## 3 Methods
### Importance-Correlated Directed Graph Convolutional Networks
#### 3.2.2 Importance-Correlated Directed Graph Convolutional Networks  
The whole graph convolutional neural networks consist of \(L\) layers of tandem graph neural networks, each layer containing a graph convolutional layer and a graph activation layer. The graph activation layer can be expressed by:  
\[\mathbf{X}_{l}=\sigma[\mathcal{A}_{l}(\mathbf{X}_{l-1},\mathbf{S}^{d})], \tag{11}\]  
where \(\mathbf{X}_{l-1}\in\mathbb{R}^{N\times F_{l-1}}\) is the input of the \(l\)th layer, \(\mathbf{X}_{l}\in\mathbb{R}^{N\times F_{l}}\) is the output of the \(l\)th layer, \(\mathcal{A}_{l}(\mathbf{X}_{l-1},\mathbf{S}^{d})\) is the output of the graph convolution layer in the \(l\)th layer, and \(\sigma\) denotes the graph activation function. The output of the last layer \(\mathbf{X}_{L}=\mathbf{U}_{n}\) is the action policy of each robot. In this paper, we choose the Tanh activation function as the graph activation function. Laplace matrix and Adjacency matrix are familiar graph shift operators that assign the same weight to each node in the subgraph. When they are employed as the graph shift operators, as long as communication relationships are established between nodes, the neighbors' information will be aggregated by equal-weight transformation. In this paper, we assume that robot nodes and links have different importance and their weights in the process of data aggregation should be distinguished, so we propose to use a weighted link contraction degree matrix to replace the traditional graph shift operator. It is worth noting that the calculation of link contraction degree only requires obtaining local information of 1-hop neighbors, in contrast to the calculation of link importance, which requires obtaining contraction degrees of all links in the swarm. Therefore, link contraction degree is selected for the real-time construction of graph shift operators. The concepts of link contraction \(\deg(e_{ij})\) and link importance decay coefficient \(\omega_{ij}\) are clarified in Section 3.1. In order to evaluate the weight of information from different nodes in a constrained communication environment, this paper defines the directed contraction degree and contraction decay coefficients related to the link length. For the directed graph, this paper specifies that the link can only be contracted along the direction of the directed edges, and the link contraction degree is calculated by counting only the in-degree of the new node after contraction, the directed link contraction is denoted as \(\deg(e_{ij}^{d})\). The directed link contraction degree decay coefficient \(\xi_{ij}^{d}\) is defined as:  
\[ \begin{cases}d_{i}=\frac{\sum\|\mathbf{v}_{ij}\|}{|\mathcal{N}^{d}(v_{i})|}, \{j|v_{j}\in\mathcal{N}^{d}(v_{i})\},\\ \xi_{ij}^{d}=1-\frac{\|\mathbf{v}_{ij}\|-d_{i}\|}{d_{max}},\{j|v_{j}\in\mathcal{ N}^{d}(v_{i})\},\end{cases}  \tag{12}\]  
where \(d_{i}\) is the average length of all directed edges ending at node \(v_{i}\), \(|\mathcal{N}^{d}(v_{i})|\) is the total number of 1-hop neighbors of node \(v_{i}\) in the directed graph, and \(d_{max}\) is the maximum communication distance. In addition, we use \(\omega_{ij}^{d}\deg(\varrho_{ji}^{d})\) to evaluate the weights of information from different links. To ensure the general performance of the control strategy, the weights of all links of the same node are normalized so that the weighted link contraction matrix can be expressed as:  
\[\left[\mathbf{S}^{d*}\right]_{ij}= \begin{cases}-\frac{w_{ij}^{d}\deg(\varrho_{ji }^{d})}{\sum_{\gamma_{ij}\in\Lambda^{d}(v_{i})}w_{ij}^{d}\deg(\varrho_{ji}^{e })},&\text{if }\varrho_{ji}^{d}\in\mathcal{E},\\ 1,&\text{if }i=j,\\ 0,&\text{otherwise},\end{cases}  \tag{13}\]  
where the value on the diagonal is set to be 1 and the value of the node's neighbors are set to be the opposite of its weighted link contraction degree, since this paper is more concerned with the observation error of the neighboring nodes versus the node itself.