In the real world, community structure has been discovered in many large-scale physical and social systems such as school friendship networks,^[1] biological networks,^[2] congressional cosponsorship networks,^[3] protein–protein interaction networks,^[4] World Wide Web,^[5] and so on. They are usually modeled by a community network, in which the nodes in the same community have the identical function or property but in different communities have nonidentical ones. For example, in protein–protein interaction networks, communities correspond to functional groups/clusters, i.e., to proteins having the same or similar functions, which are expected to be involved in the same processes.^[6] A great number of phenomena show that the current state of a node is affected by not only the current states but also the states in the previous period of its neighbor nodes. For example, in biological neural networks, it usually has a spatial extent due to the presence of parallel pathways with a variety of axon sizes and lengths.^[7,8] Thus, there is a distribution of transmission delays and it cannot be modeled with discrete time delays. In such a case, a more appropriate way is to incorporate the distributed time delays in the models.^[7,8] Until now, many valuable results about dynamical networks with distributed delays have been obtained.^[9–11] However, community networks with distributed delays have seldom been considered, thus they deserve further investigations.

In recent years, synchronization has gained extensive attention from various fields on account of its broad practical applications, such as regulation of power grid, parallel image processing, the operation of no-man air vehicle, the realization of chain detonation, etc.^[12,13] Cluster synchronization,^[14–22] a special synchronization of complex dynamical networks, denotes that synchronization of the nodes in the same community can be achieved, but synchronization of the nodes in different communities cannot be achieved. That is, the nodes in the same or different communities have identical or nonidentical goals. Further, many studies show that cluster synchronization has increasing applications in different fields.^[23,24] Kaneko pointed out that cluster synchronization is an important phenomenon in communication engineering and biological science.^[23] Yoshioka showed that neurons are divided into several clusters in neurons networks interconnected with chemical synapses and synchronization of the neurons is achieved in each cluster. From the viewpoint of real life, cluster synchronization by inner adjustment of communities and influence of different communities without external control cannot be achieved. An effective method of achieving synchronization, i.e., adding proper controllers, has been found from lots of obtained results. Thus, how to design proper controllers is a key issue.

Impulsive control is a representative control strategy and its main characteristic is working on the nodes only at inconsecutive instants.^{[21,22,25–31]} Due to the characteristic, researchers have been studying impulsive control deeply for a long time. Cai et al. considered the exponential cluster synchronization of hybrid-coupled delayed dynamical networks via impulsive control.^[21] Fan et al. considered the cluster synchronisation in non-linearly coupled networks with non-identical nodes and time-varying delays via impulsive control.^[22] As we know, the impulsive gains and intervals related to the cost and efficiency are important parameters of impulsive controllers. Therefore, designing proper impulsive controllers means choosing appropriate impulsive gains and intervals in terms of synchronization criteria. However, for different community networks with nonidentical system parameters, the needed impulsive gains and intervals need to be chosen repetitively, which undoubtedly increases workloads and reduces efficiency. In order to overcome the shortcoming, we introduce an adaptive strategy to design unified impulsive controllers. A community network combined with adaptive impulsive controllers can adjust impulsive gains or intervals adaptively and achieve cluster synchronization.

Motivated by the above discussion, the cluster synchronization of the community network with distributed time delays is investigated. On one hand, proper impulsive controllers are added to the community networks for achieving cluster synchronization, and simple criteria about the system parameters, the impulsive gains, and intervals are obtained on the basis of the Lyapunov function method and linear matrix inequality technique. On the other hand, proper controllers are improved by introducing an adaptive strategy and criteria are obtained as well. Noticeably, the adaptive law of the impulsive gain and the method for estimating the impulsive instants are provided in terms of synchronization criteria. In Section 2, the network model is introduced and some preliminaries are given. In Section 3, the impulsive controllers for achieving synchronization are designed and the synchronization criteria are provided. In Section 4, several numerical simulations are performed to verify the results. In Section 5, the conclusion for this paper is given.

