# Redundancy-Reduction-Based Hierarchical Design in Synchronization of Multi-Agent Systems
## 2 Preliminary Knowledge
For a certain single node in the networks, it is obvious that the connected edge with greater weight plays a more influential role than the smaller one. Therefore, by evaluating the influence of different connection weights of individual nodes without breaking the integrity of network structure, we could establish a rule that focuses on prior connections. The weighted edges with a strong influence could be selected and retained, and the edges with smaller weights that have trivial influence on the global synchronization target could be ignored. The structure will further save on computing resources. Based on that, the CDN with a tremendous number of nodes and connections could be partitioned by a finite number of sub-networks. **Remark 1**. : _In the undirected graph, unlike the directed graph, usually there are no necessary pinning control networks. Therefore, throughout our paper, the central node is free to choose, and the setting of node \(1\) is just for convenience._  
A simple example of CDN with 8 nodes is offered in Figure 1.  
The weight allocation matrix is  
\[\mathcal{E}= \begin{bmatrix}-68&12&8&41&0&0&0&7\\ 12&-140&0&0&125&3&0&0\\ 8&0&-69&61&0&0&0&0\\ 41&0&61&-214&0&112&0&0\\ 0&125&0&0&-149&24&0&0\\ 0&3&0&112&24&-145&6&0\\ 0&0&0&0&0&6&-6&0\\ 7&0&0&0&0&0&-7\end{bmatrix}  \tag{8}\]  
In Figure 1, we can observe information transfers between node 1 and node 2, 3 and 8. Node 3 exchanges information with node 1 and 4 at the same time. For node 3, the coupling weight with node 4 is much bigger than with node 1, such that the impact from node 4 is even greater than node 1 to 3. Therefore, without affecting the complete transmission of topology, we could try to ignore the connection between node 1 and 3, to avoid useless calculations. By neglecting some trivial connections, we can realize the partitioning of the network to a finite number of sub-networks layer after layer, such that the computation burden could be relieved and the effectiveness could be boosted. In Figure 1, nodes 7 and 8 are the isolated nodes that only have one edge connecting to other nodes. Since node 1 is the central note, node 8 could be considered as a sub-network. As mentioned above, the connection weight between node 1 and node 3 is small. It does not affect the traversal of the whole network. Therefore, we could separate nodes 3, 4, 6, and 7 as another sub-network. Additionally, nodes 2 and 5 constitute the rest of the sub-network. Consequently, the network of Figure 1 could be transferred into the following Figure 2, where the example with the form of system (7) could be rewritten as 3 parts:  
Figure 1: An example of CDN with 8 nodes. Figure 2: Structures of sub-networks in Figure 1.  
The algorithm in [13] is applied here as reference. The total computation of scalar decision variables is \(7.5m^{2}+1.5m+N-1\), where \(m=(N-1)n\), \(N=8\) in the example. Therefore, without partitioning the CDN, in Example 1, the number is \(367.5n^{2}+10.5n+7\), while after partitioning the CDN, the number is \(105n^{2}+9n+6\), which is much less than the method without partitioning the CDN. Moreover, the advantages of partitioning will become more obvious as \(N\) increases. Systematic process algorithms can be derived from the previous example.