−M d

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

M−1

N×N

λ− (N−1)a a a a ··· a a

aλ−a0 0 ··· 0 0

a0λ−a0··· 0 0

a0 0 λ−a··· 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a0 0 0 ··· λ−a0

a0 0 0 ··· 0λ−a

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌N×N

�λ(λ−Na)(λ−a)N−2(λ−a−Md)(N−2)(M−1)λ−Na +Md +Md0+��

Na +Md +Md0

 􏼁2−4(N−1)aMd +aMd0+M2dd0

􏼂 􏼃

􏽱2

⎛

⎜

⎝⎞

⎟

⎠M−1

·λ−Na +Md +Md0−��

Na +Md +Md0

 􏼁2−4(N−1)aMd +aMd0+M2dd0

􏼂 􏼃

􏽱2

⎛

⎜

⎝⎞

⎟

⎠M−1

(9)

Let λI� ((Na +Md +Md0) −

��

(Na +Md +Md0)2−4[(N−1)aMd +aMd0+M2dd0]

􏽱)/2

and λII � ((Na +Md +Md0) +

��

(Na +Md +Md0)2−4[(N−1)aMd +aMd0+M2dd0]

􏽱)/2.

erefore, the eigenvalues of Lare 0, Na,a· · · a

􏽼√√􏽻􏽺√√􏽽

N−2

a+Md · · · a+Md

􏽼√√√√√√√√􏽻􏽺√√√√√√√√􏽽

(N−2)(M−1)

,λI· · · λI

􏽼√√√􏽻􏽺√√√􏽽

M−1

,λII · · · λII

􏽼√√√􏽻􏽺√√√􏽽

M−1

, where ais the N−2

multiple roots, a+Md is the (N−2)(M−1)multiple roots,

4Discrete Dynamics in Nature and Society

and λ

are the M−1 multiple roots. rough

simpliﬁcation, one gets

λI�Na +Md +Md0−��

Na +Md +Md0

 􏼁2−4(N−1)aMd +aMd0+M2dd0

􏼂 􏼃

􏽱2

�Na +Md +Md0−��

Na +Md0−Md

 􏼁2+4Ma d −d0

 􏼁

􏽱2

�1

2Na 1+Md +Md0

Na −��

1+Md0−Md

􏼠 􏼡2

+4Ma d −d0

 􏼁

(Na)2

􏽶

􏽴

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, N ≫a, d, d0, M

≈1

2Na 1+Md +Md0

Na −��

1+Md0−Md

􏼠 􏼡2

􏽶

􏽴

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, N ≫a, d, d0, M

�Md.

(10)

erefore, for suﬃciently large N, there is λ

⟶Md. In a

similar way, there is

λII �Na +M d +Md0+��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2⟶Na +Md0, N ≫a, d, d0, M.

(11)

For suﬃciently large N, there are λ2�min λI, a

􏼈 􏼉⟶

min Md, a

{ } and R� (λmax /λ2) � (λII/ min λI, a

􏼈 􏼉)⟶

((Na +Md0)/ min M d, a

{ }). To begin with, according to the

MSF theory, we study synchronizability of multilayer star

networks with diﬀerent interlayer coupling strength (d

≠d),

as shown in Table 1.

3.2. e Eigenvalue Spectrum and Synchronizability of the

Multilayer Star Networks with d

�d. When d

�d, one gets

the characteristic polynomial of the supra-Laplacian matrix L:

|λI−L| � λ(λ−Na)(λ−a)N−2(λ−a−Md)(N−2)(M−1)

(λ−Na −Md)(M−1)(λ−Md)(M−1).

(12)

e eigenvalues of the networks are 0, Na,a···a

􏽼√√􏽻􏽺√√􏽽

N−2

a+Md · · · a+Md

􏽼√√√√√√√√􏽻􏽺√√√√√√√√􏽽

(N−2)(M−1)

,Na +Md · · · Na +Md

􏽼√√√√√√√√√√􏽻􏽺√√√√√√√√√√􏽽

M−1

,Md · · · Md

􏽼√√√√􏽻􏽺√√√√􏽽

M−1

According to the MSF theory, λ2�min Md, a

{ } and

R�λmax/λ2� ((Na +Md)/ min Md, a

{ }). en, we study

synchronizability of multilayer star networks with invariant

interlayer coupling strength (d

�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 Nand d

≠d, when the

synchronous region is unbounded, the synchronizability of

the multilayer star networks depends on the smaller of aand

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 syn-

chronous region, the synchronizability depends on N+

(Md0/a)or (Na +Md0)/Md. When a<Md, the syn-

chronizability is determined by N+ (Md0/a), and the

synchronizability is weakened with the increase of N,M, and

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

Discrete Dynamics in Nature and Society 5

strengthened with the increase of Mand dand weakened

with the increase of a,N, and d

; hence, dand ahave the

opposite eﬀect; when Nis ﬁxed and dis increased, the

synchronizability of the networks can be maintained invariant

by increasing a. For d

=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 of d

≠dand d

�d, respectively.

4.1. e Eigenvalue Spectrum and Synchronizability of Mul-

tilayer Star-Ring Networks with d

≠d. For multilayer star-

ring networks, when d

≠d, one obtains the supra-Laplacian

matrix Lof the networks:

−a

∗

−a−a−a

−a

00 00 −d0

−d0

00···00

−a−a00−d

−d

000 00···00

−a

−a−a

−a

−a−a−a

−a

−a−a−a

−a

−a−a−a−a

−a−a

−a

−a00

0 0 0000···00

.....

00 0 0 0 0000 ··· 0

0 0 0 0 0 0 0 0 ··· 0

00 .

000 0

...

00 00 00 ....

.....

....

0 0 00 .

000 0 0

.......00···00

00···00

....00 ···00

...

....

.....

....−d

−d

000··· 0

000···0

00 ···

···

··· ··· ··· ··· ···

··· ··· ··· ···

0 0 ··· ··· ··· ··· ··· ···

··· ··· ··· ··· ··· ···

00···00 ···

00··· 0000···00···0

00 ··· 0000···00···00

.....

000··· 0000 ··· 0 0 0 ···

0 0 0 ··· 0 0 0 0 ··· 0 0 ···

−a

ℒ = ,

(13)

Table 1: When d

≠d, the synchronizability of the multilayer star

networks varies with N,a,d,d

, and M.

e multilayer star networks

e case of synchronous

region unbounded

e case of

synchronous

region

bounded

Md a>Md a<

Synchronizability

With the

increase of N−−(Nis suﬃciently

large) ↓ ↓

With the increase

of M−↑ ↓ ↑

With the increase

of a↑−↑ ↓

With the increase

of d−↑−↑

With the increase

of d

− − ↓ ↓

−: unchange; ↑: strengthen; ↓: weaken.

