# A Graph Neural Network Node Classification Application Model with Enhanced Node Association
## 3 Model Analysis
### ENode-GAT Model
#### 3.1.3 Graph Convolution Layer  
The primary function of the graph convolution layer is to input the heterogeneous graph structure from the input layer and the feature matrix from the feature extraction layer into the graph convolution model for semi-supervised classification model training, and to output the corresponding convolution results at the end of the convolution layer. The graph structure of the input layer can be understood as a combination of m nodes \(V=\{v_{1},v_{2},v_{3},\ldots,v_{m}\}\) and n edges \(E=\{e_{1},e_{2},e_{3},\ldots,e_{n}\}\) that have connections between nodes. In order to implement node classification using the graph convolutional neural network model, it is necessary to calculate the adjacency matrix, the degree matrix, and the Laplace matrix of the graph structure, as shown in Equations (3)-(5). \[A=\left(a_{ij}\right) \tag{3}\]  
\[D=diag\left(\sum_{i\
eq j}a_{ij}\right) \tag{4}\]  
\[L=I_{N}-D^{-\frac{1}{2}}AD^{-\frac{1}{2}} \tag{5}\]  
In the equation above, \(A\) represents the adjacency matrix, \(D\) represents the degree matrix, \(L\) represents the Laplacian matrix, and \(I_{n}\) is a unitary array of order \(n\). For model training, the computed data are fed into two convolutional layers of the GCN model and GAT model. Figure 3 depicts the schematic diagram of the GCN+GAT model. Figure 3: Schematic diagram of the GCN+GAT model. The convolution of the two-layer graph convolution network is calculated as in Equations (6) and (7). \[H^{(1)}=ReLU\Big{(}\hat{A}XW^{(1)}\Big{)} \tag{6}\]  
\[H^{(2)}=\hat{A}H^{(1)}W^{(2)} \tag{7}\]  
In the above equation, \(H^{(1)}\) denotes the output of the first layer and the input of the second layer; \(H^{(2)}\) denotes the output of the second layer; \(ReLU\) denotes the activation function; \(X\) denotes the input of the first layer neural network; \(W\) denotes the weight of the graph neural network; \(W^{(1)}\) denotes the weight of the first layer neural network; \(W^{(2)}\) denotes the weight of the second layer neural network. \(\hat{A}\) is a single-layer convolution operation, which can be interpreted as a convolution kernel, as shown in Equation (8). \[\hat{A}=\widetilde{D}\ ^{-\frac{1}{2}}\ \widetilde{A}\ \widetilde{D}\ ^{-\frac{1}{2}} \tag{8}\]  
The GAT module begins by calculating the attention factor using the obtained calculation data. We calculate the similarity coefficients of vertex \(i\) is neighbors and itself, one by one, as illustrated by Equation (9). \[e_{ij}=a\big{(}\big{[}\ Wh_{i}||Wh_{j}\big{]}\big{)},j\in N_{i} \tag{9}\]  
In the equation above, \(h\) can be expressed as a series of vectors embedded within vectors. \(W\) represents the dimensioning of the vertices' features; \(||\) represents the stitching of the transformed features for vertices \(i,j\); and \(a()\) represents the mapping of the stitched high-dimensional features to a real number. The original attention factor is \(e_{ij}\). As shown in Equation (10), the initial attention coefficients are then subjected to a normalization operation to acquire the final attention coefficients \(a_{ij}\). \[a_{ij}=\frac{\exp\big{(}LeakyReLU\big{(}e_{ij}\big{)}\big{)}}{\sum_{k\in N_{i} }\exp(LeakyReLU\big{(}e_{ik}\big{)})} \tag{10}\]  
In the equation above, \(LeakyReLU\) denotes the activation function; \(a_{ij}\) denotes the final attention coefficient calculated. Finally, the features are weighted and summed according to the calculated attention coefficients, as shown in Equation (11). \[h_{i}^{l+1}=\sigma\Bigg{(}\sum_{j\in N_{i}}a_{ij}Wh_{j}\Bigg{)} \tag{11}\]  
In the equation above, \(W\) is the weight matrix of feature multiplication; \(a_{ij}\) denotes the final attention coefficient calculated previously; \(\sigma\) denotes the nonlinear activation function; \(j\in N_{i}\) denotes traversing all nodes adjacent to \(i\). The early stop algorithm can effectively prevent the overfitting phenomenon that occurs during small sample training.