# Few-Shot Charge Prediction with Data Augmentation and Feature Augmentation
## 3 Method
### Feature Augmentation
#### 3.2.2 Dropedge Graph Convolutional Network  
Based on the charge label heterogeneous graph established above, this paper selected the graph convolution network (GCN) for feature extraction. Compared with other graph neural network models, GCN has a simple structure and easy operation and can extract more effective features based on fewer parameters. The traditional GCN performs feature extraction on homogeneous graphs [37], while the charge label graph in this paper contains different entities and relationships, which are typical for heterogeneous graphs. By analyzing different types of information in the charge label graph, complex semantics and inter-word dependencies can be captured, which enriches the few-shot charge semantics more and effectively improves the accuracy of prediction results. Formally, consider a graph \(G=(V,E)\), where \(V(|V|=m)\) and \(E\) are sets of nodes and edges, respectively. Further, in this paper, the message passing is used to update the representation of each node. The graph convolution of the GCN model is actually a special form of Laplacian smoothing [38]. We obtained new features for each node by calculating the weighted  
Figure 4: Charge label heterogeneous graph. average of its own nodes and its neighbors. In addition, information from neighboring can be collected and integrated to update each node's own representation. The new \(d_{g}\) dimensional node feature matrix \(H^{(l+1)}\in R^{m\times d_{g}}\) is calculated as follows:  
\[H^{(l+1)}=\sigma(\widetilde{A}H^{(l)}W^{(l)}) \tag{9}\]  
where \(\widetilde{A}=\widetilde{D}^{-\frac{1}{2}}A^{\prime}\widetilde{D}^{-\frac{1}{2}}\), \(A^{\prime}=A+I\) is the adjacency matrix \(A\) with added self-connection \(I\) and \(A\) is the weight matrix calculated by formula (8). \(\widetilde{D}\) is a diagonal matrix with \(\widetilde{D}_{ii}=\sum_{j}A^{\prime}_{ij^{\prime}}\), \(H^{(l)}\) as the node representation matrix of the \(l-\)th layer. \(W^{(l)}\) is the trainable parameter matrix. \(\sigma(. )\) is the activation function, such as ReLU. Initially, \(H^{(0)}=X\). In order to obtain higher-order charge information representation, the GCN contains 4 hidden layers. The distributed representation of each node in the last layer can be learned by applying the propagation rule to our graph defined before. However, while simply increasing the depth of the GCN network, the problem of over-smoothing arises and there is convergence in the node representation, thus making the differentiation of different types of nodes poor, which hinders the distinction of confusing charges. Therefore, we introduced the Dropedge [39] technique by randomly setting some non-zero elements of the adjacency matrix \(A\) to zero, i.e., deleting a fixed proportion of edges in the original graph, we called the new model Dropedge graph convolutional network (DGCN) and made the node connections more sparse without affecting the original features, avoiding the over-smoothing problem caused by increasing the number of GCN layers. At the same time, the Dropedge technique enhances the randomness and diversity of the data and alleviates the problem of over-fitting. If we denote the resulting adjacency matrix as \(A_{drop}\), then its relation with \(A\) becomes:  
\[A_{drop}=A-A^{\prime} \tag{10}\]  
where \(A^{\prime}\) is a sparse matrix expanded by a random subset of non-zero elements of the adjacency matrix \(A\). Then, we also performed the re-normalization trick on \(A_{drop}\), leading to \(\widetilde{A}_{drop}\). We replaced \(\widetilde{A}\) with \(\widetilde{A}_{drop}\). To further improve charge information' distinguishable feature, we computed the charge graph information \(c=[c_{1},c_{2},\dots,c_{k}]\) by adopting two kinds of pooling operators to extract the distinguishing features of hidden states.