## 4 Numerical Simulation  
Under the digraph \(G\) with 8 agents as shown in Figure 1, numerical simulations of system (1) are executed. In Figure 1, for any \(i,j\in\{1,\ldots,8\}\), if \((i,j)\in E\), then there is a black arrow from the agent \(i\) pointing to the agent \(j\). Figure 1 shows that the digraph \(G\) not only contains a spanning tree but also contains a cycle and the adjacency matrix of the digraph \(G\) is particle. Hence, the digraph \(G\) do not meet the topological assumptions in [25; 26]. The communication cycles of agents are all \(T=1\) s. When the state of the agent \(i\) is updated for the \(k\)th time, the distance from the center of mass of the agent \(i\) to the starting point is \(D_{i}(k)\), and \(C_{ji}(k)\) represents the communication distance from the agent \(i\) to the agent \(j\), the maximum communication distance between agents is \(C_{max}(k)=\max_{i,j\in V,s\leq k}\{C_{ji}(s)\}\). The speed of information dissemination is \(v(m/s)\). In this paper, it is assumed that the relationship between the time delay and the communication distance is linear, so let \(M(k)=C_{max}(k)/v\). For multi-agent systems with increasing communication distances (1), by numerically simulating the convergence of the system (1) under four different conditions where the growth rate of the communication distances is different, the conclusions of this paper are verified. Among the four simulations, the first three simulations all use eight agents to explore a wide area from the same point, and each agent is responsible for a fan-shaped area with an angle of \(\pi/4\). Record the starting point as the origin. In the simulation (1), for any integer \(k\geq 0\) and any \(i\in V\), \(D_{i}^{(1)}(k)=\frac{v}{2}(-\log_{a}k^{b})\), where \(a\in(0,\tilde{c})\), \(b\in\left(0,\frac{1}{2n}\right)\). Therefore, there exist \(a_{1}\in(0,\tilde{c})\) and \(b_{1}\in\left(0,\frac{1}{2n}\right)\) such that the maximum communication distance of the system satisfies \(C_{max}^{(1)}(k)=v*(-\log_{a_{1}}k^{b_{1}})\). This system can be described as a system (1) with an unbounded maximum delay \(M_{1}(k)\). In the simulation (2), for any integer \(k\geq 0\) and any \(i\in V\), \(D_{i}^{(2)}(k)=\frac{v\cdot d_{2}k}{2}\), where \(d_{2}\in(0,1)\). Therefore, there exists \(d_{2}^{\prime}\in(0,1)\) such that the maximum communication distance of the system satisfies \(C_{max}^{(2)}(k)=vd_{2}^{\prime}k\). The system can be described as a system (1) with an unbounded maximum delay \(M_{2}(k)\). In the simulation (3), for any integer \(k\geq 0\) and any \(i\in V\), \(D_{i}^{(3)}(k)=\frac{v}{2}(-\log_{a_{1}}(k^{b/2}))\). Therefore, there exist \(a_{3}=a_{1}\) and \(b_{3}=b_{1}/2\) such that the maximum communication distance of the system satisfies \(C_{max}^{(3)}(k)=v*(-\log_{a_{3}}k^{b_{3}})\). The system can be described as a system (1) with an unbounded maximum delay \(M_{3}(k)\). The simulation (4) uses 8 agents to explore in a circle with a diameter of \(v(m)\). The starting points of agents are 8 octsection points on the circle, and each agent is responsible for the fan-shaped area with an angle of \(\pi/4\). The agent finally converges at the center of the circle. Then, for any integer \(k\geq 0\) and any \(i\in V\), \(D_{i}^{(4)}(k)=v-d_{4}vk\), where \(d_{4}>0\), and satisfies \(D_{i}^{(4)}(k)\geq 0\). Therefore, the maximum communication distance of the system satisfies \(C_{max}^{(4)}(k)\leq 2v\). This system can be described as a system (1) with a bounded maximum delay \(M_{4}(k)\). It can be noticed that in the first three simulations, the maximum communication distance \(C_{max}^{(1)}(k)\), \(C_{max}^{(2)}(k)\) and \(C_{max}^{(3)}(k)\)  
Figure 1: The topology graph \(G\) with 8 agents. tends to infinity as the number of iterations \(k\) growths, and the maximum communication distance \(C_{max}^{(4)}(k)\) in the simulation (4) is bounded.