## Appendix A Proof of Theorem 1  
In order to prove Theorem 1, the following lemma is given. Figure 8: The system error curve in the simulation (1), the simulation (3) and the simulation (4) under logarithmic coordinates. **Lemma A1**. : _Under Assumption 1, for any \(P(l)\in S(K),l\in\{1,\cdots,H_{M}(K)\}\), \(\prod_{l=1}^{H_{M}(K)}P(l)\), the product of these \(H_{M}(K)\) matrices, contains a column in which all entries are not zero._  
Proof of Lemma A1. : For any \(n(M(K)+1)\)-dimensional square matrix \(P\in S(K)\), We divide the matrix into blocks as follows  
\[P=\left[ \begin{array}{ccc}P_{[1i]}&\cdots&P_{[1(M(K)+1)]}\\ \vdots&\ddots&\vdots\\ P_{[(M(K)+1)]}&\cdots&P_{[(M(K)+1)(M(K)+1)]}\end{array} \right],\]  
where for \(i,j\in\{1,\ldots,M(K)+1\}\), \(P_{[ij]}\in R^{n\times n}\). For any \(s\geq 1\) and any \(P(l)\in S(K),l\in\{1,\cdots,s\}\), the product of any \(s\) matrices belonging to the set \(S(K)\) is represented by \(\prod_{l=1}^{s}P(l)\). Firstly, for \(\prod_{l=1}^{M(K)+1}P(l)\) and any \(i\in\{1,\ldots,M(K)+1\}\), diagonal entries of \((\prod_{l=1}^{M(K)+1}P(l))_{[i1]}\) are all non-zero. Under Assumption 1, without loss of generality, let the agent 1 be the root agent. There must exists \(V_{1}\subset V\) such that \((1,s_{1})\in E\) for all \(s_{1}\in V_{1}\). Hence, for any \(P(M(K)+2)\in S(K)\) and any \(s_{1}\in V_{1}\), \((\prod_{l=1}^{M(K)+2}P(l))_{[11]})_{s1}\) is non-zero. Then, for \(\prod_{l=1}^{2(M(K)+1)}P(l)\), any \(i\in\{1,\ldots,M(K)+1\}\), and any \(s_{1}\in V_{1}\), diagonal entries and \(s_{1}-1\) entry of \((\prod_{l=1}^{2(M(K)+1)}P(l))_{[i1]}\) are all non-zero. There also exists \(V_{2}\subset V\), which satisfies that for each \(s_{2}\in V_{2}\), and there is at least one \(s_{1}\in V_{1}\) such that \((s_{1},s_{2})\in E\). Hence, for any \(P(2M(K)+3)\in S(K)\) and any \(s\in V_{1}\cup V_{2}\), \(((\prod_{l=1}^{2M(K)+3}P(l))_{[11]})_{s1}\) is non-zero. Then, for \(\prod_{l=1}^{M(K)+1}P(l)\), any \(i\in\{1,\ldots,M(K)+1\}\), and any \(s\in V_{1}\cup V_{2}\), diagonal entries and \(s-1\) entry of \((\prod_{l=1}^{3(M(K)+1)}P(l))_{[i1]}\) are all non-zero. Since subscripts of sets \(V_{r}\) all satisfy the inequality that \(r\leq D_{G}-1\), and Assumption 1 is satisfied, we have that \(\{1\}\cup V_{1}\cup\cdots\cup V_{D_{G}-1}=V\). Therefore, for any \(P(l)\in S(K),l\in\{1,\cdots,H_{M}(K)\}\), \(\prod_{l=1}^{H_{M}(K)}P(l)\) contains a column in which all entries are not zero. Based on Lemma A1, the proof of Theorem 1 is given. Proof of Theorem 1.: For any integer \(K\geq 1\), we have (3) for all \(k\in Z\bigcap[0,K-1]\). For any integers \(k_{1},k_{2}\) satisfying \(0\leq k_{1}\leq k_{2}<K\), let \(R_{K}(k_{2},k_{1})=\prod_{s=k_{1}}^{k_{2}}A_{K}(s)\). It can be noticed that \(R_{K}(k_{1},k_{1})=I\). According to Assumption 2, there exists an integer \(K_{1}>0\) such that \(K>H_{M}(K)\) for all \(K\in Z\bigcap[K_{1},\infty)\). Under Assumption 1, Lemma A1 shows that for any integer \(K\in[K_{1},\infty)\) and all integers \(k\in[H_{M}(K),K-1]\), \(R_{K}(k,k-H_{M}(K))\) contains a column in which all entries are not zero. Hence, for any integer \(K\geq K_{1}\), we have that \(\max_{j}\min_{i}\{R_{K}(k,k-H_{M}(K))_{ij}\}\geq\varepsilon^{H_{M}(K)}>0\) for all integers \(k\in[H_{M}(K),K-1]\). We consider the Lyapunov function \(V(y_{x,K}(k))=\max_{i}\{(y_{x,K})_{i}(k)\}-\min_{i}\{(y_{x,K})_{i}(k)\}\).