Table 2: When d

�d, the synchronizability of the multilayer star

networks varies with N,a,d, and M.

e multilayer star networks

e case of

synchronous

region

unbounded

e case of

synchronous

region bounded

a<Md a >Md a <Md a >Md

Synchronizability

With the increase of N− − ↓ ↓

With the increase of M−↑ ↓ ↑

With the increase of a↑−↑ ↓

With the increase of d−↑ ↓ ↑

−: unchange; ↑: strengthen; ↓: weaken.

6Discrete Dynamics in Nature and Society

where ∗ ≙ (N−1)a+ (M−1)d0and ⊗ ≙ 3a+ (M−1)d.

According to Lemma 1, the characteristic polynomial of Lis

|λI−L| �

λ− (N−1)a−Md0aaa· · · a a

aλ−3a−M d a 0· · · 0a

a a λ−3a−M d a · · · 0 0

a0aλ−3a−M d · · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a000· · · λ−3a−M d a

a a 0 0 · · · aλ−3a−M d

M−1

N×N

λ− (N−1)a a a a · · · a a

aλ−3a a 0· · · 0a

a a λ−3a a · · · 0 0

a0aλ−3a· · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a0 0 0 · · · λ−3a a

a a 0 0 · · · aλ−3a

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌N×N

(14)

Let

|C|M−1

N×N�

λ− (N−1)a−Md0a a a · · · a a

aλ−3a−M d a 0· · · 0a

a a λ−3a−M d a · · · 0 0

a0aλ−3a−M d · · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a0 0 0 · · · λ−3a−M d a

a a 0 0 · · · aλ−3a−M d

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

M−1

N×N

|D|N×N�

λ− (N−1)a a a a · · · a a

aλ−3a a 0· · · 0a

a a λ−3a a · · · 0 0

a0aλ−3a· · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

a0 0 0 · · · λ−3a a

a a 0 0 · · · aλ−3a

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌N×N

(15)

By simplifying,

|C|M−1

n�λ−Na +M d +Md0+��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

⎛

⎜

⎝⎞

⎟

⎠M−1

·λ−Na +M d +Md0−��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

⎛

⎜

⎝⎞

⎟

⎠M−1

|H|M−1

(N−1)×(N−1),

(16)

Discrete Dynamics in Nature and Society 7

where

|H| �

1 1 1 1 · · · 1 1

aλ−3a−M d a 0· · · 0 0

0aλ−3a−M d a · · · 0 0

0 0 aλ−3a−M d · · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 · · · λ−3a−M d a

a000· · · aλ−3a−M d

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌(N−1)×(N−1)

|D|N×N�λ(λ−Na)|Q|(N−1)×(N−1)�λ(λ−Na)

1 1 1 1 · · · 1 1

aλ−3a a 0· · · 0 0

0aλ−3a a · · · 0 0

0 0 aλ−3a· · · 0 0

⋮ ⋮ ⋮ ⋮ ⋱ ⋮ ⋮

0 0 0 0 · · · λ−3a a

a0 0 0 · · · aλ−3a

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌

􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌􏼌(N−1)×(N−1)

(17)

When Nis an odd number, the values of λin |Q|are

5a, a +4asin2((kπ)/(2(N−1)))

􏽼√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√􏽽

, k �2,4,6,..., N −3.

(18)

When Nis an even number, the values of λin |Q|are

a+4asin2((kπ)/(2(N−1)))

􏽼√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√􏽽

, k �2,4,6,..., N −2.

(19)

When Nis an odd number, the values of λin |H| are

M d +5a, M d +a+4asin2((kπ)/(2(N−1)))

􏽼√√√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√√√􏽽

k�2,4,6,. . . , N −3.

(20)

When Nis an even number, the values of λin |H|are

a+4asin2((kπ)/(2(N−1)))

􏽼√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√􏽽

, k �2,4,6,..., N −2.

(21)

us, the eigenvalues of Lare as follows:

When Nis an odd number:

0,5a, Na, a +4asin2kπ

2(N−1)

􏼠 􏼡

􏽼√√√√√√√√√√􏽻􏽺√√√√√√√√√√􏽽

,(k�2,4,6,. . . , N −3), M d +5a

􏽼√√√􏽻􏽺√√√􏽽

M−1

, M d +a+4asin2kπ

2(N−1)

􏼠 􏼡

􏽼√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√􏽽

2(M−1)

,(k�2,4,6,. . . , N −3),

Na +M d +Md0−��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

􏽼√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√√􏽽

M−1

Na +M d +Md0+��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

M−1

(22)

8Discrete Dynamics in Nature and Society

When Nis an even number:

0, Na, a +4asin2kπ

2(N−1)

􏼠 􏼡

􏽼√√√√√√√√√√􏽻􏽺√√√√√√√√√√􏽽

, k �2,4,6,..., N −2,

M d +a+4asin2kπ

2(N−1)

􏼠 􏼡

􏽼√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√􏽽

2(M−1)

, k �2,4,6,..., N −2,

Na +M d +Md0−��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

M−1

Na +M d +Md0+��

Na +M d +Md0

 􏼁2−4(N−1)aM d +aMd0+M2dd0

􏼂 􏼃

􏽱2

M−1

(23)

Let λIII �Na +M d +Md0−

��

(Na +M d +Md0)2−4[(N−1)aM d +aMd0+M2dd0]

􏽱/2

and λIV �Na +M d +Md0+

��

(Na +M d +Md0)2−4[(N−1)aM d +aMd0+M2dd0]

􏽱/2.

Whether Nis an odd or even number, λmax �λ

and

λ2�min λIII, a +4asin2(π/(N−1))

􏼈 􏼉. Similar to Section

3.1, for suﬃciently large N,λ

III

⟶Md and λ

⟶Na +

; therefore, λ2�min λIII, a +4asin2(π/N−1)

􏼈 􏼉⟶

min M d, a +4asin2(π/N−1)

􏼈 􏼉and R�λmax/λ2(λIV / min

λIII, a +4asin2(π/N−1)

􏼈 􏼉)⟶(Na +Md0)/min M d, a+

{

4asin2(π/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.

4.2. e Eigenvalue Spectrum and Synchronizability of Mul-

tilayer Star-Ring Networks with d

�d. Similarly, when d

�

d, the eigenvalues of Lare as follows:

When Nis an odd number: 0, 5a,Na,

a+4asin2(kπ/2(N−1))

􏽼√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√􏽽

,(k�2,4,6,. . . , N −3),M d

􏽼√􏽻􏽺√􏽽

M−1

M d +5a

􏽼√√√􏽻􏽺√√√􏽽

M−1

,M d +a+4asin2(kπ/2(N−1))

􏽼√√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√√􏽽

2(M−1)

,(k�2,4,6,. . .

