# Redundancy-Reduction-Based Hierarchical Design in Synchronization of Multi-Agent Systems
## 3 Synchronization of Complex Dynamic Networks
The effect of reducing the computation amount is more obvious when strategies consider more layers and a shorter chain; meanwhile, the efficiency of synchronization will be reduced by more layers. Moreover, coupling weight should also be taken into account because that will appreciably impact the synchronization rate. Therefore, the balance should be taken into account of equity simultaneously. After the network-partitioning process, assuming that the new net has \(p_{i}\) and by re-arranging the weight allocation matrix, the following form would be shown:  
\[\mathcal{E}_{r}= \begin{cases}a_{11}&P_{1}&\cdots&P_{s_{2}}\\ P_{1}^{T}&P_{1,1}&\ddots&\vdots\\ \vdots&\ddots&\ddots&\vdots\\ P_{s_{2}}^{T}&\cdots&\cdots&P_{s_{2},s_{2}}\end{cases}  \tag{18}\]  
where \(P_{1},\ldots,P_{s_{2}}\) are the matrix block, and \(P_{i,i}\) is the folded portion of the \(i\)th matrix block and is a symmetric matrix. With the re-arranged weight-allocation form, the connection between different nodes could be clarified. In our results, all of the sub-networks could be grouped into finite sub-networks that are only connected to the central node. Proofs could be offered to show where the calculation savings are coming from. This is not merely an oversight of some unnecessary coupling weights. Before the Proof, the following assumption should be offered to make the network-partitioning process clearer. Each node in the complex networks only has one route connecting to the central node after the process of network partitioning. Considering the fully connected complex dynamic network, with the projection of network partitioning, under the condition of Assumption 1, the number of decision variables could be reduced such that the calculation could be saved. No matter what synchronization criterion we choose, the number of decision variables will always have following form:  
\[\sum_{i=1}^{M}c_{i}(N-1)^{i} \tag{19}\]  
where \(c_{i}\) is an integer coefficient greater than or equal to zero; also, \(n\) for the dimension of node is considered in the \(c_{i}\). \(M\) is the max order. \(N\) is the number of nodes. For all of the partitioned network, the first layer only has the central node and is represented as \(s_{1}=1\). In the \(2,3,\ldots,l\) layer, the nodes are \(s_{2},s_{3},\ldots,s_{l}\), which are all sub-central nodes. Let us start from the partitioned networks with layer 2, which means that all of the nodes are connected to the central node; therefore, we have \(s_{2}+1=N\). Additionally, the number of decision variables could be  
\[\sum_{i=1}^{M}c_{i}(s_{1}+s_{2}-1)^{i} \tag{20}\]  
which is the same as the unpartitioned networks (19). Let us extend the situation into the \(n_{l}\) layer; for the \(i\)th layer, the \(s_{i}\) sub-central node has \(s_{i}\) sub-networks, and there are nodes in each sub-network accordingly. \(m\) is the serial number from \(1,\ldots,s_{i}\). Therefore, in the \(l-1\) layer, the number of decision variables we need to calculate is  
\[\sum_{i=1}^{M}\sum_{j=1}^{s_{l-1}}c_{i}(N_{j_{i}j_{i-1}})^{i}\] \[= \sum_{i=1}^{M}c_{i}(N_{1,s_{l-1}})^{i}+\cdots+c_{i}(N_{s_{l-1}s_{ l-1}})^{i}\] \[\leq \sum_{i=1}^{M}c_{i}(\sum_{j=1}^{s_{l-1}}(N_{j_{i}j_{i-1}}))^{i} \tag{21}\]  
where \((a^{i}+b^{i})\leq(a+b)^{i}\) is implemented here. Furthermore, in each layer, the above proposition could be achieved. As the number of layers increases, the number of decision variables we need to deal with decreases at an accelerating rate. For the central node, \(N_{m,s_{1}}\) represents sub-networks connected to the central node (even if only one point is included) from \(m=1\) to \(m=s_{1}\). Additionally, the number of decision variables we have after partitioning the network is as follows:  
\[\sum_{i=1}^{M}\sum_{j=1}^{s_{1}}c_{i}(N_{1,s_{1}}-1)^{i}+\sum_{i= 1}^{M}c_{i}(s_{1}+1-1)^{i}\] \[= \sum_{i=1}^{M}c_{i}\Big{(}(N_{1,1}-1)^{i}+(N_{1,2}-1)^{i}+\cdots +(N_{1,s_{1}}-1)^{i}\Big{)}\] \[+ \sum_{i=1}^{M}c_{i}(s_{1}+1-1)^{i}\] \[\leq \sum_{i=1}^{M}c_{i}(N-1)^{i} \tag{22}\]  
Such that the proof is complete.