## 3 Methods
### Importance-Correlated Directed Graph Convolutional Networks
#### 3.2.1 Directed Graph Convolution Layer  
Adjacency matrices and Laplace matrices are often used to formally describe the communication relationships between robot nodes, and such matrices are also known as graph shift operators. Considering the time-varying local communication, the graph shift operator at \(n\)th sampling moment is uniformly denoted by \(\mathbf{S}_{n}\). The graph shift operation \(\mathbf{S}_{n}(\cdot)\) realizes the linear transfer and recombination of the graph signal along the edge at \(n\)th sampling moment.  
In this paper, the adjacency matrix based on directed graphs is expressed as:  
\[[\mathbf{S}_{n}^{d}]_{ij}=\begin{cases}1,&\text{if }e_{ij}^{d}\in\mathcal{E},\\ 0,&\text{otherwise}.\end{cases} \tag{9}\]  
For the directed graph, the graph shift operator \(\mathbf{S}_{n}^{d}\) is not guaranteed to be symmetric, which also leads to the fact that the Graph Fourier transformation cannot be executed on the graph signal based on the directed graph, and the graph convolution operation cannot be explained from the perspective of frequency filtering. Based on the assumption of directed graphs, this paper explains the directed graph convolution operation only from the spatial convolution perspective.  
We assume that local communication between the node and its neighbors can be achieved only once per sampling period, i.e., the data of \(k\)th neighbors acquired by the node in real-time come from the \((n-k)\)th sampling moment. The graph convolution operation based on the assumption of time-varying graph structure and communication delays can be defined as a weighted linear combination of graph shift operation in multidimensional subgraphs:  
\[\mathcal{A}_{n}(\mathbf{X}_{n-k},\mathbf{S}_{n-k}^{d}|k=0,1,\cdots,K-1)=\sum _{k=0}^{K-1}(\mathbf{S}_{n}^{d}\mathbf{S}_{n-1}^{d}\cdots\mathbf{S}_{n-k}^{ d})\mathbf{X}_{n-k}\mathbf{A}_{k}, \tag{10}\]where \(\mathbf{S}_{n-k}^{d}\in\mathbb{R}^{N\times N}\) describes the directed graph structure at the \((n-k)\)th sampling moment, \(\mathbf{X}_{n-k}\in\mathbb{R}^{N\times F}\)represents the observation matrix of the swarm at the \((n-k)\)th sampling moment, \(\mathbf{A}_{k}\in\mathbb{R}^{F\times D}\) is the matrix of filter coefficients to be optimized, and the \(\mathbf{A}_{k}\) are shared by each node. \(K\) is the max order of the subgraph, which is the maximum range of behavior reference for that node. The larger the value of \(K\), the larger the range for individual behavior reference, which theoretically has a positive impact on the flocking control effect. However, due to the influence of communication delay, the higher the orders of the neighboring nodes, the less reliable their data are, so the value of \(K\) needs to be chosen reasonably. The optimal value of \(K\) will be discussed in Section 4.2.4.  
Obviously, the size of the filter coefficient matrix does not change as long as the dimensionality of the observation and action policy remains unchanged. Therefore, the control strategy has the scalability, which means that the strategy obtained by training on a small-scale swarm can be directly applied to a larger-scale swarm.