..., N −3),Na +M d

􏽼√√√√􏽻􏽺√√√√􏽽

M−1

. When Nis an even number: 0, Na,

a+4asin2(kπ/2(N−1))

􏽼√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√􏽽

,(k�2,4,6,. . . , N −2),M d

􏽼√􏽻􏽺√􏽽

M−1

M d +a+4asin2(kπ/2(N−1))

􏽼√√√√√√√√√√√√√√􏽻􏽺√√√√√√√√√√√√√√􏽽

2(M−1)

,(k�2,4,6,. . . , N −2),

Na +M d

􏽼√√√√􏽻􏽺√√√√􏽽

M−1

. Hence, λmax �Na +Md and λ2�min

M d, a +4asin2(π/N−1)

􏼈 􏼉. According to the MSF theory,

we study synchronizability of multilayer star-ring networks

with invariant interlayer coupling strength (d

�d), as

shown in Table 4.

e synchronizability of two kinds of multilayer net-

works varying with the structural parameters under diﬀerent

≠d) interlayer coupling strength is basically the same as

that under the same (d

�d) interlayer coupling strength.

5. Numerical Simulation

In this section, we will investigate the synchronizability of

multilayer star and star-ring networks under diﬀerent inter-

layer coupling strength (d

>d,d

<d, and d

�d) through a

large number of numerical simulation experiments.

5.1. e Synchronizability of Multilayer Star Networks.

For diﬀerent interlayer coupling strength (i.e., d

>d,d

<d,

and d

�d.), the synchronizability of multilayer star networks

varying with the structural parameters is shown in Figures 2–8.

Table 3: When d

≠d, the synchronizability of the multilayer star-ring networks varies with N,a,d,d

, and M.

e multilayer star-ring networks

e case of synchronous region unbounded e case of synchronous region bounded

a+4asin2(π/N−1)<Md a +4asin2(π/N−1)>Md a +4asin2(π/N−1)<Md a +4asin2(π/N−1)>Md

Synchronizability

With the increase of NFirst ↓, then − − (Nis suﬃciently large) ↓ ↓

With the increase of M−↑↓↑

With the increase of a↑−↑ ↓

With the increase of d−↑−↑

With the increase of d

− − ↓ ↓

−: unchange; ↑: strengthen; ↓: weaken.

Discrete Dynamics in Nature and Society 9

When a<Md, it can be seen from the left panel of

Figure 2 that λ

remains invariant at a speciﬁc value λ

�a�

1 with increasing Nfor diﬀerent d

and diﬀerent values of d

have no eﬀect on the synchronizability of the networks, with

the unbounded synchronous region. e right panel reveals

that Rincreases with increasing N, and the observation

reveals that the synchronizability is weakened with in-

creasing the number of nodes in each layer. e smaller d

the stronger the synchronizability, with the bounded syn-

chronous region. When a>Md, from Figure 3(a), λ

re-

mains invariant at a speciﬁc value λ

�Md �10 with

increasing Nfor d

�d;λ

increases at ﬁrst and then almost

levels oﬀ at an upper bound value λ

�Md �10 for d

<d;λ

decreases at ﬁrst and then almost levels oﬀ at a lower bound

value λ

�Md �10 for d

>d, with the unbounded syn-

chronous region. From Figure 3(b), Rincreases with in-

creasing N, the synchronizability is weakened with

increasing N, and diﬀerent values of d

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

increases linearly from

zero and then remains invariant; this means that the syn-

chronizability is strengthened at ﬁrst and then remains

Table 4: When d

�d, the synchronizability of the multilayer star-ring networks varies with N,a,d, and M.

e multilayer star-ring networks

e case of synchronous region unbounded e case of synchronous region bounded

a+4asin2(π/N−1)<Md a +4asin2(π/N−1)>Md a +4asin2(π/N−1)<Md a +4asin2(π/N−1)>Md

Synchronizability

With the increase of NFirst ↓, then − − ↓ ↓

With the increase of M−↑↓↑

With the increase of a↑−↑ ↓

With the increase of d−↑↓↑

−: unchange; ↑: strengthen; ↓: weaken.

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

50 100 150 200 250 300 350 400 450 5000

(a)

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

100

200

300

400

500

600

50 100 150 200 250 300 350 400 450 5000

(b)

Figure 2: e synchronizability of multilayer star networks vs. varying the number of nodes N(a�1, M�10, d�1, and a<Md). (a) λ

with

respect to varying Nfor diﬀerent d

. (b) Rwith respect to varying Nfor diﬀerent d

10 Discrete Dynamics in Nature and Society

invariant with increasing a. When a<Md, diﬀerent values of

have no eﬀect on the synchronizability of the networks;

when a>Md, the smaller d

, the worse the synchronizability,

with the unbounded synchronous region. From Figure 4(b),

one can observe that, with increasing a, Rﬁrst 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

, the stronger

the synchronizability, and when a>Md, diﬀerent values of

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 in-

variant (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/Mfor interlayer coupling

strength to maximizing synchronizability of multilayer star

networks. When dis 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 cou-

pling strength d

, with the unbounded synchronous region;

panel (b) plots that the synchronizability is weakened with

, with the bounded synchronous region. From Figure 7(a),

when a>Md, the synchronizability is strengthened slightly

and basically maintained at λ

=Md = 10, with the un-

bounded synchronous region; from Figure 7(b), the syn-

chronizability is also strengthened slightly, with the bounded

synchronous region. Figure 8(a) shows that increasing the

number of layers Mincreases λ

(increases gradually at ﬁrst

and then approaches Md), and therefore, the synchroniz-

ability 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. e Synchronizability of Multilayer Star-Ring Networks.

For diﬀerent interlayer coupling strength (i.e., d

>d,d

<d, and

�d), the synchronizability of the multilayer star-ring networks

varies with the structural parameters, as shown in Figures 9–15.

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

λ2

50 100 150 200 250 300 350 400 450 5000

(a)

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

200

400

600

800

1000

1200

50 100 150 200 250 300 350 400 450 5000

(b)

Figure 3: e synchronizability of multilayer star networks vs. varying the number of nodes N(a�20, M�10, d�1, and a >Md). (a) λ

with

respect to varying Nfor diﬀerent d

. (b) Rwith respect to varying Nfor diﬀerent d

Discrete Dynamics in Nature and Society 11

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

λ2

510150

(a)

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

300

320

340

360

380

400

420

440

460

480

500

510150a

(b)

Figure 4: e synchronizability of multilayer star networks vs. varying the intralayer coupling strength a(N�300, M�10, and d�1). (a) λ

with respect to varying afor diﬀerent d

. (b) Rwith respect to varying afor diﬀerent d

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

(b)

Figure 5: e synchronizability of multilayer star networks vs. varying the interlayer coupling strength d(N�300, M �10, a�20, and

�2). (a) λ

with respect to varying d. (b) Rwith respect to varying d.

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.

