^{1}

^{1}

^{*}

^{1}

Multilayer network is a frontier direction of network science research. In this paper, the cluster ring network is extended to a two-layer network model, and the inner structures of the cluster blocks are random, small world or scale-free. We study the influence of network scale, the interlayer linking weight and interlayer linking fraction on synchronizability. It is found that the synchronizability of the two-layer cluster ring network decreases with the increase of network size. There is an optimum value of the interlayer linking weight in the two-layer cluster ring network, which makes the synchronizability of the network reach the optimum. When the interlayer linking weight and the interlayer linking fraction are very small, the change of them will affect the synchronizability.

Complex networks are ubiquitous in the world, such as power and transportation networks, biological networks, economic and financial networks, and social networks. The study of complex networks in many disciplines has allowed us to better understand a myriad of complex phenomena, including the spread of disease on networks of human contacts, the functioning of intricate biological pathways, and gene circuits, as well as to provide theoretical support for engineers to control or optimize artificial interacting systems [

Based on the above research, this paper extends the single-layer cluster ring networks studied by Lu et al. to a two-layer network model. In this paper, a two-layer cluster ring network model is proposed to study the effect of the change of the interlayer linking weight and linking fraction on the synchronizability of the two-layer cluster ring networks, and explores the change of synchronizability as the cluster blocks and nodes within the cluster blocks change.

This paper is organized as follows. Section 2 briefly introduces the master stability function (MSF) approach and network models used in this paper. Section 3 discusses how the route to synchronizability is affected by changing the network size for the complex cluster ring networks. Section 4 describes the relationship between the synchronizability and the interlayer linking weight and linking fraction. Finally, concluding comments are given in Section 5.

Cluster ring network refers to a network that connects m cluster blocks into a ring [

For a multiplex network consisting of M layers each consisting of N nodes, the dynamics of n-dimensional node x i α (the i-th node in the α-th layer) can be described by the following differential equation [

d x i α d t = f i α ( x i α ) + ∑ j = 1 N a i j α Γ 1 ( x j α − x i α ) + ∑ β = 1 M ω i α β Γ 2 ( x i β − x i α ) , (1)

where 1 ≤ i ≤ N ,1 ≤ α ≤ M , x i a ∈ ℜ n is the state vector of the i-th node in the α-th layer, f i α : ℜ n → ℜ n governs the dynamics of the i-th node in the α-th layer, Γ 1 : ℜ n → ℜ n is the inner coupling function defining the interaction between nodes within any particular layer, and Γ 2 : ℜ n → ℜ n is the inner coupling function defining the interaction between nodes on separate layers. To apply the master stability framework to the composite multiplex, it is necessary to assert identical nodal dynamics and identical coupling functions: f i α = f and Γ 1 = Γ 2 = Γ . The intralayer linking weight a i j α is positive if and only if there is a link from node j to node i ( j ≠ i ) within the α-th layer. Otherwise, a i j α = 0 . The interlayer linking weight ω i α β is similarly positive if and only if there is a link between node i in layer α and node i in layer β . Otherwise, ω i α β = 0 . Note that there are no links between node i on layer α and a different node j ( j ≠ i ) on a different layer β ( α ≠ β ) .

Two-layer cluster ring network dynamics model is as follows:

d x i α d t = f i α ( x i α ) + ∑ j = 1 N a i j α Γ 1 ( x j α − x i α ) + ∑ β = 1 2 ω i α β Γ 2 ( x i β − x i α ) , (2)

where 1 ≤ i ≤ N ,1 ≤ α ≤ 2 .

Corresponding to a network, the eigenvalues of Laplacian matrices are very important for studying network dynamics. A Laplacian matrix of a multilayer network is called a super-Laplacian matrix £ , it can be decomposed into two parts: the intralayer super-Laplacian matrix £ L and the interlayer super-Laplacian matrix £ I [

£ = £ L + £ I (3)

As for £ L , it can be represented by the direct sum of the Laplacian matrix within each layer, namely,

£ L = ( a 1 L ( 1 ) 0 ⋯ 0 0 a 2 L ( 2 ) ⋯ 0 ⋮ ⋮ ⋱ ⋮ 0 0 ⋯ a M L ( M ) ) = ⊕ α = 1 M a α L ( α ) (4)

where a α is the intralayer linking weight in the α-layer. L ( i , j ) ( α ) = − 1 , if there is a link from node j to node i in the α-th layer. Otherwise, L ( i , j ) ( α ) = 0 , and all rows of L ( α ) sum up to 0. For a two-layer network, when the interlayer linking weight is ω , the interlayer super-Laplacian matrix £ I is:

£ I = L I ⊗ I = ( ω − ω − ω ω ) ⊗ I = ( ω I − ω I − ω I ω I ) (5)

For a two-layer network with N nodes per layer, the super-Laplacian matrix is

£ = £ L + £ I = ( a 1 L ( 1 ) 0 0 a 2 L ( 2 ) ) + ω ( I − I − I I ) = ( a 1 L ( 1 ) + ω I − ω I − ω I a 2 L ( 2 ) + ω I ) (6)

where ω is the interlayer linking weight, a 1 and a 2 are the intralayer linking weight in the first and second layers, respectively. L ( 1 ) and L ( 2 ) are the Laplacian matrix in the first and second layers, respectively. I is the N × N identity matrix.

