## 4 Numerical Simulation
Thus, an asynchronous DoS attack is more general, and a synchronous DoS attack can be seen as a special case of asynchronous DoS attack. Second, [43] makes use of feedback control to achieve group consensus, which is a continuous control method. However, we use event-triggered control in this paper, which is a discontinuous control method. Considering the limited resources in practical engineering, event-triggered control can avoid unnecessary control signal processing, which can greatly save resources. \begin{table}
\begin{tabular}{l c c c c c c} \hline
**PerformanceAgent** & **Agent 1** & **Agent 2** & **Agent 3** & **Agent 4** & **Agent 5** & **Agent 6** \\ \hline trigger number (method 1) & 397 & 193 & 136 & 443 & 121 & 422 \\ \hline trigger number (method 2) & 918 & 387 & 257 & 912 & 184 & 940 \\ \hline \end{tabular}
\end{table} 
Table 2: The comparison of two event-triggered functions. Figure 5: The positive states of followers and leaders in group 2 for different event-triggered functions. Figure 6: The positive states of followers and leaders in group 3 for different event-triggered functions. Figure 7: The velocity states of followers and leaders in group 1 for different event-triggered functions. Figure 8: The velocity states of followers and leaders in group 2 for different event-triggered functions. Figure 9: The velocity states of followers and leaders in group 3 for different event-triggered functions. Figure 11: The event-triggered time instants of agents for different event-triggered functions. Figure 10: The switching signals of agents. **Example 2**: _In this example, we increase the number of agents and increase the DoS attack intensity. Suppose that there are nine nodes in the communication topology, which is shown in Figure 12. The adjacency matrix of this topology is given as follows:_  
\[A^{0}= \begin{pmatrix}0&1&0&0&0&0&0&0&0\\ 0&0&1&1&0&0&0&0&0\\ 1&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\\ 0&0&0&0&0&1&0&0&0\\ 0&0&0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1&0\\ \end{pmatrix} \]  
We choose a nonlinear function similar to _Example 1_; then, we can obtain \(l_{1}=l_{2}=0.001\); thus, \(\delta=1.002\). From the original communication topology, we can see that agents 3 and 4 are inter-act agents, and agent 8 is an intra-act agent with zero degree. So we should choose agents 3, 4 and 8 to be pinned. Furthermore, we divide the nine agents into three group, where \(\mathcal{V}_{1}=\{1,2,3\}\), \(\mathcal{V}_{2}=\{4,5,6\}\), and \(\mathcal{V}_{3}=\{7,8,9\}\). Choosing the system parameters in Table 3, we have the pinning matrix \(D^{0}=diag\{0,0,4.2,4.2,0,0,0,4.2,0\}\) and the coupling strength of the multi-agent system \(c=15\); then, by some calculations, we can obtain that \(\lambda_{max}((\delta+\frac{\delta}{2})I_{N}-c\hat{D}^{0}-cD^{0}+2\alpha(H^{0} )^{T}H^{0})=-0.4686<0\), which satisfies condition (1) of Theorem 1. Furthermore, we can obtain \(\rho_{1}=0.9325,\rho_{2}=1.24\). Moreover, there are two types of asynchronous DoS attacks, which are shown in Figure 13.  
Figure 19: The velocity states of followers and leaders in group 3 under DoS attack. Figure 17: The velocity states of followers and leaders in group 1 under DoS attack. Figure 18: The velocity states of followers and leaders in group 2 under DoS attack.