## 3 Proposed Method
### Deep Learning Framework  
#### 3.3.1 Graph Convolution Module  
Spatial convolution and frequency-domain convolution are the two main forms of graph convolutional neural networks. Similar to convolutional neural networks, spatial convolution is a convolution operation directly defined in space. However, the way to find nodes changes from adjacent relationships in Euclidean space to adjacency matrices in non-Euclidean space, which does not require graph theory. As shown in Figure 5(a), there exists a graph structure containing 6 nodes \((A,B,C,D,E,F)\), and the convolution operation of non-European structures can be completed by using the spatial domain convolution on the graph as shown in the first part of Figure 6. This simple graph convolutional network consists of an input layer and two graph convolutional layers. The second part of Figure 6 marks the feature propagation path when node A is the target node of the last layer. It can be seen that nodes synthesize information from nodes \((B,C,D)\) according to the adjacency matrix of graph structure in the first part of Figure 5(a), and similarly, nodes \((B,C,D)\) synthesize information from nodes \((((A,C),(A,B,E,F),(A))\), respectively. In summary, the propagation formula of graph neural networks based on spatial convolution can be summarized as follows:  
\[h_{v}^{l+1}=U^{l+1}(h_{v}^{l},\sum_{u\in ne[v]}M^{l+1}(h_{v}^{l},h_{u}^{l},x_{ uu})\]  
where \(h\) represents the graph convolution layer of the spatial convolution; \(v\) represents \(v\) nodes; \(l\) represents layer \(l\); \(ne[v]\) represents the adjacent nodes of node \(v\); and \(x_{uu}\) represents edges of nodes \(v\) and \(u\). The propagation of the entire network mainly involves two functions, \(U^{l}(x)\) and \(M^{l}(x)\), denoting the state transfer function and state confluence function of layer \(l\), respectively. Although graph spatial convolution looks similar to ordinary convolutional neural networks and is more intuitive, it lacks theoretical verification on the one hand and needs to be applied to data structures with known graph structures. As the topological relationship between electrodes of epileptic EEG acquisition devices is artificially defined and not unique, the second convolution method is adopted in this study, namely frequency domain convolution, which uses the Fourier transform of the graph to perform convolution operations. The paper [29] presents the propagation rules for graph convolution networks:  
\[H^{(l+1)}=\sigma(\tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}H^{ (l)}W^{(l)})\]  
Here, \(\tilde{A}=A+I\) is an undirected graph G with self-connections; I is the identity matrix; \(\tilde{D}\) is the diagonal matrix of \(\tilde{A}\), that is, \(\tilde{D}_{ij}=\sum_{j}\tilde{A}_{ij}\,\ W^{(l)}\) is the trainable parameter  
Figure 6: Structure of spatial convolutional neural network. matrix of the layer \(l\); \(H^{(l)}\) is the transformation matrix of layer \(l\); and \(\sigma\) is the activation function. Then, GCN is analyzed by using spectral convolution, and the Laplace operator matrix of graph G is defined as \(L=D-A\), and after normalization, it becomes \(L=I-D^{-\frac{1}{2}}AD^{-\frac{1}{2}}\), and \(\bar{L}\) feature decomposition \(\bar{L}=U\lambda U^{\tau}\). \(U\) is orthogonal eigenvector matrix, and \(\Lambda=diag(\lambda_{1},\ldots,\lambda_{n})\) is diagonal matrix of corresponding features. For an input signal, the graph Fourier transform is as follows:  
\[\bar{X}=U^{\tau}X\]  
The inverse Fourier transform is  
\[X=U\bar{X}\]  
The convolution operation on the graph is defined as the product of signal X and filter \(g\theta\) in the Fourier domain:  
\[g\theta*X=U((U^{T}g\theta)\odot(U^{T}X))=Ug\theta(\Lambda)U^{T}X\]  
The operator \(\odot\) is the convolution operation of the corresponding position of the matrix; \(g\theta(\Lambda)\)is the diagonal matrix of spectral filter coefficients, that is, the convolution kernel. Directly calculating the above equation requires a large amount of computation and time. Therefore, [PERSON] et al.