## 2 Materials and Methods
### Preliminaries
#### 2.2.3 Model
When the input is negative, \(\alpha_{k}\) is the attention vector of the \(k\)-th head and \(\parallel\) represents the vector concatenation operation. This mechanism enables the model to focus on relevant features and relationships within the graph, enhancing its ability to extract meaningful representations for downstream tasks. Generating edge and structural features is crucial in enhancing the model's understanding of the graph's topology. Both the edge feature generator and node feature generator are implemented as Multi-Layer Perceptrons (MLPs) to capture intricate patterns and relationships within the data. The edge feature \(e_{i,j}^{t}\) for nodes \(i\) and \(j\) in layer \(t\) is computed using the following equation:  
\[e_{i,j}^{t}=ReLU\Big{(}W_{e}^{t}\Big{[}h_{i}^{t-1}\parallel h_{j}^{t-1}\Big{]} \Big{)} \tag{5}\]  
Here, \(W_{e}^{t}\) represents the weight matrix associated with the edge feature generator, \(h_{i}^{t-1}\) and \(h_{j}^{t-1}\) denote the feature representations of nodes \(i\) and \(j\) in the previous layer (\(t-1\)), respectively, and \(\parallel\) represents the vector concatenation operation. The structural feature vector \(x_{i}\) for node \(i\) is then formed by combining its node feature representation \(h_{i}^{t}\) and the mean of edge feature features across its neighbors \(\mathcal{N}_{i}\). \[x_{i}=\left[h_{i}^{t}\parallel\Big{\langle}e_{i,j}^{t}\Big{\rangle}_{j\in \mathcal{N}_{i}}\right] \tag{6}\]  
Here, \(\Big{\langle}e_{i,j}^{t}\Big{\rangle}\) represents the mean edge feature of neighboring nodes. This combination enables the model to leverage both node-centric and neighborhood-wide information when constructing the structural feature vector. The mean operation ensures a comprehensive representation that considers the collective influence of neighboring nodes on the target node. Once the structural feature vector \(x\) is computed for each node, a new structural feature matrix \(X^{s}\in R^{N\times N}\) is constructed to encapsulate the structural information of the heterogeneous graph. This matrix \(X^{s}\) is defined as follows:  
\[X^{s}=\left\{ \begin{array}{l}X_{[i,j]}=0,(i\
eq j)\\ X_{[i,j]}=x_{[i,j]},(i=j)\end{array} \right. \tag{7}\]  
In essence, each diagonal element \(X_{[i,j]}\) of \(X^{s}\) represents the structural feature vector \(x_{i}\) of the corresponding node \(i\), while all off-diagonal elements are set to zero. After obtaining the structural feature matrix \(X^{s}\), it is multiplied with the initial adjacency matrix A to compute the structural representation \(Z\):  
\[Z=AX^{s} \tag{8}\]  
This operation integrates the structural information derived from node features into the adjacency matrix, thereby enriching the representation of edges in the graph. The resulting structural representation \(Z\) captures the combined influence of node attributes and graph topology, facilitating more informed predictions in subsequent stages of the model. To incorporate higher-order neighbor information and enhance the structural representation \(Z\), we refine the computation outlined in Equation (8). The improved formulation is expressed as \(Z=f\Big{(}\Phi,\sum_{l=1}^{L}\Big{\{}\text{A}_{l},X_{\text{s}}^{l}\Big{\}}\Big{)}\). Here, \(f\) is a function, \(\Phi\) is a parameter, \(\text{A}_{l}\) denotes the \(l\)-th adjacency matrix, and \(X_{\text{s}}^{l}\) represents the structural feature matrix at the \(l\)-th hop. This adjustment aims to capture the influence of higher-order neighbors on the structural representation. Specifically, we iteratively update \(Z\) by applying a MLP denoted as \(g\):  
\[Z=g\Big{(}Z^{(0)},Z^{(1)},\dots,Z^{(l)}\Big{)} \tag{9}\]  
In this equation, \(Z^{(0)}\) represents the initial structural features, and \(Z^{(l)}\) is computed as \(Z^{(l)}=\sum_{i=0}^{l-1}\beta^{i}\text{A}_{i}\text{X}_{\text{s}}^{l}Z^{(0)}\). Here, \(\beta\) is a hyperparameter controlling the weight ratio between nearby and distant neighbors.