Consider a general complex cluster network with m clusters, where each cluster contains n nodes and all clusters arrange into a ring. The linking between neighboring clusters is specific intercluster linking, involving selecting a specific node from each cluster. The cluster block is composed of random, small world or scale-free structures. In this paper, we discuss the change of synchronizability of two-layer cluster ring network when the network size increases. The increase in the size of the network can be divided into two situations: the first case is to keep the cluster block of the two-layer cluster ring network unchanged ( m = 20 ) , and increase the number of nodes in each block (n from 10 to 100), the numerical simulation results are shown in

[

We consider the impact of the interlayer linking weight and interlayer linking fraction on synchronizability of two-layer cluster ring network. Each layer nodes of the network is fixed to N = 1000 . We recorded λ 2 and log 2 R of the Laplacian matrix obtained by running 50 times for each network model. The experimental simulation data was taken as the average of 50 experiments.

First, we consider the impact of the interlayer linking weight on network synchronizability. We assume that the interlayer linking fraction γ = 1 , that is, every node in one layer is linked to its counterpart in the other layer. The interlayer linking weight ω is varied from 0.0002 to 2.002. The simulation results are shown in

Now, we explore the effect of interlayer linking fraction on synchronizability. The interlayer linking weight ( ω = 1 ) is fixed, we randomly connect the node pairs in different layers and change the connection probability. The interlayer linking fraction γ is varied from 0.001 to 1. The simulation results are shown in

Phase diagrams for λ 2 and log 2 R with respect to both ω and γ are displayed in Figures 6-11 to illustrate the impact of the two parameters on the

synchronizability of two-layer cluter ring networks, for the three different two-layer network models. ω takes a change between 0.0002 and 0.02, and γ takes a change from 0.01 to 1. BA-cluster ring network denotes that the internal structure of the cluster blocks in the two-layer cluster ring network is scale-free. ER-cluster ring network denotes that the internal structure of the cluster blocks in the two-layer cluster ring network is random. WS-cluster ring network denotes that the internal structure of the cluster blocks in the two-layer cluster ring network is small-world.

In conclusion, we find that ω ( ω ∈ [ 0.0002,0.0018 ] ) is maintained very small, synchronizability is greatly affected by γ , and the synchronizability of the network will increase as γ increases. When ω ( ω ∈ [ 0.002,0.02 ] ) is slightly

larger, γ is getting bigger and bigger, and the synchronizability is not affected by γ . γ ( γ ∈ [ 0.01,0.09 ] ) is maintained very small, synchronizability is greatly

affected by ω , and the synchronizability of the network will increase as ω increases. When γ ( γ ∈ [ 0.1,1 ] ) is slightly larger, ω is getting bigger and bigger, and the synchronizability is not affected by ω .

space ( ω , γ ) for randomly correlated two-layer BA-cluster ring networks.

γ values with ω .

As shown in Figures 8-11, the effects of varying ω and γ on the two-layer WS-cluster ring and the two-layer ER-cluster ring tend to be consistent with the two-layer BA-cluster ring. The simulation results are consistent with the simulation results of two-layer BA-cluster ring.

In this paper, a two-layer cluster ring network model is established. Through a large number of simulation experiments, the effects of two-layer cluster ring network size variation, interlayer linking weight and interlayer linking fraction change on synchronizability are studied. The experimental results show that such networks are not scalable with respect to synchronizability. Network synchronizability decreases with increasing network size for any cluster structures. Also, the interlayer linking weight and interlayer linking fraction has a significant influence on the network synchronizability. There is an optimum value of the interlayer linking weight in the network, which makes the synchronizability of the network reach the optimum. ω is maintained very small; synchronizability is greatly affected by γ , and the synchronizability of the network will increase as γ increases. When ω is slightly larger, γ is getting bigger and bigger, and the synchronizability is not affected by γ . γ is maintained very small, synchronizability is greatly affected by ω , and the synchronizability of the network will increase as ω increases. When γ is slightly larger, ω is getting bigger and bigger, and the synchronizability is not affected by ω . The smaller interlayer linking weight and the smaller interlayer linking fraction will affect the synchronizability of the two-layer cluster ring network.

This project is supported by National Natural Science Foundation of China (Nos. 61563013, 61663006) and the Natural Science Foundation of Guangxi (No. 2018GXNSFAA138095).

The authors declare no conflicts of interest regarding the publication of this paper.

Deng, Y., Jia, Z. and Liao, L. (2019) Synchronizability of Two-Layer Cluster Ring Networks. Communications and Network, 11, 35-51. https://doi.org/10.4236/cn.2019.112004