Consider a community network consisting of k (2 ≤ k ≪ N) communities and N nodes with distributed time delays. As we know, in many real networks with community structure, such as neural, social, biological, and communication networks, the individual nodes in the same community can be viewed as the identical functional units, while any pair of nodes in different communities are essentially different according to their functions.^[21] Therefore, we assume that the node dynamics and the distributed delays in the same community are identical, while those in different communities are nonidentical. Further, the distributed delays between different pairs of communities are nonidentical as well. According to these settings, the community network can be described as

Let η_πi(t) be a solution of an isolated node in the π_i-th community, i.e., . The objective here is to achieve synchronization of the network (1) by designing proper controllers, i.e.,

For achieving the synchronization, proper impulsive controllers are designed and applied onto the network (1). The controlled network is

Let e_i(t) = y_i(t) − η_πi(t) be the synchronization errors. Then the error systems are

Let G_m = {γ_m−1 + 1,…,γ_m} (m = 1,2,…,k) with γ₀ = 0 and γ_k = N. suppose that B ∈ R^{N × N} can be divided into the following block form:

Definition 1 If B_pp is a zero-row-sum, symmetric, and irreducible matrix with non-negative off-diagonal elements, we say that B_pp belongs to class 𝐶₁ and denote B_pp ∈ 𝐶₁, p = 1,2,…,k.

Definition 2 If B_pm is a zero-row-sum matrix, we say that B_pm belongs to class 𝐶₂ and denote B_pm ∈ 𝐶₂, p,m = 1,2,…,k.

Definition 3 If B_pp ∈ 𝐶₁, B_pm ∈ 𝐶₂, p,m = 1,2,…,k (p ≠ m), we say that B belongs to class 𝐶₃ and denote B ∈ 𝐶₃.

Assumption 1 Suppose that there exists a positive constant M such that

Lemma 1 (see Ref. [7]) Let A be a symmetric n × n real matrix and A > 0. Then

Let , be the impulsive intervals,, and be the minimum and the maximum of , λ be the largest eigenvalue of , μ(t) = (1 + φ(t))². From the definition of φ(t), one has μ(t) = 1 for t ≠ t_υ. The outer coupling matrix B considered in this paper is supposed to belong to 𝐶₃.

Theorem 1 Suppose that Assumption 1 holds. If there exists a constant δ > 0 such that

Proof Consider the following Lyapunov function candidate:

When t ∈ (t_υ−1,t_υ),

When t = t_υ,

Then, for t ∈ (t_υ,t_υ+1],

Remark 1 In particular, if the set is empty set, the Lyapunov function is

Remark 2 From condition (4), for any given community network, the needed impulsive gains and intervals for achieving cluster synchronization can be estimated by some calculations. Similar to many existing results about the impulsive control, when the impulsive gains are fixed, the upper bound of the impulsive intervals should be smaller than a certain positive constant. On the other hand, for different community networks, the needed values are different. That is, condition (4) is conservative. For solving this problem, an adaptive strategy is adopted to design adaptive impulsive controllers.

Theorem 2 Suppose that Assumption 1 holds. If there exists a constant δ > 0 such that

Proof Consider the following Lyapunov function candidate:

When t ∈ (t_υ−1,t_υ),

Thus, similar to the proof of Theorem 1, the proof is obtained.

Remark 3 Generally, for any given ρ_υ and δ, the conditions (7) hold when choosing

Remark 4 In practical application, when φ(t_υ) and δ are fixed, the control intervals should be chosen to be as large as possible to reduce the cost. Thus, the impulsive instants t_υ can be chosen by finding the maximum value of t_υ subject to with υ = 1,2,….

Remark 5 From conditions (7), the constant λ in condition (4) with respect to the system parameters does not need to be calculated beforehand and can be estimated by when the cluster synchronization is achieved. Remarks 3 and 4 show that the impulsive gains or instants can be estimated adaptively through choosing proper parameters. Specially, from Remark 4, the upper bound of the impulsive intervals does not need to be calculated beforehand. That is, the conditions (7) are not conservative to some extent.

In this section, we use three numerical examples to demonstrate the effectiveness of the theoretical results. Choose the density function as ϕ_pm(θ) = 1/ξ_pm, the inner coupling matrix as the identity matrix, and ξ₁₁ = 0.2, ξ₁₂ = 0.1, ξ₁₃ = 0.2, ξ₂₁ = 0.4, ξ₂₂ = 0.4, ξ₂₃ = 0.1, ξ₃₁ = 0.2, ξ₃₂ = 0.4, ξ₃₃ = 0.1 in the following examples. In addition, choose the initial values of y_i(t) and η_πi(t) randomly.

Example 1 Consider a community network consisting of 20 nodes in 3 communities, whose topology is shown in Fig. 1. Choose the node dynamics of the first community as the Lorenz system^[32]

Fig. 1.

