## 3 Convergence Analysis
The proof of Theorem 3 is given in Appendix C.  
Combing (5), (6), and (8), we have that under Assumptions 1-3, for all integers \(k\geq\tilde{K}\), the convergence rate of the consensus system (1) is equal to the rate of \(\left(1-\tilde{c}^{H_{M}(K)}\right)^{\frac{k}{H_{M}(K)}}\) tending to \(0\). When \(k\) tends to infinity, the faster \(M(k)\) tends to infinity, the slower \(\left(1-\tilde{c}^{H_{M}(K)}\right)^{\frac{k}{H_{M}(K)}}\) tends to \(0\). Hence, we directly obtain that the convergence rate of the consensus system (1) and the growth rate of the maximum delay \(M(k)\) are negatively correlated. In other words, for the multi-agent system (1) with increasing communication distances, the convergence rate of the consensus system (1) and the growth rate of the maximum communication distance are negatively correlated. **Theorem 4**. : _Under Assumptions 1-3, for the constant \(\tilde{c}\), there exist an integer \(\hat{K}>0\) such that for all integers \(k\geq\hat{K}\), the rate of system (1) achieving a consensus and the rate of the maximum delay \(M(k)\) tending to infinity are negatively correlated._  
Next, we use some corollaries to support Theorem 4. When the maximum time delay satisfies some special forms, we have following corollaries. The first form is the bounded maximum time delay. In other words, the maximum communication distance is bounded. Actually, Theorem 3 is suitable to consensus systems with bounded delays. For the bounded time delay \(m_{ij}(K)\), its maximum time delay satisfies \(M(K)\leq M\) for all integers \(K\geq 0\). Then \(K-m_{ij}(K)\to\infty\) as \(K\to\infty\), and, for all integers \(K\geq 0\), \(\left(1-\tilde{c}^{H_{M}(K)}\right)^{\frac{K}{H_{M}(K)}}=\mathrm{O}\big{(}a^{K} \big{)}\), where \(a=\left(1-\tilde{c}^{DG(M+1)}\right)^{\frac{1}{DC(M+1)}}\in(0,1)\). In other words, system (1) with the bounded maximum delay \(M(k)\) satisfies Assumptions 2 and 3 at the same time. As direct consequences of Theorem 3, we have the following corollary. **Corollary 1**. : _(Exponential Consensus) Suppose that \(M(k)\leq M\) for all integers \(k\geq 0\), where \(M>0\) is a constant integer. Under Assumption 1, system (1) achieves a consensus. Furthermore, for the constant \(\tilde{c}\), there exist constants \(a=\left(1-\tilde{c}^{DG(M+1)}\right)^{\frac{1}{DC(M+1)}}\in(0,1)\) and \(x^{*}\in[\min_{i\in V}\{x_{i}(0)\},\max_{i\in V}\{x_{i}(0)\}]\) such that_  
\[|x_{i}(k)-x^{*}|=O\Big{(}a^{k}\Big{)} \tag{9}\]  
_for all \(i\in V\) and all integers \(k\in[0,\infty)\)._  
The second form is the unbounded maximum time delay. Let \(M(k)=\lfloor-\log_{a}k^{b}+M\rfloor\) for all integers \(k\geq 0\), where \(a\in(0,\tilde{c})\), \(b\in(0,\frac{1}{2 DC})\), and \(M\) is a constant. In other words, the growth rate of the maximum communication distance is not faster than the growth rate of the function \(\lfloor-\log_{a}k^{b}+M\rfloor\). Obviously, the maximum time delay \(M(k)\) is unbounded and the time delay \(m_{ij}(k)\) corresponding to \(M(k)\) satisfies Assumption 2. Furthermore, Theorem 2 shows that the unbounded maximum time delay \(M(k)=\lfloor-\log_{a}k^{b}+M\rfloor\) satisfies Assumption 3. Then, combining Theorems 2 and 3, we directly have the following corollary. **Corollary 2**. : _(Asymptotical Consensus) Suppose that \(M(k)=\lfloor-\log_{a}k^{b}+M\rfloor\) for all integers \(k\geq 0\), where \(a\in(0,\tilde{c})\), \(b\in(0,\frac{1}{2\cdot DC_{\hat{K}}})\), and \(M\) is a constant. Under Assumption 1, system (1) achieves a consensus. Furthermore, for the constant \(\tilde{c}\) and any \(e\in(\hat{h},1)\) shown in Lemma A2, there exist an integer \(\hat{K}=\max\{K_{1},K_{2}\}\), a constant \(x^{*}\in[\min_{i\in V}\{x_{i}(0)\},\max_{i\in V}\{x_{i}(0)\}]\), and a constant \(a_{1}\in(0,1)\) related to \(M(\hat{K})\) and \(\tilde{c}\) such that_  
\[|x_{i}(k)-x^{*}|= \begin{cases}O\Big{(}a_{1}^{k}\Big{)}&k\in Z\bigcap[0,\hat{K })\\ O\Big{(}e^{\sqrt{k}}\Big{)}&k\in Z\bigcap[\hat{K},\infty)\end{cases}