Figure 10 displays that the synchronizability remains

invariant with increasing Nfor d

�d; the synchronizability

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.9999

1.0001

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

300

310

320

330

340

350

360

370

380

390

400

(b)

Figure 6: e synchronizability of multilayer star networks vs. varying the interlayer coupling strength d

(N�300, M�10, a�1, d�1, and

a<Md). (a) λ

with respect to varying d

. (b) Rwith respect to varying d

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

550

560

570

580

590

600

610

620

630

640

650

(b)

Figure 7: e synchronizability of multilayer star networks vs. varying the interlayer coupling strength d

(N�300, M�10, a�20, d�1,

and a>Md). (a) λ

with respect to varying d

. (b) Rwith respect to varying d

Discrete Dynamics in Nature and Society 13

λ2

12 14 16 18 20

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0>d

d0= 0.5, d= 1, d0>d

d0= 0.3, d= 1, d0>d

d0= 0.1, d= 1, d0>d

4 6 8 1012141618202

(a)

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0>d

d0= 0.5, d= 1, d0>d

d0= 0.3, d= 1, d0>d

d0= 0.1, d= 1, d0>d

12 14 16 18 20

300

302

304

306

308

310

312

314

316

318

320

200

400

600

800

1000

1200

1400

1600

4 6 8 1012141618202

(b)

Figure 8: e synchronizability of multilayer star networks vs. varying the number of the layers M(N�100, a�10, and d�1). (a) λ

with

respect to varying Mfor diﬀerent d

. (b) Rwith respect to varying Mfor diﬀerent d

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0>d

d0= 0.5, d= 1, d0>d

d0= 0.3, d= 1, d0>d

d0= 0.1, d= 1, d0>d

0.5

1.5

2.5

3.5

50 100 150 200 250 300 350 400 450 5000

(a)

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0>d

d0= 0.5, d= 1, d0>d

d0= 0.3, d= 1, d0>d

d0= 0.1, d= 1, d0>d

100

200

300

400

500

600

50 100 150 200 250 300 350 400 450 5000

(b)

Figure 9: e synchronizability of multilayer star-ring networks vs. varying the number of nodes N(a�1, M�10, d�1, and a <Md). (a) λ

with respect to varying Nfor diﬀerent d

. (b) Rwith respect to varying Nfor diﬀerent d

14 Discrete Dynamics in Nature and Society

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0<d

d0= 0.5, d= 1, d0<d

d0= 0.3, d= 1, d0<d

d0= 0.1, d= 1, d0<d

λ2

50 100 150 200 250 300 350 400 450 5000

(a)

d0= 10, d= 1, d0>d

d0= 8, d= 1, d0>d

d0= 5, d= 1, d0>d

d0= 3, d= 1, d0>d

d0=d=1

d0= 0.8, d= 1, d0>d

d0= 0.5, d= 1, d0>d

d0= 0.3, d= 1, d0>d

d0= 0.1, d= 1, d0>d

200

400

600

800

1000

1200

50 100 150 200 250 300 350 400 450 5000

(b)

Figure 10: e synchronizability of multilayer star-ring networks vs. varying the number of nodes N(a�20, M�10, d�1, and a>Md). (a)

with respect to varying Nfor diﬀerent d

. (b) Rwith respect to varying Nfor diﬀerent d

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

λ2

150510

(a)

d0 = 0.8, d = 1, d0 > d

d0 = 0.5, d = 1, d0 > d

d0 = 0.3, d = 1, d0 > d

d0 = 0.1, d = 1, d0 > d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

280

300

320

340

360

380

400

420

440

460

480

500

151050

(b)

Figure 11: e synchronizability of multilayer star-ring networks vs. varying the intralayer coupling strength a(N�300, M�10, and d�1).

(a) λ

with respect to varying afor diﬀerent d

. (b) Rwith respect to varying afor diﬀerent d

Discrete Dynamics in Nature and Society 15

is strengthened at ﬁrst and then almost remains invariant

with increasing Nfor d

<d; and the synchronizability is

weakened at ﬁrst and then almost remains invariant with

increasing Nfor d

>d, with the unbounded synchronous

region. e inﬂuences of the number of nodes on syn-

chronizability 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 basi-

cally similar to those for the multilayer star networks; a few

diﬀerent points between the two will be introduced in Section 5.3.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

(a)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

(b)

Figure 12: e synchronizability of multilayer star-ring networks vs. varying the interlayer coupling strength d(N�300, M�10, a�20, and

�2). (a) λ

with respect to varying d. (b) Rwith respect to varying d.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

1.0007

1.0008

1.0009

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

300

310

320

330

340

350

360

370

380

390

400

(b)

Figure 13: e synchronizability of multilayer star-ring networks vs. varying the interlayer coupling strength d

(N�300, M�10, a�1, d�

1, and a<Md). (a) λ

with respect to varying d

. (b) Rwith respect to varying d

16 Discrete Dynamics in Nature and Society

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

(a)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

550

560

570

580

590

600

610

620

630

640

650

(b)

Figure 14: e synchronizability of multilayer star-ring networks vs. varying the interlayer coupling strength d

(N�300, M�10, a�20, d�

1, and a>Md) (a) λ

with respect to varying d

. (b) Rwith respect to varying d

2468101214161820

11 12 13 14 15 16 17 18 19 20

10.0044

d0 = 0.8, d = 1, d0 < d

d0 = 0.5, d = 1, d0 < d

d0 = 0.3, d = 1, d0 < d

d0 = 0.1, d = 1, d0 < d

d0 = 10, d = 1, d0 > d

d0 = 8, d = 1, d0 > d

d0 = 5, d = 1, d0 > d

d0 = 3, d = 1, d0 > d

d0 = d = 1

(a)

2 4 6 8 10 12 14 16 18 20

200

400

600

800

1000

1200

1400

1600

11 12 13 14 15 16 17 18 19 20

295

300

305

310

315

320

$R$

d0 = 0.8, d = 1, d0 < d

d0 = 0.5, d = 1, d0 < d

d0 = 0.3, d = 1, d0 < d

d0 = 0.1, d = 1, d0 < d

d0 = 10, d = 1, d0 < d

d0 = 8, d = 1, d0 < d

d0 = 5, d = 1, d0 < d

d0 = 3, d = 1, d0 < d

d0 = d = 1

(b)

Figure 15: e synchronizability of multilayer star-ring networks vs. varying the number of the layers M(N�100, a�10, and d�1). (a) λ

with respect to varying Mfor diﬀerent d

. (b) Rwith respect to varying Mfor diﬀerent d

Discrete Dynamics in Nature and Society 17

5.3. Comparison of Two Types of Network Synchronizability.

In this section, we will discuss the diﬀerence of synchro-

nizability between multilayer star and star-ring networks

under the same structural parameters.

Figure 16 shows λ

and eigenratio Rnumerically cal-

culated from supra-Laplacian matrices of multilayer star and

star-ring networks with varying scale size Nin the case of a<

Md. When Nis small (approximately N<50), as can be

observed from Figure 16(a), λ

for multilayer star-ring

networks is larger than that for multilayer star networks;

Figure 16(b) shows Rfor multilayer star networks is larger

than that for multilayer star-ring networks. is implies that,

when Nis small, the multilayer star-ring network has better

synchronizability as that of the multilayer star network.

