# Redundancy-Reduction-Based Hierarchical Design in Synchronization of Multi-Agent Systems
## 2 Preliminary Knowledge  
Consider an undirected graph \(\mathcal{G}(\mathcal{N},\mathcal{E},\mathcal{A})\), where \(\mathcal{N}=\{n_{1},n_{2},\ldots,n_{N}\}\) represents \(N\) notes in the graph, \(\mathcal{A}=[e_{ij}]_{N\times N}\) is the adjacency matrix, and \(e_{ij}\) represents an undirected edge in \(\mathcal{G}(\mathcal{N},\mathcal{E},\mathcal{A})\) of the ordered pair of nodes \(n_{i}\) and \(n_{j}\). \(\mathcal{E}\) is a set of undirected edges and \(\mathcal{E}\subseteq\mathcal{N}\times\mathcal{N}\). Without orientation, the edge of node \(\big{(}n_{i},n_{j}\big{)}\) is identical to the edge \(\big{(}n_{j},n_{i}\big{)}\), and the information from different nodes \(n_{i}\) and \(n_{j}\) share the same weight. Diagonal element \(e_{ii}\) is defined as \(e_{ii}=-\sum_{j=1,j\
eq 1}^{N}e_{ij}\). It is easy to surmise that \(\sum_{j=1}^{N}e_{ij}=0\), \(\sum_{j=1}^{N}e_{ij}=0\). We consider a common CDN with \(N\) identical nodes of the form  
\[\dot{x}_{i}(t)=Ax_{i}(t)+\tilde{f}(x_{i}(t),t)+c\sum_{j=1}^{N}e_{ ij}\Gamma x_{j}(t_{k}) \tag{1}\] \[t\in[t_{k},\,t_{k+1}),\,i=1,2,\ldots,N\]  
where we have \(\Gamma\) is an inner-coupling matrix of appropriate dimensions, \(t_{0}=0\) as initial instant. \(t_{k}\) are the communication instants and satisfy  
\[0=t_{0}<t_{1}<\cdots<t_{k}<\cdots<\lim_{k\to+\infty}t_{k}=+\infty\] \[k=0,1,2,\ldots \tag{2}\]  
\(x_{i}(t)=[x_{i1}(t),\,x_{i2}(t),\,\ldots,\,x_{in}(t)]^{\mathrm{T}}\in\mathbb{R }^{n}\) is the state vector of node \(i\). \(d\) is a constant of time delay. When \(t<0\), \(x_{ij}(t-d)=0\) holds at the initial condition. To satisfy the synchronization requirement, all nodes must track the central node. In this paper, node 1 is always chosen as the central node. Let  
\[x(t)= \begin{bmatrix}x_{1}(t)\\ x_{2}(t)\\ \vdots\\ x_{N}(t)\end{bmatrix} \in\mathbb{R}^{Nn},\,g(t)= \begin{bmatrix}\tilde{f}(x_{1}( t),t)\\ \tilde{f}(x_{2}(t),t)\\ \vdots\\ \tilde{f}(x_{N}(t),t)\end{bmatrix} \in\mathbb{R}^{Nn}\]  
Therefore, we could rewrite Equation (1) as follows:  
\[\dot{x}(t)=(I_{N}\otimes A)x(t)+g(t)+c(\mathcal{E}\otimes\Gamma)x(t_{k}),\,t \in[t_{k},\,t_{k+1}) \tag{3}\]  
Let  
\[y(t)= \begin{bmatrix}x_{1}(t)-x_{2}(t)\\ x_{1}(t)-x_{3}(t)\\ \vdots\\ x_{1}(t)-x_{N}(t)\end{bmatrix}  \tag{4}\]  
then the synchronization process could be described as \(\lim_{t\to\infty}y(t)=0\). To achieve the tracking goal, following assumption and lemma would be used. **Assumption 1** ([28]). : _A diagonal matrix \(\Lambda>0\) exists, and then the nonlinear part of the node system satisfies the following condition:_  
\[(\tilde{g}(x_{1},t)-\tilde{g}(x_{2},t))^{\mathrm{T}}(\tilde{g}(x_{1},t)- \tilde{g}(x_{2},t))\leqslant(x_{1}-x_{2})^{\mathrm{T}}\Lambda(x_{1}-x_{2}) \tag{5}\]  
**Lemma 1** ([28]). : _The following equation holds if matrix \(G\in\mathbb{R}^{N\times N}\) satisfies the same sum for each row._  
\[\tilde{M}G=\tilde{M}G\Phi\tilde{M} \tag{6}\]_where_  
\[\bar{M}= \begin{bmatrix}1&-1&0&\cdots&&0\\ 1&0&-1&0&\cdots&0\\ \vdots&&&\ddots&&\vdots\\ 1&0&\cdots&0&-1&0\\ 1&0&\cdots&&0&-1\end{bmatrix} \in\mathbb{R}^{(N-1)\times N}\]  
\[\Phi= \begin{bmatrix}0&0&0&\cdots&0\\ -1&0&0&\cdots&0\\ 0&-1&0&\cdots&\vdots\\ 0&0&\ddots&\ddots&0\\ \vdots&&\ddots&-1&0\\ 0&\cdots&0&0&-1\end{bmatrix} \in\mathbb{R}^{N\times(N-1)}\]  
Equation (3) could be represented as  
\[\dot{y}(t) =(I_{N-1}\otimes A)y(t)+g(t)\] \[+cM(\mathcal{E}\Phi\otimes\Gamma)y(t_{k}),\,t\in[t_{k},\,t_{k+1}) \tag{7}\]  
where \(M=\bar{M}\otimes I_{n}\). In practical neural networks, usually \(N\) would be a huge number; therefore, there would be a tremendous amount of \(\mathcal{E}\). The computation burden would be enormous, although mathematical work could alleviate the problem, but the ceiling may have been encountered. Therefore, some structural approaches may be applied that may be more effective than extensive algorithms. In our model, the structure of network is known and the central node is confirmed.