## 4 Numerical Examples  
**Example 1**. : _Consider the CCPNs as shown in Figure 2, which have five nodes (node 5 is an isolated node). The parameters are chosen as follows:_  
\[A=\left[ \begin{array}{ccc}-5&4&7\\ -3&-7&-4\\ -2&-7&-9\end{array} \right],G=\left[ \begin{array}{ccc}1&-1&0&0\\ 0&1&-1&0\\ 0&0&1&-1\\ -1&0&0&1\end{array} \right],B=\left[ \begin{array}{c}0.4\\ 1.7\\ 0.3\end{array} \right],\]  
\(\Gamma=diag(0.1,0.1,0.1),\epsilon_{\min}=7.236,\epsilon_{1}=1,\gamma_{i}=0.5,q _{1}=6.8541,q_{2}=5.2361,q_{3}=2.7058,\epsilon_{2}=6.236,\alpha=19.9552\). _When the complex cyber-physical networks are not attacked by DoS attacks, one can obtain the H matrix by the communication topologies as shown in Figure 2._  
\[H=\left[ \begin{array}{cccc}3&-1&0&-1\\ -1&1&0&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{array} \right].\]  
_By Theorem 1, we obtain the controller gain K and Lyapunov variable P as follows:_  
\[K=\left[ \begin{array}{cccc}-2.4163&8.8224&-4.9069\end{array} \right],\]  
\[P=\left[ \begin{array}{cccc}1.0221&-0.5332&0.5421\\ -0.5332&0.9725&-0.7241\\ 0.5421&-0.7241&1.1136\end{array} \right].\]  
_The initial states of nodes are \(x_{1}=\left[ \begin{array}{cccc}-2&0.5&0.5\end{array} \right]^{T}\), \(x_{2}=\left[ \begin{array}{cccc}2&-3&4\end{array} \right]^{T}\), \(x_{3}=\left[ \begin{array}{cccc}2&3&3\end{array} \right]^{T}\), \(x_{4}=\left[ \begin{array}{cccc}-3.5&-0.5&1\end{array} \right]^{T}\)._  
_As shown in Figure 3, the trajectories of error states between follower nodes and the isolated node approach zero when the DoS attacks are dormant. Figures 4 and 5 show the trajectories of synchronization errors based on the ETC strategy and transmission errors of nodes, which prove that all nodes of the systems are synchronized with the distributed controller. The transmission attempts of nodes based on the ETC strategy are shown in Figure 6. It means that each node's triggering rules are independent. It can be seen from Figures 3, 4 and 6 that the proposed ETC strategy saves communication resources and reduces the unnecessary transmission of the systems while ensuring the convergence effect. The above simulation results show that when the external attacks are zero, the system realizes the synchronization of all nodes based on the distributed event-triggered controller, which verifies the validity of Theorem 1 in this paper._  
_When the systems are attacked by the DoS attacks, it is supposed that two edges can be attacked separately or simultaneously. Figure 7b-d show several network topologies generated by DoS attacks. Figure 8 shows the DoS signal, where mode 0 denotes DoS dormancy and mode 1 denotes DoS activation. Suppose that three topologies caused by DoS attacks are shown as in Figure 8. We have_  
\[\overline{H_{1}}=\left[ \begin{array}{cccc}2&-1&0&0\\ -1&1&0&0\\ 0&0&1&-1\\ 0&0&0&1\end{array} \right],\overline{H_{2}}=\left[ \begin{array}{cccc}2&-1&0& 0\\ -1&1&0&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{array} \right],\]  
\[\overline{H_{3}}=\left[ \begin{array}{cccc}3&-1&0&-1\\ -1&1&0&0\\ 0&0&1&-1\\ 0&0&-1&2\end{array} \right].\]_The parameters are chosen as \(\beta_{1}=15\), \(\beta_{2}=0.3843\), \(\beta_{3}=1\). By Theorem 2, we obtain the controller gain \(F\) and Lyapunov variable \(Q\) as follows:_  
\[F=\left[ \begin{array}{ccc}-0.7125&6.5357&-5.1397\end{array} \right],\]  
\[Q=\left[ \begin{array}{ccc}0.7845&-0.6392&0.9091\\ -0.6392&2.8395&-2.7592\\ 0.9091&-2.7592&3.4799\end{array} \right].\]  
_Then, we obtain \(\theta_{\max}(P)=2.2434,\theta_{\min}(P)=0.3125,\theta_{\max}(Q)=6.1641,\theta _{\min}(Q)=0.3172,\mu=19.7251\). Let \(\theta_{1}^{*}=0.8448\), then, one can obtain_  
\[F_{a_{(t_{0})}}=\frac{N_{a_{(t_{0})}t}}{t-t_{0}}\leq\frac{\theta_{1}^{*}}{2 \ln(\mu)+(\alpha+\beta_{3})\Lambda_{*}}=0.316,\tau_{a}>\frac{\alpha+\beta_{3} }{\alpha+\theta_{1}^{*}}=1.001.\]  
_This means that the DoS attacks cannot occur more than 0.316 s during a unit of time and duration smaller than 1.001 s. Figure 9 shows the trajectories of error states between follower nodes and the isolated node converge to zero under DoS attacks.