When Nis suﬃciently large, the multilayer star-ring net-

work has the same synchronizability as that of the mul-

tilayer 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 syn-

chronizability 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 sim-

ulation results are consistent with the theoretical results,

for the case of a<Md; when Nis small enough, λ2�

a+4asin2(π/N−1)for multilayer star-ring networks is

larger than λ

=afor multilayer star networks; when Nis

suﬃciently large, λ2�a+4asin2(π/N−1)⟶aof mul-

tilayer star-ring networks is almost equal to that of mul-

tilayer star networks.

Figure 17 shows λ

and eigenratio Rnumerically cal-

culated from supra-Laplacian matrices of multilayer star

and star-ring networks with varying scale size Nin 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 net-

works is determined by the number of the layers Mand the

interlayer coupling strength dbetween leaf nodes and

invariant with increasing N, for the unbounded syn-

chronous region; the synchronizability of the two net-

works is weakened with increasing N, for the bounded

synchronous region. e simulation results are consistent

with the theoretical results.

6. Discussion and Conclusion

In this paper, the eigenvalue spectrum of the supra-

Laplacian matrices of multilayer star and star-ring net-

works is strictly derived theoretically, and the synchro-

nizability 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 ais relatively weak,

the synchronizability of the networks only depends on

the intralayer coupling strength a. When the interlayer

coupling strength dbetween the leaf nodes is weak, the

0.5

1.5

2.5

0.5

1.5

2.5

100 20304050

Multilayer star networks

Multilayer star-ring networks

0 50 100 150 200 250 300 350 400 450 500

(a)

Multilayer star networks

Multilayer star-ring networks

100

200

300

400

500

600

100203040

0 50 100 150 200 250 300 350 400 450 500

(b)

Figure 16: e synchronizability of multilayer star and star-ring networks, where a�1, d�1, M�10, and a<Md. (a) λ

with respect to

varying Nfor multilayer star and star-ring networks. (b) Rwith respect to varying Nfor multilayer star and star-ring networks.

18 Discrete Dynamics in Nature and Society

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 dis 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 Nare unchanged), the interlayer coupling strength d

increases, we can increase the intralayer coupling strength ato

maintain the synchronizability. e multilayer star-ring net-

works 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

, and ato maintain or improve the syn-

chronizability 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 in-

cluded within the supplementary information ﬁle.

Conflicts of Interest

e authors declare that they have no conﬂicts of interest.

Acknowledgments

is project was supported by the Natural Science Foun-

dation of Guangxi (no. 2018GXNSFAA138095) and the

National Natural Science Foundation of China (no.

61563013).

Supplementary Materials

e procedures are MATLAB language code, and the data

are obtained by running the code provided. (Supplementary

Materials)

References

[1] L. Tang, J. Lu, X. Wu, and J. L¨u, “Impact of node dynamics

parameters on topology identiﬁcation of complex dynamical

networks,” Nonlinear Dynamics, vol. 73, no. 1-2,

pp. 1081–1097, 2013.

[2] X. F. Wang et al., “Complex networks: topology, dynamics

and synchronization,” International Journal of Bifurcation

and Chaos, vol. 12, no. 5, pp. 885–916, 2002.

50 100 150 200 250 300 350 400 450 500

9.9

9.92

9.94

9.96

9.98

10.02

10.04

10.06

10.08

Multilayer star networks

Multilayer star-ring networks

(a)

Multilayer star networks

Multilayer star-ring networks

50 100 150 200 250 300 350 400 450 500

200

400

600

800

1000

1200

(b)

Figure 17: e synchronizability of multilayer star and star-ring networks, where a �20, d�1, M�10, and a>Md. (a) λ

with respect to

varying Nfor multilayer star and star-ring networks. (b) Rwith respect to varying Nfor multilayer star and star-ring networks.

Discrete Dynamics in Nature and Society 19

[3] A. Barrat, M. Barth´

elemy, and A. Vespignani, “Weighted

evolving networks: coupling topology and weight dynamics,”

Physical Review Letters, vol. 92, no. 22, Article ID 228701, 2004.

[4] W. Wang, B. Wang, B. Hu, G. Yan, and Q. Ou, “General

dynamics of topology and traﬃc on weighted technological

networks,” Physical Review Letters, vol. 94, no. 18, Article ID

188702, 2005.

[5] H. Kohestani, M. Totonkuban, L. Di Paola, V. Todde, and

A. Giuliani, “e basic principles of topology-dynamics rela-

tions in networks: an empirical approach,” Physica A: Statistical

Mechanics and its Applications, vol. 508, pp. 584–594, 2018.

[6] G. Mei, X. Wu, D. Ning, and J.-A. Lu, “Finite-time stabili-

zation of complex dynamical networks via optimal control,”

Complexity, vol. 21, no. 1, pp. 417–425, 2016.

[7] F. Liu, Q. Song, and J. Cao, “Pinning synchronization of one-

sided lipschitz complex networks,” Discrete Dynamics in

Nature and Society, vol. 2014, Article ID 627060, 8 pages, 2014.

[8] F. Zhang, J. Jia, and L. Yang, “Adaptive synchronization in

complex network with diﬀerent order node dynamics,” Dis-

crete Dynamics in Nature and Society, vol. 2015, Article ID

919146, 7 pages, 2015.

[9] X. Liu and T. Chen, “Cluster synchronization in directed

networks via intermittent pinning control,” IEEE Transactions

on Neural Networks, vol. 22, no. 7, pp. 1009–1020, 2011.

[10] Y. Yang, G. Xie, and J. Xie, “Mining important nodes in

directed weighted complex networks,” Discrete Dynamics in

Nature and Society, vol. 2017, Article ID 9741824, 7 pages,

2017.

[11] S. Weber, M.-T. H ¨utt, and M. Porto, “Pattern formation and

eﬃciency of reaction-diﬀusion processes on complex net-

works,” EPL (Europhysics Letters), vol. 82, no. 2, Article ID

28003, 2008.

[12] G. Siudem and J. A. Hołyst, “Diﬀusion on hierarchical systems

of weakly-coupled networks,” Physica A: Statistical Mechanics

and its Applications, vol. 513, pp. 675–686, 2019.

[13] M. Rounaghi and F. Zadeh, “Investigation of market eﬃciency

and ﬁnancial stability between S&P 500 and London stock

exchange: monthly and yearly forecasting of time series stock

returns using ARMA model,” Physica A: Statistical Mechanics

and its Applications, vol. 456, pp. 10–21, 2016.

[14] A. Arenas, A. D´

ıaz-Guilera, J. Kurths, Y. Moreno, and

C. Zhou, “Synchronization in complex networks,” Physics

Reports, vol. 469, no. 3, pp. 93–153, 2008.

[15] M. Denker, M. Timme, M. Diesmann, F. Wolf, and T. Geisel,

“Breaking synchrony by heterogeneity in complex networks,”

Physical Review Letters, vol. 92, no. 7, Article ID 074103, 2004.

