# Redundancy-Reduction-Based Hierarchical Design in Synchronization of Multi-Agent Systems
## 3 Synchronization of Complex Dynamic Networks
The sub-central node \(x_{i,s_{i}^{j},1}\) represents the \(j\)th sub-central node in the \(i\)th layer. \(t_{0}=0\) is the initial time. The aim is to prove that if \(\delta>0\), for any \(\|x_{i,s_{i}^{j},1}(t_{0})-x_{i,s_{i}^{j},k}(t_{0})\|<\delta\) exists, and \(\lim_{t\to\infty}\|x_{i,s_{i}^{j},1}(t)-x_{i,s_{i}^{j},k}(t)\|=0\). Then, the \(\lim_{t\to\infty}\|x_{1,1,1}(t)-x_{i,s_{i}^{j},k}(t)\|=0\) also set up. The proof could be achieved by the iterative method. Firstly, with the last layer, sub-networks (\(n_{l}\)th layer) asymptotically synchronize to the sub-central node \(x_{n_{l},s_{n_{l}}^{j},k^{\prime}}\), then, the corresponding Lyapunov functional could exist and be represented as \(V_{n_{l},s_{n_{l}}^{j}}(t)\), and we have  
\[\dot{V}_{n_{l},s_{n_{l}}^{j}}(t)<0 \tag{13}\]  
for all \(s_{n_{l}}^{j}\). Then, for the one up layer, a similar Lyapunov functional could be established with \(V_{n_{l}-1,s_{n_{l}-1}^{j}}(t)\) to prove that for the \(n_{l}-1\) layer, the sub-network could achieve asymptotic synchronization. We combine the \(V_{n_{l},s_{n_{l}}^{j}}(t)\) and \(V_{n_{l}-1,s_{n_{l}-1}^{j}}(t)\) to be a new Lyapunov functional \(V_{n_{l},s_{n_{l}}^{j}}(t)+V_{n_{l}-1,s_{n_{l}-1}^{j}}(t)\), such that  
\[\hat{V}_{n_{l},s_{n_{l}}^{j}}(t)+\hat{V}_{n_{l}-1,s_{n_{l}-1}^{j}}(t)<0 \tag{14}\]  
By iterating this method over the entire network, the Lyapunov functionals could always exist  
\[V(t)=\sum_{i=1}^{n_{l}}V_{i,s_{i}^{j}}(t) \tag{15}\]  
and  
\[\hat{V}(t)=\sum_{i=1}^{n_{l}}\hat{V}_{i,s_{i}^{j}}(t)<0 \tag{16}\]  
Such that there exist \(\delta>0\) to satisfy  
\[\|x_{1,1,1}(t_{0})-x_{i,s_{i}^{j},k}(t_{0})\|<\delta\] \[\Rightarrow\lim_{t\rightarrow\infty}\|x_{1,1,1}(t)-x_{i,s_{i}^{j },k}(t)\|=0 \tag{17}\]  
**Remark 2**. : _In the proof, in Theorem 1, no concrete Lyapunov functional is needed to complete the proof; hence, it can be ignored._  
**Remark 3**. : _Note that in the definition of \(x_{i_{1},s_{n_{l}}^{j_{1}},k^{\prime}}x_{i_{2},s_{n_{l}}^{j_{2}},k}\) could be the same node because nodes in the \(i_{1}\)th layer could the sub-central node in the \(i_{2}\)th layer. The representation method does not affect the proof process._  
One of the available partitioning network method is proposed with following steps:  
* _Step 1_: Lists all of the notes connected to the central node. * _Step 2_: Find nodes with only one connection and list them out; name these nodes \"initial points\". * _Step 3_: Start the chain with the initial nodes. Find the node connected to the initial points, then continue the chain with the connected node to find the next step. The strategy is to connect the greater coupling weight. If the initial node only communicates with the central note, stop the search and mark the connection node as a sub-network. * _Step 4_: If the two chains meet at one node (not the central node), then two chains could be combined to be one chain and continue the path. * _Step 5_: When all of the chains are connected to the central node, mark all the chains and continue dealing with the rest of the nodes. * _Step 6_: Start the path from each node of the marked chain to build the sub-chain; like _Step 1_, the sub-central node could be set here if needed. Use the greater coupling weight strategy to connect all the other nodes. * _Step 7_: If two sub-chains connect to one node, the connection with stronger coupling weight could be kept as one node of the subchain. The weaker coupling weight connected node ignores this connection and searches for the second great coupling weight node. * _Step 8_: Repeat _Step 6_ until all nodes are traversed. * _Step 9_: After traversing all nodes in the network, the search could stop and the sub-networks could be built. The algorithm is only a simple solution; it is not optimal. Strategies such as dynamical programming, prim's algorithm, and others could be adopted here. Additionally, event-triggered samplers could be applied to offer a threshold.