## 2 Preliminaries and Problem Statement
### Preliminaries  
For a finite set \(\mathcal{S}\), \(|\mathcal{S}|\) denotes the number of its elements. For a vector or a matrix \(A\), \(\|A\|\) stands for its Euclidean norm, \(\|A\|_{\infty}\) stands for its \(\infty\)-norm and \(A^{T}\) is its transpose. \(\mathbf{1}_{N}\) and \(\mathbf{0}_{N}\) are the \(N\) dimension column vectors with all entries \(1\) and \(0\), respectively. The matrix \(\mathrm{diag}\{a_{1},a_{2},...,a_{N}\}\) denotes a diagonal matrix with diagonal entries \(a_{1},a_{2},...,a_{N}\). \(\rho(.)\) denotes the spectral radius of matrix, and \(\rho_{2}(.)\) stands for the minimum positive eigenvalue of matrix with positive eigenvalues.  
For a directed graph \(\mathcal{G}\), \(\mathcal{G}=(\mathcal{V},\mathcal{E},\mathcal{A})\), where \(\mathcal{V}=\{1,2,...,N\}\) is a set of nodes, \(\mathcal{E}=\mathcal{V}\times\mathcal{V}\) denotes a set edges, and \(\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}\) stands for a weighted adjacency matrix, in which \(\mathbb{R}\) denotes a set of real numbers. In the directed graph \(\mathcal{G}\), for all \(i\in\mathcal{V}\), \((i,i)\
ot\in\mathcal{E}\). Namely, for edge \((j,i)\in\mathcal{E}\), with an start of node \(j\) and an end of node \(i\), agent \(i\) can get state information from agent \(j\). To be specific, the edge \((i,j)\) links with the elements \(a_{ij}\) of a weighted adjacency matrix \(\mathcal{A}\), \(a_{ij}>0\) if and only if \((i,j)\in\mathcal{E}\), otherwise \(a_{ij}=0\). Although the opposite direction of transmission may not exist or have a different weight value \(a_{ij}\). Given any two distinct nodes \(i\) and \(j\), if there exists a directed path from \(j\) to \(i\), the directed graph \(\mathcal{G}\) is strongly connected.  
Moreover, agent \(j\) is called an in- and out-neighbor of agent \(i\). \(\mathcal{N}_{i}^{in}=\{j\in\mathcal{V}\,|\,a_{ij}>0\}\) and \(\mathcal{N}_{i}^{out}=\{j\in\mathcal{V}\,|\,a_{ji}<0\}\) are used to demonstrate the in- and out-neighbor set of agent \(i\), respectively. Both the in- and out-degree of \(i\)th agent are denoted by \(d_{i}^{in}=\sum_{j=1}^{N}a_{ji}\), and \(d_{i}^{out}=\sum_{j=1}^{N}a_{ij}\), respectively. We define Laplacian matrix of \(\mathcal{G}\) by \(\mathcal{L}=\mathcal{D}-\mathcal{A}\), where \(\mathcal{D}=\mathrm{diag}\{d_{1}^{out},d_{2}^{out},...,d_{N}^{out}\}\). Accordingly, the eigenvalues of \(\mathcal{L}\) can be listed in a descending order as \(\lambda_{N}\geq...\geq\lambda_{2}\geq\lambda_{1}=0\), where \(\lambda_{N}=\rho(\mathcal{L})\) denotes the spectral radius of \(\mathcal{L}\).  
The lemmas listed below are applied to facilitate the analysis in the following paper.  
Lemma 1: _For any \(x,y\in\mathbb{R}\) and \(a>0\), the following properties are applied:_  
\[\begin{split} 1.xy\leq\frac{a}{2}x^{2}+\frac{1}{2a}y^{2};\\ 2.(x^{2}+y^{2})\leq(x+y)^{2},if\quad xy\geq 0.\end{split} \tag{1}\]  
Lemma 2 ([29]): _For a strong connected digraph \(\mathcal{G}\), composed of a spanning tree, the vector \(\xi=[\xi_{1},\xi_{2},...,\xi_{N}]^{T}>0\) is the left eigenvector corresponding to zero eigenvalue of Laplacian matrix \(\mathcal{L}\), that satisfies \(\sum_{i=1}^{N}\xi_{i}=1\) and \(\xi^{T}\mathcal{L}=\mathbf{0}_{N}\). Furthermore, \(\mathcal{L}^{T}\Theta+\Theta\mathcal{L}^{T}\) is semi-positive definite with all rows sum up to zeros and has zero eigenvalue with algebraic dimension one, where \(\Theta=\mathrm{diag}\{\xi_{1},\xi_{2},...,\xi_{N}\}\)._