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A novel approach is brought forward for synchronization of a clustered network in this paper, the objective of which is twofold. The first one is to study cluster synchronization by analyzing the inner coupling matrices of the individual clusters instead of the one of the whole network. The other is to show that full synchronization can be ensured by several types of cluster synchronization, the partitions of which are connected together. Compared with the classical method for full synchronization, our approach reduces the network size to the cluster size and additionally obtains the thresholds for different types of cluster synchronization. As a numerical example, cluster and full synchronization in a special clustered network are investigated through our approach. It turns out that we obtain the same threshold for full synchronization as the one obtained by the classical method. Numerical simulations confirm the validity of our approach.

Collective behavior of complex networks has become a focal subject due to the important and extensive applications in various fields of science and technology. Full and cluster synchronization are two types of typical and fundamental collective behavior. The former means that all oscillators in a network acquire identical behavior, while the latter means that the coupled oscillators split into subgroups called clusters, and all the oscillators in the same cluster behave in the same fashion. Research on cluster and full synchronization has attracted increasing attention in the past decades.

Several effective methods have been applied to study full synchronization, which is also called complete synchronization. Pecora and Carroll proposed the famous master stability function method to study the local stability of the synchronous state [

Cluster structures can be found to exist widely in real-world networks such as circles of friends or colleagues in social networks [

However, to the best of our knowledge, most of the previous research on cluster and full synchronization focused on the topology of the whole network, which may be very complex. In this paper, the complexity of a clustered network is simplified by partitioning the whole network into clusters under a certain hypothesis. It is proved that cluster synchronization can be ensured by suitable inner couplings of the clusters, and sufficient conditions, which are independent of the outer couplings between different clusters, are obtained theoretically. Based on this result, a novel method for full synchronization is derived. If there exist two or several partitions connected together along some arrangement of all the oscillators, which imply that the intersection of the cluster synchronization manifolds corresponding to those partitions is equal to the full synchronization manifold, then full synchronization occurs if cluster synchronization corresponding to every partition is ensured. The method declares that both cluster and full synchronization can be studied by the inner topologies of the individual clusters. Obviously, the network size reduction provides convenience for the studies on synchronization in clustered networks with great mounts of oscillators.

The rest of the paper is organized as follows. Section

Consider a network composed of

Suppose that the index set

Denote the cardinal number of cluster

Suppose that all oscillators in the cluster

Suppose that

We will discuss sufficient conditions for the

The synchronization submanifold of the cluster

The cluster synchronization manifold of the partition

The transverse subspace for

The transverse space for

In case

Definitions of cluster and full synchronization in the network (

The cluster synchronization manifold

where

In case of

Before the results on cluster synchronization are carried out, two common hypotheses in previous related research should be introduced.

At first, a synchronization manifold is always supposed to be an invariant manifold in order to discuss its attractiveness. The following lemma gives a sufficient and necessary condition for a cluster synchronization manifold being an invariant manifold.

Partition the coupling matrix

According to Lemma

In order to study the inner couplings of the cluster

The second crucial hypothesis is the individual oscillator dynamics satisfying

Hypothesis (

Now, the preliminaries above, together with Lyapunov function method, bring us to the following theorem.

Suppose that hypotheses

For a rigorous proof of Theorem

As a special case, if all the row sums of

Suppose that there are a set of partitions

Rearrange the numbers

For example, the set

From the definition above, we obtain the following lemma.

A set of partitions

The proof of Lemma

Now, we are in a position to carry out the following theorem on full synchronization of the network (

There are a set of partitions

The proof of Theorem

As a special case, if all the row-sums of

Consider the system (

Define the coupling matrix

Topology structure corresponding to the coupling matrix (

Obviously, partitions

If

If

If

These results can be seen more clearly in Figure

Thresholds for the attractiveness of the synchronization manifolds

In order to be compared with the previous classical results on full (complete) synchronization, the second-largest eigenvalue of

Define the following cluster errors:

Dependence of the cluster errors

In fact, these results can be forecasted in Figure

This paper has investigated cluster and full synchronization in a clustered network. In order to study cluster synchronization, we propose the concept of principal quasi-submatrices corresponding to the individual clusters, which represent the inner couplings of the individual clusters. Theoretically, sufficient conditions independent of the outer couplings between different clusters are obtained for cluster synchronization. In order to study full synchronization, we propose the concept of partitions connected together along some arrangement. If all types of cluster synchronization of those partitions are ensured, it is proved that full synchronization occurs. The results are more advantageous than the classical results. Firstly, it allows us to divide a network composed of great amounts of oscillators into some smaller subnetworks. The network size reduction provides convenience to reduce the great amounts of calculations. Secondly, our approach can be applied to study cluster synchronization corresponding to any possible partitions. In summary, this paper has proposed a novel, convenient, and double purpose approach for both cluster and full synchronization in clustered networks.

Denoting

Denote

In order to utilize the QUAD

As we know, the symmetric matrix

Finally, some techniques in [

This project was supported the by the National Natural Science Foundation of China (Grant nos. 11162004 and 60964006), Zhejiang Provincial Natural Science Foundation of China (no. LQ12a01003), the Science Foundation of Guangxi Province (no. 2013GXNSFAA019006), and the Deanship of Scientific Research (DSR), King Abdulaziz University, under Grant no. 3-130/1434/HiCi. The authors, therefore, acknowledge the technical and financial support of KAU.