## 3 Convergence Analysis  
This section studies the consensus problem of system (1). Firstly, under Assumption 2, properties of system states in the interval \([0,K]\) for any finite integer \(K>0\) is studied. Let \(\tilde{c}=\min\{\min_{i,j\in V}\{a_{ij}|a_{ij}\
eq 0\}c,1-c\}\in(0,1)\), \(D_{G}\triangleq\max_{i,j\in V}\{\text{dis}(i,j)\}+1\), and \(H_{M}(K)\triangleq D_{G}(M(K)+1)\). Furthermore, the state error of the system (1) satisfies following theorem. **Theorem 1**. : _Under Assumptions 1 and 2, for the constant \(\tilde{c}\), there exist an integer \(K_{1}>0\) such that the state error of system (1) satisfies_  
\[|x_{i}(K)-x_{j}(K)|=O\bigg{(}\left(1-e^{H_{M}(K)}\right)^{\frac{K}{H_{M}(K)}} \bigg{)} \tag{5}\]  
_for all \(i,j\in V\) and all integers \(K\geq K_{1}\)._  
The proof of Theorem 1 is given in Appendix A. For system (1), Theorem 1 shows that the convergence rate of the state error and the growth rate of the maximum delay \(M(K)\) are related. Since the relationship between the maximum delay \(M(K)\) and the maximum communication distance is linear, the rate of states achieving a consensus and the growth rate of the maximum communication distances are related. It is worth noting that when the topology of system (1) satisfies Assumption 1, system (1) cannot achieve a consensus if the delay only satisfies Assumption 2. Next, inorder to make system (1) satisfy Definition 1, the growth rate of \(M(k)\) needs to be reasonably restricted by the following assumption, which also is the constraint on the growth rate of the maximum communication distance. For the constant \(\tilde{c}\), there exists an integer \(K_{2}>0\), such that  
\[\left(1-\tilde{c}^{H_{M}(K)}\right)^{\frac{K}{H_{M}(K)}}=O(f(K)) \tag{6}\]  
for all integers \(K\geq K_{2}\), where \(f(K)=o\left(\frac{1}{K}\right)\) as \(K\to\infty\). The following theorem shows the sufficient condition for the maximum delay \(M(k)\) to satisfy Assumption 3. For the constant \(\tilde{c}\), any \(a\in(0,\tilde{c})\), any \(b\in(0,\frac{1}{2\cdot\mathrm{D}\tilde{c}})\), and any constant \(M\), if \(M(k)\leq-\log_{a}k^{b}+M\) for all integers \(k\geq 0\), then for the constant \(\tilde{c}\) and any \(\epsilon\in(\tilde{h},1)\), there exists an integer \(K_{2}>0\) such that  
\[\left(1-\tilde{c}^{H_{M}(K)}\right)^{\frac{K}{H_{M}(K)}}<e^{\sqrt{K}} \tag{7}\]  
for all \(k\in Z\bigcap[K_{2},\infty)\), which means that the maximum delay \(M(k)\) satisfies Assumption 3. In Appendix B, the proof of Theorem 2 and \(\hat{h}\) are given. In the multi-agent system (1) with increasing communication distances, since the relationship between the maximum delay and the maximum communication distance is linear, if the growth rate of the maximum communication distance is not faster than the growth rate of the function \(-\log_{a}k^{b}+M\), the maximum delay \(M(k)\) of system (1) satisfies conditions in Theorem 2. When the maximum delay \(M(k)\) satisfies Assumptions 2 and 3, the following theorem is given. Under Assumptions 1-3, system (1) achieves a consensus. Furthermore, for the constant \(\tilde{c}\), there exist an integer \(\tilde{K}>0\), a constant \(x^{*}\in[\min_{i\in V}\{x_{i}(0)\},\max_{i\in V}\{x_{i}(0)\}]\), and a constant \(a\in(0,1)\), which is related to \(M(\tilde{K})\) and \(\tilde{c}\), such that  
\[|x_{i}(k)-x^{*}|= \begin{cases}O\left(a^{k}\right)&k\in Z\bigcap[0,\tilde{K}) \\ O\left(f(K)\right)&k\in Z\bigcap[\tilde{K},\infty)\end{cases}  \tag{8}\]  
for all \(i\in V\). For the multi-agent system (1) with increasing communication distances, Theorems 2 and 3 show that when the growth rate of the maximum communication distance is not faster than the growth rate of the function \(-\log_{a}k^{b}+M\), system (1) achieves a consensus if the Assumption 1 for topology is satisfied.