# A Domain-Oriented Entity Alignment Approach Based on Filtering Multi-Type Graph Neural Networks
## 4 The Filtering Multi-Type Graph Neural Networks
### The Embedding Module
#### 4.2.1 The Structure-Aware Entity Embedding  
There is a lack of relationships between the entities in the domain KGs. To address this issue, we utilize the attribute values (represented by white dots) as the bridges and conduct iterative computations to progressively learn the multi-hop structural features of the entities through the multiple hidden layers of the SA. We first construct the input of the SA according to the attribute triples. Subsequently, we employ the SA with multiple hidden layers to learn the high-dimensional representation vectors for each entity and attribute value. These representation vectors can capture the semantic information between entities and attribute values, and the relevant information between the entities gradually spreads to the representation vectors of the other entities through the virtual relationship of attribute values. Finally, the indirect relationship (represented by a dotted green line with an arrow) between the entities can be captured through these representation vectors. The aggregation process of the SA is shown in Figure 4. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(\sigma\) & The nonlinear activation function \\ \(\mathbf{A}\) & The adjacency matrix \\ \(\mathbf{I}\) & The identity matrix \\ \(\mathbf{D}\) & The degree matrix corresponding to the adjacency matrix \\ \(W^{(l)}\) & The weight coefficient matrix of the \(l\)-th layer \\ \(\bar{A}=A+I\) & The adjacency matrix with self-connection \\ \(N_{i}\) & The collection of neighboring entities of the \(i\)-th entity in KGs \\ \(\mathbf{H}\) & The embedding of domain KGs \\ \(b^{(l)}\) & The bias term of the \(l\)-th layer \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parts of the parameters of the SA and AA. The computation of the hidden layer plays a crucial role in the aggregation process of the SA. Due to the sparsity or even absence of relationship triplets in the domain knowledge graphs, we utilize attribute triplets as a substitute for relationship triplets for feature embedding. Multiple layers of the SA are employed to indirectly establish associations between entities by learning multi-hop neighborhood information. The neighborhood information can be quantified in the form of multiple hidden layers. Equation (1) encapsulates a comprehensive formulation for the update of the \(l\)-th hidden layer, \(\mathbf{H}^{(l)}\), within the SA. The function, \(f(\cdot)\), is denoted as the update mechanism that utilizes the input from the previous layer, \(\mathbf{H}^{(l-1)}\), and the adjacency matrix, \(\mathbf{A}\). \[\mathbf{H}^{(l)}=f\Big{(}\mathbf{H}^{(l-1)},\mathbf{A}\Big{)} \tag{1}\]  
In Equation (1), the parameters of the feature matrix, \(\mathbf{H}^{(0)}\), and the adjacency matrix, \(\mathbf{A}\), in the SA are initialized according to the information about the dishes and related ingredients. Equation (2) presents a specific instantiation of the update mechanism, where an activation function, \(\sigma(\cdot)\), is applied and the propagation of information from the previous layer, denoted as \(\mathbf{A}\mathbf{H}^{(l-1)}\mathbf{W}^{(l-1)}\), is incorporated. In addition, our approach selects LeakyReLu as the specific activation function to mitigate gradient vanishing. \[\mathbf{H}^{(l)}=\sigma\Big{(}\mathbf{A}\mathbf{H}^{(l-1)}\mathbf{W}^{(l-1)}+b^{(l-1)}\Big{)} \tag{2}\]  
In the equation above, \(b^{(l-1)}\) represents the bias, which is a parameter adjusted through back-propagation. To effectively capture the bidirectional relationships among the entities and the inherent features of the entities during the process of feature learning, we perform symmetric normalization and self-connection processing on the adjacency matrix, \(\mathbf{A}\), as indicated by Equation (3):  
\[\widehat{\mathbf{A}}=\mathbf{D}^{-\frac{1}{2}}\mathbf{A}\mathbf{D}^{-\frac{1}{2}}+\mathbf{I} \tag{3}\]  
Equation (4) is a variation of Equation (2).