## 4 Numerical Simulation
Figure 2 describes the motion trajectories of eight agents in four simulations, in which the abscissa and ordinate describe the position of any point in the two-dimensional plane, and the unit is \(m\). The colored curves are the motion trajectories of all agents. Starting from the purple trajectory and rotating clockwise to the green trajectory, corresponding agent numbers are \(1,2,3,4,5,6,7,8\), respectively; and the black dashed line is used to separate the detection areas of the eight agents. In Figure 2, the circle enclosed by the black dashed line in the sub-figure simulation (4) is the boundary of the detection area in the simulation (4). Figures 3, 4, 6 and 7 show the variation of the maximum communication distances between agents \(C_{max}^{(s)}(k),s=1,2,3,4\) and the maximum delays \(M_{\rm s}(k),s=1,2,3,4\) with the number of iterations \(k\) of four simulations, respectively. It can be seen that the maximum communication distance \(C_{max}(k)\) and the maximum communication delay \(M(k)\) meet the same changing trend. Figure 5 is a comparison figure of change curves of the agent state value in the simulation (1) and the simulation (2). Figure 8 describes the logarithmic function curve of the state error between the agents in the simulation (1), the simulation (3) and the simulation (4) in logarithmic coordinates. First, compare the simulation (1) and the simulation (2). Since the growth rate of \(D_{i}^{(2)}(k)\) in the simulation (2) is faster than the growth rate of \(D_{i}(k)^{(1)}\) in the simulation (1), Figure 2 shows that the range of agent detection in the simulation (2) obviously exceeds the range of agent detection in the simulation (1). Figure 3: The maximum communication distances \(C_{max}^{(1)}(k)\) between agents and the maximum delays \(M_{1}(k)\) in the simulation (1). Figure 2: The motion trajectories of 8 agents. Figure 3 shows that since the maximum communication distance satisfies \(C^{(1)}_{max}(k)=v*(-\log_{a_{1}}k^{b_{1}})\), the curve of the maximum communication delay \(M_{1}(k)\) closely follows the logarithmic function curve \(g_{1}(k)=-\log_{a_{1}}k^{b_{1}}\). Figure 4 shows that since the maximum communication distance satisfies \(C^{(2)}_{max}(k)=vd_{2}^{\prime}k\), the curve of the maximum communication delay \(M_{2}(k)\) closely follows the linear function curve \(g_{2}(k)=d_{2}^{\prime}k\). Therefore, the unbounded maximum delay \(M_{1}(k)\) satisfies the condition in Corollary 2, and the unbounded maximum delay \(M_{2}(k)\) only satisfies Assumption 2. Figure 5 shows that the system with the maximum delay \(M_{1}(k)\) is convergent, but the system with the maximum delay \(M_{2}(k)\) is divergent. In other words, when the growth rate of the maximum communication distance can ensure that the maximum delay satisfies both Assumption 2 and Assumption 3, the system (1) can reach a consensus under Assumption 1 for topology, which is more relaxed than the topological assumptions in [25; 26]. Next, compare the simulation (1), the simulation (3) and the simulation (4). Since \(D^{(1)}_{i}(k)\) growths faster than \(D^{(3)}_{i}(k)\), Figure 2 shows that the detection range of agents in the simulation (1) is larger than the detection range of agents in the simulation (3). Furthermore, since \(D^{(3)}_{i}(k)\) increases with the growth of \(k\), \(D^{(4)}_{i}(k)\) decreases with the growth of \(k\), there exists \(K>0\) such that \(D^{(3)}_{i}(k)>D^{(4)}_{i}(k)\) for all integers \(k>K\). In other words, due to the limited detection range of the simulation (4), as long as the detection time is long enough, the detection range of agents in the simulation (1) and the simulation (3) will always be larger than the detection range of agents in the simulation (4). Figure 6 shows that due to the maximum communication distance satisfies \(C^{(3)}_{max}(k)=v*(-\log_{a_{3}}k^{b_{3}})\), the curve of the maximum communication delay \(M_{3}(k)\) closely follows the logarithmic function curve \(g_{3}(k)=-\log_{a_{3}}k^{b_{3}}\).