## 3 Materials and Methods
### Classification  
Graph Convolutional Residual Network (GCN):  
The proposed GCN model can be divided into three sequential modules: Input, ResNet, and Classification. The architecture of the proposed model is displayed in Figure 3. A k-fold cross-validation is performed, where (k-1) folds are used for training and validating the model, and the remaining fold is used for testing the model. Each module is discussed in this section. ```
Inputs: G: Initial graph structure Steps: The number of iterations for the algorithm. Data: A matrix of observed variables (EEG data) with dimensions (N, T), where N is the number of variables (EEG channels) and T is the number of time points. Window size: The size of the time window to use for each iteration. Overlap: The overlap between adjacent time windows is a fraction of the window size. Outputs: Graphs: A list of updated graph structures--one for each time window. ```  
**Algorithm 1**Bayesian-Structure Learning Algorithm  
Input module: The input module is designed for graph data, particularly for handling the intricacies of irregular and non-uniform graph structures. By integrating the graph convolutional network (GCN) design, the module efficiently captures both the localized and broad characteristics present in the graph. It achieves this by assimilating data from proximate nodes and their respective edges, rendering a comprehensive understanding of the graph's topology [9]. The core operation in the GCN layer is represented by:  
\[z=f(X,A)=\Big{(}\hat{A}ReLU\Big{(}\hat{A}\times w^{(0)}\Big{)}w^{(1)}\Big{)} \tag{15}\]  
where  
\[\hat{A}=\overset{\sim}{D}^{-\frac{1}{2}}\overset{\sim}{AD}^{-1/2} \tag{16}\]  
Here:  
* \(X\) is an input feature matrix with dimensions \(N\times D\). \(N\) is the node count, and \(D\) indicates the number of initial features. * \(A\) symbolizes the weighted adjacency matrix of the graph. * The resultant \(Z\) is a node-level output matrix of size \(N\times F\), with \(F\) denoting the feature count for each node. * \(w^{(0)}\) signifies the weight matrix transitioning from input to hidden layer, designed for a hidden layer comprising \(H\) feature maps. * \(w^{(1)}\) is the weight matrix transitioning from the hidden layer to the eventual output features. * The term \(\Big{(}\hat{A}ReLU\Big{(}\hat{A}\times w^{(0)}\Big{)}w^{(1)}\Big{)}\) serves as a nonlinear ReLU activation function. The hidden neural network is described by:  
\[H^{(l+1)}=f\Big{(}H^{(l)},A\Big{)} \tag{17}\]  
Starting with \(H^{(0)}=X\) and culminating in \(H^{(L)}=Z\), where \(L\) is the total number of layers. ResNet module: The performance of deep neural networks, such as convolutional neural networks (CNNs), began to degrade due to the complications associated with training. These impediments, attributed mainly to vanishing and exploding gradient problems, hindered the convergence of the generalizability of profoundly deep networks. Residual network (ResNet) architectures often incorporate a \"shortcut\" structure in the network to counteract these challenges [14]. These structures provide an unimpeded gradient flow throughout the network. The shortcut structure facilitates feature learning using optimal learning and congruent mapping as follows:  
\[F(y)=H(x)-x \tag{18}\]  
Figure 3: Architecture of GCN. The model consists of the input, ResNet, and classification modules. The input module contains graph convolutional layers that receive functional-connectivity features. The ResNet module includes a “shortcut” network structure in convolutional and identity blocks. The classification module includes a fully connected layer followed by SoftMax activation. The proposed GCN model consists of an identity block and a convolutional block. Each block consists of dropout and batch-normalization layers. The dropout layer is employed against overfitting; the dropout layers intermittently neutralize a subset of neurons, instilling a degree of randomness and thereby preventing co-adaptation of features.