## 4 Numerical Simulation  
In this section, we will give two simulation examples to demonstrate the efficacy of the results. _Example 1:_ Suppose that there are six nodes in the communication topology, which is shown in Figure 2. The adjacency matrix of this topology is given as follows:  
\[A^{0}= \begin{pmatrix}0&1&0&0&0\\ 0&0&0&1&0&0\\ 1&0&0&0&0&0\\ 0&0&1&0&0&0\\ 1&0&0&0&0&0\\ 0&0&0&0&1&0\end{pmatrix} \]Let the nonlinear function be \(f(x_{i}(t),v_{i}(t))=6 cos(4t)+0.01 cos(2x_{i}(t))+0.01 sin(4v_{i}(t))\). From Assumption 1, we can obtain \(l_{1}=l_{2}=0.001\); thus, \(\delta=1.002\). From the original communication topology, we can see that agents 1 and 4 are inter-act agents, and agent 6 is an intra-act agent with zero degree. So we should choose agents 1, 4, and 6 to be pinned. Furthermore, we divide the six agents into three group, where \(\mathcal{V}_{1}=\{1,2\}\), \(\mathcal{V}_{2}=\{3,4\}\), and \(\mathcal{V}_{3}=\{5,6\}\). Choosing the system parameters in Table 1, we have the pinning matrix \(D^{0}=diag\{4.2,0,0.4.2,0.4.2\}\) and the coupling strength of the multi-agent system \(c=15\); then, by some calculations, we can obtain that \(\lambda_{max}((\delta+\frac{c}{2})I_{N}-c\tilde{D}^{0}-cD^{0}+2c\alpha(H^{0}) ^{T}H^{0})=-0.6348<0\), which satisfies condition (1) of Theorem 1. Furthermore, we can obtain \(\rho_{1}=0.7557,\rho_{2}=1.56\). Moreover, there are three types of asynchronous DoS attacks, which are shown in Figure 3.  
Furthermore, choosing \(T=2,f=5.5\), based on condition (2) of Theorem 1, we can obtain a persistent dwell time \(\tau=5.8\). The switching signal of the changing communication topology can be seen in Figure 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{5}{c}{**System Parameters**} \\ \hline c & \(c=15\) & \(\alpha=0.05\) & \(\mu=1.1\) & \(T=2\) & \(f=5.5\) & \(d_{i}=4.2,i=1,4,6\) \\ \hline
**3** & **5** & **3** & **5** & **3** & **5** \\
**4** & **6** & **4** & **6** & **4** & **6** \\ \hline \end{tabular}
\end{table} 
Table 1: System parameters. Figure 3: Three types of DoS attacks. Figure 2: Original topology. Here, we depict two types of event-triggered functions. The first one is the event-triggered function designed using the combinational measurements in this paper. The second one is the event-triggered function implemented by utilizing a sampled-event detector, which mainly depends on the state error, such as in [42]. Figures 5-10 demonstrate the comparison of the two different event-triggered functions. From Figures 5-10, we can see that the multi-agent system achieves consensus at around 3 s with the event-triggered function designed using the combinational measurements; however, the multi-agent system achieves consensus at around 6 s with the event-triggered function designed using the state error. Further, Figure 11 demonstrates the event-triggered time instants of two different event-triggered functions. Moreover, in Table 1, we can see that the multi-agent system will trigger less frequently with the event-triggered function designed using combinational measurement, which can greatly save resources. (For a brief review, in Table 2, method 1 stands for the event-triggered function designed using combinational measurement, and method 2 stands for the event-triggered function designed using the state error.) Thus, regardless of the time the agent's position and velocity states achieve consensus or the event-triggered time instants for a multi-agent system, it is clear that the method presented in this paper is more effective. Figure 4: The positive states of followers and leaders in group 1 for different event-triggered functions. Furthermore, compared with [43], which also studies group consensus of a multi-agent system under DoS attack, there are two improved aspects in this paper. First, we consider asynchronous DoS attacks in this paper, but [43] studies synchronous DoS attacks. An asynchronous DoS attack means part of the communication links will be interpreted, but a synchronous DoS attack means all communication links are interpreted.