Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators

doi:10.1007/s11467-023-1324-0

Front. Phys.

2023, Vol. 18

Issue (4) : 45302 https://doi.org/10.1007/s11467-023-1324-0

RESEARCH ARTICLE

Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators

Huawei Fan^1,², Ya Wang², Xingang Wang²(

)

¹. School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
². School of Physics and Information Technology, Shaanxi Normal University, Xi’an 710062, China

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Abstract

Whereas topological symmetries have been recognized as crucially important to the exploration of synchronization patterns in complex networks of coupled dynamical oscillators, the identification of the symmetries in large-size complex networks remains as a challenge. Additionally, even though the topological symmetries of a complex network are known, it is still not clear how the system dynamics is transited among different synchronization patterns with respect to the coupling strength of the oscillators. We propose here the framework of eigenvector-based analysis to identify the synchronization patterns in the general complex networks and, incorporating the conventional method of eigenvalue-based analysis, investigate the emergence and transition of the cluster synchronization states. We are able to argue and demonstrate that, without a prior knowledge of the network symmetries, the method is able to predict not only all the cluster synchronization states observable in the network, but also the critical couplings where the states become stable and the sequence of these states in the process of synchronization transition. The efficacy and generality of the proposed method are verified by different network models of coupled chaotic oscillators, including artificial networks of perfect symmetries and empirical networks of non-perfect symmetries. The new framework paves a way to the investigation of synchronization patterns in large-size, general complex networks.

Keywords cluster synchronization complex networks network symmetry coupled oscillators

Corresponding Author(s): Xingang Wang

Issue Date: 10 August 2023

Cite this article:

Huawei Fan,Ya Wang,Xingang Wang. Eigenvector-based analysis of cluster synchronization in general complex networks of coupled chaotic oscillators[J]. Front. Phys. , 2023, 18(4): 45302.

URL:

https://academic.hep.com.cn/fop/EN/10.1007/s11467-023-1324-0
https://academic.hep.com.cn/fop/EN/Y2023/V18/I4/45302

Fig.1 CS in small-size networks of perfect symmetries. (a1) The structure of the 6-node network. The network dynamics is described by Eq. (15). (a2) By model simulations, the CS states observed in synchronization transition.

δ x i = ? | x i ? x 2 | ? T

is the synchronization error between oscillator

i

and the 2nd oscillator.

ε

denotes the uniform coupling strength. In the range of

ε ∈ (1.8, 2.7)

, two synchronization pairs, (2, 6) and (3, 5), are generated. In the range of

ε ∈ (2.7, 4.2)

, two synchronization clusters, (1, 2, 6) and (3, 4, 5), are formed. Global network synchronization is achieved when

ε > ε c ′ ≈ 4.2

. (b1) The network structure of the Nepal power grid. The network nodes are partitioned into three non-trivial clusters,

C 1 = {1, …, 5}

C 2 = {9, …, 13}

and

C 3 =

{6, 7, 8}

, and two trivial clusters,

C 4 = {14}

and

C 5 = {15}

. (b2) By numerical simulations, the variation of the synchronization errors of the non-trivial clusters,

δ x m c = ∑ i, j ∈ C m ? | x i ? x j | ? T / n m (n m ? 1)

(with

m = 1, 2, 3

the cluster index and

n m

the number of nodes in cluster

m

), with respect to

ε

. The 1st, 2nd and 3rd clusters are synchronized at about

ε

= 0.9, 0.7 and 0.32, respectively.

Fig.2 CS in complex community networks of non-perfect symmetry. (a1) The structure of the community network, which contains

N = 90

nodes and

M = 3

communities of equal size. (a2) The elements of the eigenvectors

v 2

v 3

and

v 4

. (b1) The matrix of eigenvector distance,

{δ e i, j} i, j = 1, …, N

, when the mode of

λ 2

is unstable. (c1) The matrix of eigenvector distance when the modes of

λ 2

and

λ 3

are unstable. (d1) The matrix of eigenvector distance when the modes of

λ 2

λ 3

and

λ 4

are unstable. (b2) The matrix of synchronization error,

{δ x i, j} i, j = 1, …, N

, for

ε = 7.6

. (c2) The matrix of synchronization error for

ε = 6

. (d2) The matrix of synchronization error for

ε = 0.5

Fig.3 (a1) The structure of the community network consisting of

N = 180

nodes and

M = 6

communities. (a2) The elements of the eigenvectors

v 2

v 3

and

v 4

. (b1) The matrix of eigenvector distance when the mode of

λ 2

is unstable. (c1) The matrix of eigenvector distance when the modes of

λ 2

and

λ 3

are unstable. (d1) The matrix of eigenvector distance when the modes of

λ 2

λ 3

and

λ 4

are unstable. (b2?d2) are the matrices of synchronization error,

{δ x i, j} i, j = 1, …, N

, obtained by numerical simulations for the coupling strengths

ε = 10

(b2),

ε = 5.65

(c2) and

ε = 4

(d2).

Fig.4 CS in the cortico-cortical network of the cat brain. (a1) The network structure. The nodes are divided into four functional divisions:

\embassy C V = (1, …, 16)

(visual division),

\embassy C A = (17, …, 23)

(auditory division),

\embassy C S M = (24, …, 39)

(somatomotor division) and

\embassy C F L = (40, …, 53)

(frontolimbic division). (a2) The components of the eigenvectors

v 2

and

v 3

. (b1) The matrix of eigenvector distance,

{δ e i, j} i, j = 1, …, N

, when the mode of

λ 2

is unstable. (c1) The matrix of eigenvector distance when the modes of

λ 2

and

λ 3

are unstable. (b2) The matrix of synchronization error,

{δ x i, j} i, j = 1, …, N

, for the coupling strength

ε = 2.2

. (c2) The matrix of synchronization error for

ε = 1.9

Fig.5 CS in the cortical network of the human brain. The synchronization patterns predicted by the theory under different coupling strengths are plotted in the first row, and the corresponding results obtained by model simulations are plotted in the second row. (a1, a2) show the results for

ε =

15

, by which only the mode of

λ 2

is unstable. (b1, b2) are the results for

ε = 8.5

, by which the modes of

λ 2, 3, 4

are unstable. (c1, c2) are the results for

ε = 5.0

, by which the modes

λ 2, …, 6

are unstable. In each subplot, nodes in the non-synchronizable regions are represented by grey symbols, and nodes in the synchronizable regions are represented by colored symbols. Nodes with the same color (except the grey-colored nodes) form a synchronization cluster. See the context for more details.

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