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Synchronization for Complex Networks with Multiple Weights Under Recoverable Attacks

1.1 Introduction

During the last decade, the dynamical behavior of complex networks (CNs) has aroused increasing attention because CNs prevalently exist in the real world. Particularly, synchronization has been an appealing research topic in CNs, and many meaningful results have been obtained [1–16]. By choosing appropriate adaptive state-feedback controllers and Lyapunov functional, Zhou et al. [1] discussed the global and local synchronization in a CN with uncertain coupling functions. In [4], the synchronization problem for a CN with switching disconnected topology was addressed, and some synchronization conditions were established for such a network model. Lv et al. [5] tackled the exponential synchronization problem for CNs with coupling delay based on the impulsive control and event-triggered control techniques. In [11], the synchronization problem for stochastic CNs was discussed via pinning control technique and graph theory, in which the topology structure may be unknown. Zhu et al. [14] used the adaptive control method to deal with the synchronization problem for a type of CNs with time-varying delay, in which the restriction that time delay is differentiable is removed.

For some practical networks, such as urban population flow networks, food webs, etc., may be better described by CNs with multiple weights (CNMWs). More recently, some authors have addressed the problem of synchronization for CNMWs [17–26]. Wang et al. [17] not only investigated the pinning synchronization in the CNMWs with undirected and directed topologies but also presented several feedback gains and coupling strengths adjustment schemes. In [18], a criterion of synchronization for output-strictly passive CNMWs was obtained, and the synchronization problem of CNMWs was further discussed based on the nodes- and edges-based pinning control approaches, and the output-strict passivity. Zhao et al. [23] introduced a multiple delayed CN model with uncertain inner coupling matrices and developed a criterion of synchronization through the adaptive control scheme for such a network model. Dong et al. [24] took into account the exponential synchronization of multiple delayed CNs with switching and fixed topologies by employing the scramblingness property for adjacency matrix. Qin et al. [26] analyzed the robust synchronization of multiple delayed CNs, and a criterion for guaranteeing the robust synchronization was also developed by employing the adaptive state-feedback controller.

It is well known that the network topology may be destroyed owing to the various forms of attacks (e.g., power grids, military communication networks, and so on [27–29]), which might lead to undesirable dynamical behavior in the CNs. Consequently, it is very meaningful to study the dynamical behavior for CNs under attacks. Recently, some researchers have studied the synchronization problem of CNs suffering the attacks [30,31]. Wang et al. [30] investigated the synchronization for multiple memristive neural networks with the communication links subject to attacks and developed several synchronization criteria based on inequality techniques, -matrix properties, and event-triggered control method. Wang et al. [31] gave a global synchronization criterion for a network model suffering the successful but recoverable attacks by exploiting the switching system theory and derived the upper bounds of the average recovering time and the attack frequency. Regretfully, the network models with single coupling were discussed in these existing works about the synchronization for CNs under attacks [30,31], and the synchronization for CNMWs subject to attacks has not yet been explored. Obviously, it is very valuable and significative to further address the synchronization problem of CNMWs suffering the attacks.

This chapter discusses the synchronization for CNs with multiple state couplings (CNMSCs) or CNs multiple delayed state couplings (CNMDSCs) under recoverable attacks, respectively. The main contributions of our work are summarized as follows. First, we not only give a sufficient condition for ensuring the synchronization of directed CNMSCs suffering the attacks but also further study the synchronization problem by selecting the suitable state-feedback controller. Second, the analysis and control for the synchronization problem of undirected CNMSCs subject to attacks are also discussed, and several synchronization criteria are presented based on some inequality techniques. Third, we not only develop several synchronization criteria for CNMDSCs under attacks by constructing appropriate Lyapunov functional but also devise the suitable state-feedback controller to ensure the network synchronization.

1.2 Preliminaries

1.2.1 Notations

Let ; for any real square matrix , ; and denote the smallest and the largest eigenvalues of real symmetric matrix.

1.2.2 Lemmas

Lemma 1.1 (See [32]) Define

and let the sum of each row in the matrix be equal. Then, satisfying

Remark 1.2  The matrices and are very important for us to discuss the synchronization problem of CNMSCs and CNMDSCs, which will be utilized throughout this chapter.

Lemma 1.3  (See [33]) The Kronecker product has the following properties:

1. (i)
2. (ii)
3. (iii)
4. (iv)

where , and are matrices with suitable dimensions.

1.2.3 Network Models

In this chapter, two kinds of network models are considered as follows:

where ; denotes the state vector of the th node; stands for the coupling strength; is a continuous function; denotes the inner coupling matrix; represents the time delay; stands for the outer coupling matrix satisfying the following condition: if there is an edge from node to node , then ; otherwise, ; and

In this chapter, the function meets the following condition (see [34]):

for some constant matrices and , and a positive constant , where .

Remark 1.4  In the networks (1.1) and (1.2), the different coupling forms are required to have the same topology. In fact, this situation commonly exists in some real-life networks, such as inter-city population flow networks, urban public traffic networks, and so on. For instance, in the inter-city population flow networks, choosing each city as a node, the edge represents the population flow from any city to any other city. Obviously, the changes of the urban population depend on many factors, such as economic development, climate change, and education. Therefore, the intercity population flow networks should be modeled by CNMWs, in which each influencing factor corresponds to a coupling form. Apparently, the different coupling forms in the intercity population flow networks have the same topology.

Remark 1.5  In this chapter, the topology subject to the “successful” but recoverable attacks is discussed in CNMSCs (1.1) and CNMDSCs (1.2). Namely, the attacks happen at and thus makes the topology to be broken, and the broken topology is recovered after , . In practice, this phenomenon exists in many real networks, such as military communications networks, and power grids [35,36]. Therefore, some authors have studied the synchronization of CNs suffering the successful but recoverable attacks [30,31]. However, the synchronization for CNMWs under the successful but recoverable attacks has not yet been discussed.

When , the attacks happen and the topologies of the networks (1.1) and (1.2) are destroyed. After , the broken topology can be recovered. In this chapter, we assume that the networks (1.1) and (1.2) suffering the attacks have different types of topologies, and .

Therefore, one has

where , , , represents the outer coupling matrix of the networks (1.1) and (1.2) subject to the attacks, in which has the same definition as .

Denote

where .

Then, we have

in which .

Next, the synchronization definition for the network (1.4) [or (1.5)] is introduced as follows.

Definition 1.6  The network (1.4) [or (1.5)] can achieve synchronization if

Denote

1.3 Synchronization of CNMSCs Under Recoverable Attacks

1.3.1 Synchronization of CNMSCs with Directed Topology

(1) Synchronization analysis

Evidently, (1.4) can be rewritten as

Theorem 1.7 If there are two positive constants and satisfying

1. (i)
2. (ii)

in which , , the network (1.4) is...