## 3 Main Results
 \tag{38}\]  
Let \(y=\frac{\|e_{i}\|}{\|\dot{\xi}_{i2}\|}\), and then we have  
\[\dot{y}\leq\max\{b_{1},b_{2}\}(1+y) \tag{39}\]  
It is concluded that \(y\leq\phi(t,\phi_{0})\), where \(\phi(t,\phi_{0})\) is the solution of \(\dot{\phi}=\max\{b_{1},b_{2}\}*(1+\phi)\) satisfying \(\phi(t,\phi_{0})=\phi_{0}\). From (39), we have  
\[\phi(t,\phi_{0})=e^{\max\{b_{1},b_{2}\}t}-1 \tag{40}\]  
It yields from (22), that  
\[y=\frac{\|e_{i}\|}{\|\dot{\xi}_{i2}\|}\leq\phi(\tau,0)=e^{\max\{b_{1},b_{2} \}\tau}-1 \tag{41}\]  
Combining (40) and (41), we can obtain  
\[\tau>\max\left\{b_{1},b_{2}\right\}^{-1}\ln\left\{\sqrt{\frac{\gamma_{i}(q_{1 }-q_{2}\rho)}{q_{2}\rho^{-1}+q_{3}}}+1\right\} \tag{42}\]  
which means that Zeno behavior does not occur. For CCPNs (1) without DoS attacks, under the event-triggered controller (14), no node exhibits singular triggering behavior. Proof. : From (4) and (16), we have  
\[ \begin{split}\dot{\xi}_{i2}(t)-\xi_{i2}(t)&=\sum \limits_{j=1}^{N}a_{ij}^{0}\big{(}x_{j}\big{(}t_{k}^{i}\big{)}-x_{i}\big{(}t_ {k}^{i}\big{)}\big{)}+b_{i}^{0}\big{(}x_{N+1}\big{(}t_{k}^{i}\big{)}-x_{i} \big{(}t_{k}^{i}\big{)}\big{)}\\ &\quad-\left\{\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}x_{j}(t)-x_{i }(t)\big{)}+b_{i}^{0}(x_{N+1}(t)-x_{i}(t))\right\}\\ &=\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}e_{j}(t)-e_{i}(t)\big{)}- b_{i}^{0}e_{i}(t)\end{split}  \tag{43}\]  
Then, one can obtain  
\[\dot{\xi}_{2}-\xi_{2}=(-H\otimes I_{N})e \tag{44}\]Combining the event-triggering condition \(\|e_{i}\|\leq\sqrt{\frac{\gamma_{i}(q_{1}-q_{2}\rho)}{q_{2}\rho^{-1}+q_{3}}}\|\xi_{ 2}\|\) and (44), it yields  
\[\Big{|}\Big{\|}\xi_{i2}\Big{(}t_{k}^{i}\Big{)}\Big{\|}-\|\xi_{i2}(t)\|\Big{|}\leq \Gamma_{\max}\{-H\otimes I_{N}\}\sqrt{\frac{\gamma_{i}(q_{1}-q_{2}\rho)}{q_{2} \rho^{-1}+q_{3}}}\|e_{i}(t)\|=\sigma c_{i}\|e_{i}(t)\| \tag{45}\]  
where \(c_{i}=\sqrt{\frac{\gamma_{i}(q_{1}-q_{2}\rho)}{q_{2}\rho^{-1}+q_{3}}}\), \(\sigma=\Gamma_{\max}\{-H\otimes I_{N}\}\), \(\sigma c_{i}=\xi_{i}\), and  
\[\
u_{1}=\frac{\big{\|}\xi_{i2}\big{(}t_{k}^{i}\big{)}\big{\|}}{1+\xi_{i}}\leq \|\xi_{i2}(t)\|\leq\frac{\big{\|}\xi_{i2}\big{(}t_{k}^{i}\big{)}\big{\|}}{1- \xi_{i}}=\
u_{2} \tag{46}\]  
From (25), we have  
\[\xi_{i2}(t) =\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}x_{j}(t)-x_{i}(t)\big{)}+b_ {i}^{0}\big{(}x_{N+1}(t)-x_{i}(t)\big{)} \tag{47}\] \[=\sum\limits_{j=1}^{N}a_{ij}^{0}\big{(}\delta_{j}(t)-\delta_{i}( t)\big{)}-b_{i}^{0}\delta_{i}(t)\] \[\Rightarrow\xi_{2}(t) =(-H\otimes I_{N})\delta\Rightarrow\|\xi_{2}\|\leq\sigma\|\delta _{i}\|\leq\sqrt{\frac{V_{a}(t)}{\theta_{\min}(P)}}\]  
It is obvious to show that \(\|\xi_{i2}(t)\|\) always stays between \(\
u_{1}\) and \(\
u_{2}\) from (46), and \(V_{a}(t)\) strictly decreases to zero from (17) and (32), that is, \(\|\xi_{i2}(t)\|\) will eventually decrease to \(\
u_{1}\) based on (46) and (47). The proof is completed. Remark 4: _Lemmas 2 and 3 present the feasibility of the event-triggered strategy proposed in this paper, which means that every node of the systems is effectively transmitted under this transmission strategy. It is obvious to obtain that \(\gamma_{i}\) can affect the convergence rate from (30). Different from [25, 28, 30, 36], singular triggering behavior of is excluded in Lemma 3. It means that there exists the instant \(t_{k+1}^{i}\) after the current triggering time \(t_{k}^{i}\) so that at least one event will be triggered. That is, Lemma 3 excludes the singular triggering behavior that there is no transmission attempt for a long time to ensure that the proposed distributed ETC scheme is effective._  
The communication topologies of CCPNs (1) are easily changed by DoS attacks. However, there are few results focusing on switching topologies of CCPNs caused by DoS attacks, which motivates us to discuss this issue.