[16] A. E. Motter, C. Zhou, and J. Kurths, “Network synchroni-

zation diﬀusion and the paradox of heterogeneity,” Physical

Review E, vol. 71, no. 1, Article ID 016116, 2005.

[17] M. Xue, E. Yeung, A. Rai et al., “Initial-condition estimation

in network synchronization processes: algebraic and graphical

characterizations of the estimator,” Complex Systems, vol. 21,

pp. 297–333, 2013.

[18] Z. Jia, L. Zeng, Y.-Y. Wang, and P. Wang, “Optimization

algorithms for spectral coarse-graining of complex networks,”

Physica A: Statistical Mechanics and its Applications, vol. 514,

pp. 925–935, 2019.

[19] Y.-S. Long, Z. Jia, and Y.-Y. Wang, “Coarse graining method

based on generalized degree in complex network,” Physica A:

Statistical Mechanics and its Applications, vol. 505, pp. 655–

665, 2018.

[20] P. Wang and S. Xu, “Spectral coarse grained controllability of

complex networks,” Physica A: Statistical Mechanics and its

Applications, vol. 478, pp. 168–176, 2017.

[21] D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-

world’ networks,” Nature, vol. 393, no. 6684, pp. 440–442,

1998.

[22] N. Dame, “Statistical mechanics of complex networks,” Re-

view of Modern Physics, vol. 74, no. 1, pp. 47–97, 2002.

[23] L. Gallos and P. Argyrakis, “Absence of kinetic eﬀects in

reaction-diﬀusion processes in scale-free networks,” Physical

Review Letters, vol. 92, no. 13, Article ID 138301, 2004.

[24] S. Boccalettia, V. Latorab, Y. Morenod, M. Chavez, and

D. Hwang, “Complex networks: structure and dynamics,”

Physics Reports, vol. 424, no. 4-5, pp. 175–308, 2006.

[25] S. G´

omez, A. D´

ıaz-Guilera, J. G´

omez-Gardeñes, C. J. P´

erez-

Vicente, Y. Moreno, and A. Arenas, “Diﬀusion dynamics on

multiplex networks,” Physical Review Letters, vol. 110, no. 2,

Article ID 028701, 2013.

[26] C. Granell, S. G´

omez, and A. Arenas, “Dynamical interplay

between awareness and epidemic spreading in multiplex

networks,” Physical Review Letters, vol. 111, no. 12, Article ID

128701, 2013.

[27] J. Aguirre, R. Sevilla-Escoboza, R. Guti´

errez, D. Papo, and

J. M. Buld´u, “Synchronization of interconnected networks:

the role of connector nodes,” Physical Review Letters, vol. 112,

no. 24, Article ID 248701, 2014.

[28] M. Xu, J. Lu, and J. Zhou, “Eigenvalue spectrum and syn-

chronization ability of two-layer star networks,” Acta Physica

Sinica, vol. 65, 2016, in Chinese.

[29] M. Xu, J. Zhou, J. Lu, and X. Wu, “Synchronizability of two-

layer networks,” e European Physical Journal B, vol. 88,

no. 9, p. 240, 2015.

[30] Y. Li, X. Wu, J. Lu et al., “Synchronizability of duplex net-

works,” IEEE Transactions on Circuits and Systems II Express

Briefs, vol. 63, no. 2, pp. 206–210, 2017.

[31] J. Sun, X. Li, J. H. Zhang et al., “Synchronizability and ei-

genvalues of multilayer star networks through unidirection-

ally coupling,” Acta Physica Sinica, vol. 52, p. 188901, 2017, in

Chinese.

[32] J. Wei, X. Wu, J.-A. Lu, and X. Wei, “Synchronizability of

duplex regular networks,” EPL (Europhysics Letters), vol. 120,

no. 2, Article ID 20005, 2017.

[33] L. Tang, X. Wu, J. L¨u, J. Lu, and R. M. D’Souza, “Master

stability functions for complete, intralayer, and interlayer

synchronization in multiplex networks of coupled R¨

ossler

oscillators,” Physical Review E, vol. 99, no. 1, Article ID

012304, 2019.

[34] Y. Deng, Z. Jia, G. Deng, and Q. Zhang, “Eigenvalue spectrum

and synchronizability of multiplex chain networks,” Physica

A: Statistical Mechanics and its Applications, vol. 537, Article

ID 122631, 2019.

20 Discrete Dynamics in Nature and Society

Available via license: CC BY

Content may be subject to copyright.

Synchronizability of Generalized Two-layer Networks with Different Layer Topologies

Article

Mar 2023

Linfan Lian

This paper focuses on a generalized two-layer network and its synchronizability, which randomly generate different topologies at each layer. This kind of network can better describe some irregular networks in reality. From the master stability function method of network synchronization analysis, we estimate the largest eigenvalue and the lowest nonzero eigenvalue of the supra-Laplacian matrix. Then, the influence of node coupling strength on the synchronizability of generalized two-layer networks is analyzed. We obtain that the enhancement of node coupling strength can promote network synchronization in bounded and unbounded synchronization regions. In the end, we perform numerical simulations based on theoretical analysis. The numerical results also show that the more nodes, the stronger the synchronizability under the unbounded synchronization region, and the opposite is true for the bounded synchronization region. The results have a certain guiding significance for the synchronous application of general network in reality.

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

Article

Full-text available

Jun 2021
DISCRETE DYN NAT SOC

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.

Synchronizability of Multi-Layer Dual-Center Coupled Star Networks

Article

Full-text available

Dec 2021

In the research on complex networks, synchronizability is a significant measurement of network nature. Several research studies center around the synchronizability of single-layer complex networks and few studies on the synchronizability of multi-layer networks. Firstly, this paper calculates the Laplacian spectrum of multi-layer dual-center coupled star networks and multi-layer dual-center coupled star–ring networks according to the master stability function (MSF) and obtains important indicators reflecting the synchronizability of the above two network structures. Secondly, it discusses the relationships among synchronizability and various parameters, and numerical simulations are given to illustrate the effectiveness of the theoretical results. Finally, it is found that the two sorts of networks studied in this paper are of the same synchronizability, and compared with that of a single-center network structure, the synchronizability of two dual-center structures is relatively weaker.

Synchronizability of multilayer star-ring networks with variable coupling strength

Article

Full-text available

Jan 2023

We investigate the synchronizability of multilayer star-ring networks. Two types of multilayer networks, including aggregated coupling and divergent coupling, are established based on the connections between the hub node and the leaf nodes in the subnetwork. The eigenvalue spectrum of the two types of multilayer networks is strictly derived, and the correlation between topological parameters and synchronizability is analyzed by the master stability function framework. Moreover, the variable coupling strength has been investigated, revealing that it is significantly related to the synchronizability of the aggregated coupling while having no influence on the divergent coupling. Furthermore, the validity of the synchronizability analysis is obtained by implementing adaptive control on the multilayer star-ring networks previously mentioned. Calculations and comparisons show that the differences caused by the sizes of multilayer networks and interlayer coupling strength are not negligible. Finally, numerical examples are also provided to validate the effectiveness of the theoretical analysis.

