## 2 Problem Formulation and Preliminaries  
The CCPNs with \(N\) nodes are described as follows:  
\[\dot{x}_{i}(t)=Ax_{i}(t)+\sum\limits_{j=1}^{N}\mathfrak{g}_{ij}\Gamma x_{j}(t )+Bu_{i}(t) \tag{1}\]  
where \(x_{i}(t)=(x_{i1}(t),\ x_{i2}(t),\ldots,\ x_{in}(t))^{T}\in\Re^{n}\) is the state vector of node \(i\), and \(u_{i}\in\Re^{q}\) is the control input of node \(i\). \(A\in\Re^{n\times n}\), \(B\in\Re^{n\times q}\) are constant matrices. \(\Gamma\) and \(G=(\mathfrak{g}_{kl})_{N\times N}\) are inner-coupling matrix and outer-coupling matrix, respectively. \(\mathfrak{g}_{kl}=\mathfrak{g}_{lk}=1\), if node \(k\) and node \(l\) (\(k\
eq l\)) are connected; otherwise, \(\mathfrak{g}_{kl}=\mathfrak{g}_{lk}=0\ (k\
eq l)\). And \(\sum\limits_{l=1,l\
eq k}^{N}\mathfrak{g}_{kl}=-\mathfrak{g}_{kk}\). Let \(x_{N+1}(t)\) be the isolated node whose dynamics are given as follows: \(\dot{x}_{N+1}(t)=Ax_{N+1}(t)\). Denote the tracking error as  
\[\delta_{i}(t)=x_{i}(t)-x_{N+1}(t) \tag{2}\]  
For node \(i\), designing a distributed ETC protocol is as follows:  
\[u_{i}(t)=K\xi_{i}\Big{(}t_{k}^{i}\Big{)},\quad t\in\Big{[}t_{k}^{i},t_{k+1}^{ i}\Big{)} \tag{3}\]\[\xi_{i}\Big{(}t_{k}^{i}\Big{)}=\sum\limits_{j=1}^{N}a_{ij}\Big{(}x_{j}(t_{k}^{i})- x_{i}\Big{(}t_{k}^{i}\Big{)}\Big{)}+b_{i}\Big{(}x_{N+1}\Big{(}t_{k}^{i}\Big{)}-x_{i} \Big{(}t_{k}^{i}\Big{)}\Big{)} \tag{4}\]  
where \(\big{\{}t_{k}^{i}\big{\}}_{k\in\mathcal{N}}\) denotes the event-triggering time sequence for node \(i\). If node \(i\) can obtain information from the isolated node, then \(b_{i}=1\); otherwise \(b_{i}=0\). **Definition 1**. : _All nodes of CCPNs (1) are said to be synchronized if the following condition is satisfied:_  
\[\lim_{t\to\infty}[\|x_{i}(t)-x_{N+1}(t)\|]=0 \tag{5}\]  
The proposed attack model in this paper is a time-constrained DoS attack, which mainly attacks the nodes and communication links of the systems in the physical layer. The attack behavior is described below. \(J_{s}=\left[ \begin{smallmatrix}j_{s}^{ij},j_{s}^{ij}+A_{s}^{ij}\end{smallmatrix}  \right],s\in\mathcal{N}\) defines the \(s\)-th DoS time interval, where the DoS attack begins at \(j_{s}^{ij}\) and \(\Delta_{s}^{ij}>0\) represents the attack duration of the \(s\)-th attack. Then, one can obtain  
\[\Theta_{a}(t_{1},t_{2})=[t_{1},t_{2}]\cap\cup\Theta_{a}^{(i,j)}(t_{1},t_{2}) \tag{6}\]  
and  
\[\Theta_{s}(t_{1},t_{2})=[t_{1},t_{2}]\backslash\Theta_{a}(t_{1},t_{2}) \tag{7}\]  
where \(\Theta_{a}^{(i,j)}(t_{1},t_{2})=[t_{1},t_{2}]\cap\underset{s=1}{\overset{ \infty}{\cup}}J_{s}^{(i,j)}\) is a set in which the DoS attacks are active. \(\Theta_{a}(t_{1},t_{2})\) and \(\Theta_{s}(t_{1},t_{2})\) denote the sets that the DoS attacks are active and dormant, respectively. Figure 1 is a demonstration of a CCPN with six nodes under DoS attacks, in which node 6 is the isolated node. As shown in Figure 1, the malicious attacks are active at \(t_{0}\), and the systems are repaired at \(\tilde{t}_{0}\). The system transmission channel can be restored at time \(t_{1}\). That is, the communication graphs are discontinuous between \(\tilde{t}_{0}\) to \(t_{1}\). The communication topologies of networks can be restored to the initial state at \(t_{2}\). **Remark 1**. : _DoS attacks destroy the nodes or communication edges of networks such that communication links fail or nodes do not work normally. Meanwhile, without loss of generality, suppose that at least one communication link or one node in dynamic network systems is attacked at time \(t_{2l}^{ij}\) for \(l=0,1,\ldots,N\). \(t_{2l+1}^{ij}\) is the time instant when the attack has been completely restored and the entirely topology network has been restored. It is worth noticing that the CCPNs are paralyzed during \(t\in\left[ \begin{smallmatrix}t_{2l}^{ij},t_{2l+1}^{ij}\end{smallmatrix} \right)\), and the CCPNs work well during \(t\in\left( \begin{smallmatrix}t_{2l+1}^{ij},t_{2(l+1)}^{ij}\end{smallmatrix}