## 2 Chebyshev Graph Convolutional Neural Network Modeling Method
### Feature Extraction Module Based on Fused Graph Neural Network
#### 2.2.1 ChebyNet Symmetry Processing Optimizes Feature Extraction  
The ChebyNet [15] serves as both the initial and final layer of the feature extraction network, updating the global node representation by determining the neighborhood range and iteratively aggregating information from adjacent nodes. When the value is set to 1, the node aggregates information from all directly adjacent nodes (i.e., nodes directly connected by edges) to update its node representation. When the value is set to 2, the node aggregates information from both directly adjacent nodes and nodes that are indirectly connected to update its node representation. Finally, the learned node representation is utilized to perform node classification. The iterative update process of the ChebyNet node features is illustrated in Figure 2. Here, \(h_{i}\) denotes the feature of node \(i\), \(h_{Ne}\) represents the domain node feature set, \(W\) is the weight matrix, and \(h^{\prime}_{i}\) is the updated feature of node \(i\). The symmetry processing of the ChebyNet feature extraction process is enhanced to ensure a balanced consideration of the relationships between nodes and effectively address the challenge of uneven information transmission during feature extraction under variable working conditions. The process of introducing the symmetry processing mechanism into the ChebyNet framework is elaborated in detail across three parts: the description of the original ChebyNet feature update mechanism, the implementation of the symmetry processing mechanism, and the description of the improved ChebyNet feature update mechanism. (1) Original ChebyNet feature update process: The initial node feature matrix \(X\) is represented in Equation (4), and the Laplacian matrix \(\hat{L}\) is used to propagate the features  
Figure 2: Node feature update mechanism of ChebyNet when the aggregation neighborhood range is 1 and 2, respectively. of the domain after one Chebyshev polynomial iteration is shown in Equation (5). The node features after two Chebyshev polynomial iterations are shown in Equation (6). Each iteration combines the previous and next results and recursively captures multi-order neighbor information to optimize the propagation and aggregation process of node features. The initial node feature information is introduced by subtracting the initial node feature matrix, thereby enhancing its propagation effect. The recursive update and the original features are combined to enhance the model's ability to capture complex relationships in the graph. Similarly, the node features after one iteration of the Chebyshev polynomial can be obtained, as shown in Equation (7). \[Z_{0}=X \tag{4}\]  
\[Z_{1}=\hat{L}\cdot X \tag{5}\]  
\[Z_{2}=2\hat{L}\cdot Z_{1}-Z_{0} \tag{6}\]  
\[Z_{k}=2\hat{L}\cdot Z_{\{k-1\}}-Z_{\{k-2\}},k\geq 2 \tag{7}\]  
Here, \(Z_{0}\) represents the initial node feature matrix; \(X\) denotes the input data feature matrix; \(Z_{1}\) denotes the node feature after one iteration of the Chebyshev polynomial; \(\hat{L}\) represents the scaled and standardized Laplacian matrix; \(Z_{2}\) denotes the node feature after two iterations of the Chebyshev polynomial; and \(Z_{k}\) denotes the node feature after the \(k\) iteration of the Chebyshev polynomial. (2) Implementation of symmetry processing: Because the internal data distribution of the initial node feature matrix shows significant differences, to better characterize the data features, symmetry processing is introduced in the calculation of the node features after two Chebyshev polynomial iterations by adding and averaging the intermediate matrix representation and its transpose, as shown in Equation (8). After symmetry processing, the node features obtained through recursive operations after introducing symmetry processing and iterating the Chebyshev polynomial are shown in Equation (9).