Synchronizability of Multilayer Networks With K-nearest-neighbor Topologies

Article

Full-text available

Oct 2020

In this paper, the synchronizability of multilayer K-nearest-neighbor networks is studied by using the master stability function method. The analytical expressions for the eigenvalues of the supra-Laplacian matrix are given for two-layer and multilayer K-nearest-neighbor networks. In addition, the impacts of various topological parameters (such as the network size, the node degree, the number of layers, the intra-layer and the inter-layer coupling strengths) on the network synchronizability are discussed. Finally, the theoretical results are verified through numerical simulation.

Synchronizability of Multilayer Directed Dutch Windmill Networks

Article

Full-text available

Sep 2022

This paper investigates the synchronizability of multilayer directed Dutch windmill networks with the help of the master stability function method. Here, we propose three types of multilayer directed networks with different linking patterns, namely, inter-layer directed networks (Networks-A), intra-layer directed networks (Networks-B), and hybrid directed networks (Networks-C), and rigorously derive the analytical expressions of the eigenvalue spectrum on the basis of their supra-Laplacian matrix. It is found that network structure parameters (such as the number of layers and nodes, the intra-layer and the inter-layer coupling strengths) have a significant impact on the synchronizability in the case of the two typical synchronized regions. Finally, in order to confirm that the theoretical conclusions are correct, simulation experiments of multilayer directed network are delivered.

Synchronizability of Multi-Layer Variable Coupling Windmill-Type Networks

Article

Full-text available

Oct 2021

The system model on synchronizability problem of complex networks with multi-layer structure is closer to the real network than the usual single-layer case. Based on the master stability equation (MSF), this paper studies the eigenvalue spectrum of two k-layer variable coupling windmill-type networks. In the case of bounded and unbounded synchronization domain, the relationships between the synchronizability of the layered windmill-type networks and network parameters, such as the numbers of nodes and layers, inter-layers coupling strength, are studied. The simulation of the synchronizability of the layered windmill-type networks are given, and they verify the theoretical results well. Finally, the optimization schemes of the synchronizability are given from the perspective of single-layer and multi-layer networks, and it was found that the synchronizability of the layered windmill-type networks can be improved by changing the parameters appropriately.

Synchronizability of double-layer dumbbell networks

Article

Jul 2021

Synchronization of multiplex networks has been a topical issue in network science. Dumbbell networks are very typical structures in complex networks which are distinguished from both regular star networks and general community structures, whereas the synchronous dynamics of a double-layer dumbbell network relies on the interlink patterns between layers. In this paper, two kinds of double-layer dumbbell networks are defined according to different interlayer coupling patterns: one with the single-link coupling pattern between layers and the other with the two-link coupling pattern between layers. Furthermore, the largest and smallest nonzero eigenvalues of the Laplacian matrix are calculated analytically and numerically for the single-link coupling pattern and also obtained numerically for the two-link coupling pattern so as to characterize the synchronizability of double-layer dumbbell networks. It is shown that interlayer coupling patterns have a significant impact on the synchronizability of multiplex systems. Finally, a numerical example is provided to verify the effectiveness of theoretical analysis. Our findings can facilitate company managers to select optimal interlayer coupling patterns and to assign proper parameters in terms of improving the efficiency and reducing losses of the whole team.

两条层间连边下的双层链状网络同步能力分析

Article

Full-text available

Apr 2021

Synchronizability of duplex regular networks

Article

Full-text available

Jan 2018
EPL-EUROPHYS LETT

In this paper, we investigate the synchronizability of duplex regular networks based on the master stability function framework. We study potential factors influencing synchronizability, including the network size, the coupling strength and the inter-layer connection density. From both theoretical and numerical results, we obtain that with the same size, same coupling strength and same inter-layer connection density, duplex fully-connected networks have the best synchronizability, the next is duplex stars, followed by duplex rings, and duplex chain networks have the worst synchronizability. The order of synchronizability on duplex regular networks is similar to that on single networks. We find that synchronizability of duplex regular networks is always worse than that of its isolated layer. Furthermore, for duplex fully-connected or star networks with fixed coupling strength, we find that the more the inter-layer links are, the better the synchronizability is. However, for duplex ring or chain networks, partial inter-layer links can yield the same synchronizability as that caused by one-to-one inter-layer links. Though these findings are obtained by using duplex regular networks, they will provide insight into understanding the synchronizability of general multiplex networks, and facilitate the selection of network parameters for obtaining optimum synchronizability.

Synchronizability and eigenvalues of two-layer star networks

Article

Full-text available

Jan 2016
ACTA PHYS SIN-CH ED

From the study of multilayer networks, scientists have found that the properties of the multilayer networks show great difference from those of the traditional complex networks. In this paper, we derive strictly the spectrum of the Supra-Laplace matrix and the synchronizability of two-layer star networks by applying the master stability method. Through mathematical analysis of the eigenvalues of the Supra-Laplace matrix, we study how the node number, the inter-layer and the intra-layer coupling strengths influence the synchronizability of a two-layer star network. We find that when the synchronous region is unbounded, the synchronizability of a two-layer star network is only related to the intra-layer coupling strength between the leaf nodes or the inter-layer coupling strength of the entire network. If the synchronous region of a two-layer star network is bounded, not only the inter-layer coupling strength of the network and the intra-layer coupling strength between the leaf nodes, but also the intra-layer coupling strength between the hub nodes and the network size have influence on the synchronizability of the networks. Provided that the same inter-layer and intralayer coupling strengths are concerned, we would further discuss the optimal ways of strengthening the synchronizability of a two-layer star network. If the inter-layer and intra-layer coupling strengths are far less than unity, changing the intra-layer coupling strength is the best way to enhance the synchronizability no matter what the synchronous region is. While if the coupling strengths are the same as, less than or more than unity, there will be different scenarios for the network with bounded and unbounded synchronous regions. Besides, we also discuss the synchronizability of the multilayer network with more than two layers. And then, we carry out numerical simulations and theoretical analysis of the two-layer BA scale-free networks coupled with 200 nodes and obtain very similar conclusions to that of the twolayer star networks. Finally, conclusion and discussion are given to summarize the main results and our future research interests.

Eigenvalue spectrum and synchronizability of multiplex chain networks

Article

Sep 2019
PHYSICA A

Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. In this paper, the problem of synchronization of two multiplex chain networks is investigated, according to the master stability function method. We define two kinds of multiplex chain networks according to different coupling modes: one is a class of the multiplex chain networks with one-to-one undirected coupling between layers(Networks-A), and the other is a class of the multiplex chain networks with one-to-one unidirectional coupling between layers(Networks-B). The eigenvalue spectrum of the supra-Laplacian matrices of two kinds of the networks is strictly derived theoretically, and the relationships between the structural parameters and synchronizability of the networks are further revealed. The structural parameter values of the networks to achieve the optimal synchronizability are obtained. Numerical examples are also provided to verify the effectiveness of theoretical analysis.

