## 3 Materials and Methods
### Variants of Relgraph  
The Relgraph framework can be seamlessly integrated with TransE, RotatE, and other transitive representation learning methods by utilizing distinct scoring functions. Based on the scoring function utilized, Relgraph can be classified into distinct variants. We have developed the Relgraph-RotatE and Relgraph-TransE versions of the model, along with the GAT-RotatE version for comparative experiments that do not involve entity links.  
* Relgraph-TransE. In the Relgraph-TransE version, we use the following formula as the scoring function: \[\begin{array}{ll}SC(h,r,t)=&|f_{h}+p_{r}-f_{t}|,\\ &\forall(h,r,t)\in\mathcal{K}\cup\mathcal{R}\mathcal{K}.\end{array}\] (19) Here, \(f_{hr},f_{t}\) represent the embeddings of \(h,t\), respectively, and \(p_{r}\) represents the embedding of \(r\).
* Relgraph-RotatE. In the original RotatE model, entity embeddings consist of both the real and imaginary parts, which are operated separately with the predicate embeddings representing the rotation angle in the complex space. Consequently, the dimensionality of entity embeddings needs to be twice that of the predicates. However, in Relgraph, there exists a duality between entities and predicates, requiring the dimensions of entities and predicates to be consistent. This contradiction necessitates a special design for Relgraph-RotatE. Let \(p_{r}\) be the embedding of a predicate or entity link \(r\) with a dimensionality of \(D\). We can represent \(p_{r}\) as the concatenation of two parts: \(p_{r}^{u}\) and \(p_{r}^{d}\). Here, \(p_{r}^{u}\) represents the first \(\frac{D}{2}\) dimensions of \(p_{r}\), and \(p_{r}^{d}\) represents the last \(\frac{D}{2}\) dimensions of \(p_{r}\). Mathematically, we can express it as \(p_{r}=Cat(p_{r}^{u},p_{r}^{d})\), where \(Cat\) represents the concatenation operation of vectors. In Relgraph-RotatE, the rotation angle from the head vertex to the tail vertex in a triplet is calculated using the following equation: \[p_{r}^{\theta}=\min(p_{r}^{u}+p_{r}^{d},p_{r}^{u}-p_{r}^{d}),\] (20) where \(p_{r}^{\theta}\) represents the measurement of the rotation angle from the head entity to the tail entity in RotatE. The scoring function used in Relgraph-RotatE is as follows: \[\begin{array}{ll}SC(h,r,t)=&|f_{h}\circ p_{r}^{\theta}-f_{t}|,\\ &\forall(h,r,t)\in\mathcal{K}\cup\mathcal{R}\mathcal{K},\end{array}\] (21)where \(f_{h},f_{t}\) denotes the embeddings of \(h,t\). The reason for this design is due to the discontinuous nature of the following equation (Equation (23)):  
\[p_{r}^{\theta}=\left\{\begin{array}{cc}p_{r}^{u}+p_{r}^{d}&p_{r}^{u}<0\wedge p _{r}^{d}<0,\\ p_{r}^{u}-p_{r}^{d}&else.\end{array}\right. \tag{22}\]  
This discontinuous nature allows the representation learning model to better distinguish between entities and predicates that result in true or false triple assignments. It also helps to maintain a relatively clear and stable geometric interpretation of the dimensions in the embeddings.
* GAT-RotatE and GAT-TransE. We employ GAT-RotatE and GAT-TransE as benchmark models to assess the impact of introducing entity links on model performance. By setting \(\eta\) in Equation (18) to 0, we obtain GAT-RotatE/GAT-TransE from Relgraph-RotatE/Relgraph-TransE.