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	Fig. 1. A community network consisting of 20 nodes in 3 communities. The top, bottom-left, and bottom-right are the first, the second, and the third communities, respectively. The outer coupling matrix B is defined as: if nodes i and j (j ≠ i) are connected by a solid line, then bij = 0.001; if nodes i and j (j ≠ i) are connected by a dashed line, then bij = −0.001.

According to the discussion in Ref. [35], we can choose M = 53 such that Assumption 1 holds. Choose the impulsive gains φ(t_υ) = −0.999 and intervals ρ_υ = 0.12. Then the maximum eigenvalue of is λ = 109.0001, and ln μ(t_υ) + δ + λρ_υ = −0.7345 < 0 with δ = 0.001, i.e., the condition 4 holds. Figure 2 shows the orbits of synchronization errors in 3 communities.

Fig. 2.

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	Fig. 2. The orbits of synchronization errors in 3 communities.

Example 2 Consider the same community network in Example 1 via adaptive impulsive control.

First, choose ρ_υ = 0.12, δ = 0.001, ω = 0.0003, and . According to Remark 3, choose with α = 0.001. Figure 3 shows the orbits of synchronization errors in 3 communities. Figure 4 shows the impulsive gain φ(t_υ).

Fig. 3.

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	Fig. 3. The orbits of synchronization errors in 3 communities.

Fig. 4.

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	Fig. 4. The impulsive gain φ(tυ).

Second, choose the impulsive gain φ(t_υ) = −0.999, ω = 0.0003, , δ = 0.001. According to Remark 4, we can estimate the impulsive instants. Figure 5 shows the orbits of synchronization errors in 3 communities. Figure 6 shows the impulsive interval ρ_υ.

Fig. 5.

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	Fig. 5. The orbits of synchronization errors in 3 communities.

Fig. 6.

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	Fig. 6. The impulsive interval ρυ.

In Example 1, the impulsive interval is ρ_υ = 0.12. Figure 6 shows that the network achieves synchronization when the impulsive interval converges to 0.848. Obviously, the impulsive interval with the adaptive impulsive strategy is larger than that without the strategy.

Example 3 We generate the first community network topology as the WS small-world network^[36] with N₁ = 50, K = 5, and p = 0.2, the second as the WS small-world network^[36] with N₂ = 50, K = 6, and p = 0.1, and the third as the NW small-world network^[37] with N₃ = 50, K = 2, and p = 0.1. Further, we add a few connections among the three communities. Choose the same node dynamics as Example 1.

First, choose ρ_υ = 0.2, δ = 0.001, ω = 0.00005, and . According to Remark 3, choose with α = 0.001. Figure 7 shows the orbits of synchronization errors in 3 communities. Figure 8 shows the impulsive gain φ(t_υ).

Fig. 7.

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	Fig. 7. The orbits of synchronization errors in 3 communities.

Fig. 8.

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	Fig. 8. The impulsive gain φ(tυ).

Second, choose the impulsive gain φ(t_υ) = −0.9, ω = 0.00005, , δ = 0.001. According to Remark 4, we can estimate the impulsive instants. Figure 9 shows the orbits of synchronization errors in 3 communities. Figure 10 shows the impulsive interval ρ_υ.

Fig. 9.

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	Fig. 9. The orbits of synchronization errors in 3 communities.

Fig. 10.

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	Fig. 10. The impulsive interval ρυ.

Examples 2 and 3 indicate that the cluster synchronization of networks with nonidentical node dynamics and topological structures can be achieved without calculating the constants with respect to the system parameters in advance. Thus, the adaptive impulsive controllers are unified for different community networks.

Community networks with distributed time delays and nonidentical node dynamics are investigated. Further, the coupling delays in different communities are also assumed to be nonidentical. Through designing proper impulsive controllers, the nodes in the different communities are synchronized onto different orbits and the nodes in the same community are synchronized with each other. That is, the cluster synchronization of the community network is achieved. With the linear matrix inequality technique and Lyapunov function method, some synchronization criteria with respect to the impulsive gains, instants, and system parameters are obtained. Noticeably, an adaptive strategy is adopted to make the impulsive controllers possess universality and the corresponding synchronization criteria are obtained as well. When the impulsive intervals are fixed, the impulsive gain can adaptively approach to the proper value according to the updating law. When the impulsive gains are fixed, the impulsive instants can be estimated by solving a sequence of maximum value problems. Finally, the effectiveness of the theoretical results are demonstrated by three numerical examples.