Initial-Condition Estimation in Network Synchronization Processes: Algebraic and Graphical Characterizations of the Estimator

Article

Dec 2012

Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled Rössler oscillators

Article

Jan 2019
PRE

Synchronization phenomena are of broad interest across disciplines and increasingly of interest in a multiplex network setting. For the multiplex network of coupled Rössler oscillators, here we show how the master stability function, a celebrated framework for analyzing synchronization on a single network, can be extended to certain classes of multiplex networks with different intralayer and interlayer coupling functions. We derive three master stability equations that determine, respectively, the necessary regions of complete synchronization, intralayer synchronization, and interlayer synchronization. We calculate these three regions explicitly for the case of a two-layer network of Rössler oscillators and show that the overlap of the regions determines the type of synchronization achieved. In particular, if the interlayer or intralayer coupling function is such that the interlayer or intralayer synchronization region is empty, complete synchronization cannot be achieved regardless of the coupling strength. Furthermore, for any network structure, the occurrence of intralayer and interlayer synchronization depends mainly on the coupling functions of nodes within a layer and across layers, respectively. Our mathematical analysis requires that the intralayer and interlayer supra-Laplacians commute. But, we show this is only a sufficient, and not necessary, condition and that the results can be applied more generally.

Optimization algorithms for spectral coarse-graining of complex networks

Article

Sep 2018
PHYSICA A

Coarse-graining techniques of large-scale complex networks have been an important approach to reduce network size, which merge nodes who share the same or similar properties while preserving some significant functions or properties of the original networks. However, reducing network size is often at the cost of worsening network performance. Thus, there is a trade-off between the coarse-grained network sizes and network performance. To find the balance between the two sides and based on the spectral coarse-graining approach (SCG), we propose two optimization algorithms, which are called variable step size optimization algorithm (VSSOA) and variable scale optimization algorithm (VSOA). The two algorithms can calculate the optimal coarse-grained step size and the optimal scale to reduce the share size of the network. The two algorithms are applied to the coarse-graining of several typical networks. And the feasibility and validity of the proposed algorithms are further verified by phase synchronization of coupled Kuramoto oscillators on typical networks. The related investigation provides a deep insight to the coarse-graining of large-scale complex networks.

Diffusion on hierarchical systems of weakly-coupled networks

Article

Aug 2018
PHYSICA A

We analyzed diffusion dynamics on weakly-coupled networks (interconnected networks) by means of separation of time scales. Using an adiabatic approximation we reduced the system dynamics to a Markov chain with aggregated variables and derived a transport equation that is analogous to Fick's First Law and includes a driving force. Entropy production is a sum of microscopic entropy transport, which results from the particle's migration between networks of different topologies and macroscopic entropy production of the Markov chain. Equilibrium particles partition between different sub-networks depends only on internal sub-network parameters. By changing structure of networks one can not only modify diffusion constants but can also induce or reverse the direction of the particles’ flow between different networks. Our framework, confirmed by numerical simulations, is also useful for considering diffusion in nested systems corresponding to hierarchical networks with several different time scales thus it can serve to uncover hidden hierarchy levels from observations of diffusion processes.

The basic principles of topology-dynamics relations in networks: An empirical approach

Article

May 2018
PHYSICA A

Inferring functional properties of a node from its topological role in the network is of utmost importance in many application fields. In this work, we try to give some hints on this issue focusing on the discovery of empirical topology/function rules located at different degrees of generalization going from a population of heterogeneous biological networks to a single wiring architecture. Here we report a simulation approach to this problem based on the analysis of the relationship between dissipation kinetics across the network of a perturbative stimulus applied on a single node and its topological properties. The empirical character of the study prompted us to rely on real biological networks instead of ‘ideal architectures’ and to adopt a statistical attitude with a relatively low number of affected nodes for each network that is much more realistic in real-world biological applications. The analysis allowed us to recognize some general topology-dynamics correlation rules variously modulated by global wiring architecture.

Coarse graining method based on generalized degree in complex network

Article

Apr 2018
PHYSICA A

Coarse graining technology is one of the important methods to study large-scale complex networks currently. Here, we propose a generalized-degree-based coarse graining (GDCG) approach to extract respectively the undirected or directed coarse-grained networks by merging the nodes with same or similar generalized degree. The new approach provides an adjustable generalized degree by parameter p for preserving some significant properties of the initial networks during the coarse-graining processes. Compared with the existing coarse-graining methods, the GDCG method is only based on the generalized degree, which is not only simple and operable, but also keeps some statistical properties and the synchronizability of the original networks. Moreover, the size of the coarse-grained networks can be chosen freely in the proposed method. Finally, extensive numerical simulations demonstrate the effectiveness of our approach.

Synchronizability and eigenvalues of multilayer star networks through unidirectionally coupling

Article

Sep 2017
ACTA PHYS SIN-CH ED

Previous studies on multilayer networks have found that properties of multilayer networks show great differences from those of the traditional complex networks. In this paper, we derive strictly the spectra of the Supra-Laplace matrix of three-layer star networks and multilayer star networks through unidirectionally coupling by using the master stability method to analyze the synchronizability of these two networks. Through mathematical analyses of the eigenvalues of the Supra-Laplace matrix, we explore how the node number, the intra-layer coupling strength the inter-layer coupling strength, and the layer number influence the synchronizability of multilayer star networks through unidirectionally coupling in two different ways. In particular, we focus on the layer number and the inter-layer coupling strength between the hub nodes, and then we conclude that the synchronizability of networks is greatly affected by the layer number. We find that when the synchronous region is unbounded, the synchronizability of the two different coupling multilayer star networks is related to not only the intra-layer coupling strength or the inter-layer coupling strength between the leaf nodes of the entire network, but also the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the intra-layer coupling strength is weak, the synchronizability of the two different coupling multilayer star networks is different with the changing of the intra-layer coupling strength and the inter-layer coupling strength between the leaf nodes and the layer number. If the synchronous region of two different coupling multilayer star networks is bounded, and the inter-layer coupling strength between the hub nodes is weak, the two different coupling multilayer star networks are consistent with the changing of the intra-layer coupling strength and the layer number while different from the inter-layer coupling strength between the leaf nodes and the inter-layer coupling strength between the hub nodes. We find that the node number has no effect on the synchronizability of multilayer star networks through coupling from the hub node to the leaf node. The synchronizability of the network is directly proportional to the layer number, while inversely proportional to the inter-layer coupling strength between the hub nodes. Finally, the effects of the coupling strength, the layer number and the node number on the synchronizability of the two different coupling star networks can be extended from three-layer network to multilayer networks.