## 3 Research Methods
### Model Structures  
For each relation predicate p in the triple, TransR sets a projection matrix \(\mathbf{M}_{p}\ \in\ \mathbb{R}^{k\times d}\), which can project entities from entity space to relation space. Based on this idea, we extend the time dimension on the basis of relational space to form a time-space that is perpendicular to relational space. Define the Vertical projection matrix \(\mathbf{V}_{t}\ \in\ \mathbb{R}^{d\times d}\), \(V_{t}\) can map the entity set and relation set to the time space, and reconstruct the triples under the limitation of the time dimension, so that the relation predicate contains It can be limited by duration. The overall structure of TBRm is shown in Figure 4.  
Figure 3: Knowledge Graphs of requirements based on IIoT network security.  
The scoring function \(f\) is used for the embedding method of KG completion. The function works on the embedding of the subject \(\mathbf{e}_{s}\), the object \(\mathbf{e}_{o}\), and the predicate \(\mathbf{e}_{p}\) of the triple in the time dimension, in order to represent the duration (time span) of a relational predicate, let \(\mathbf{t}=t_{e}-t_{b}\), it is worth noting that when the time represents the relational predicate in the form of timestamp, only \(t_{b}\) is taken as the current time vector, that is, when \(t_{e}=t_{b}\), \(\mathbf{t}=\mathbf{t}_{b}\). The value of a scoring function is proportional to the probability that a triple is true; a classic example of a scoring function is:  
1. TransE [10]: \[f_{t}(s,p,o)=\left\|\mathbf{e}_{s}+\mathbf{e}_{p}-\mathbf{e}_{0}\right\|_{2}\] (1) where \(\mathbf{e}_{s}\), \(\mathbf{e}_{o}\in\mathbb{R}^{d}\) are the embeddings of subject and object entities, and \(\mathbf{e}_{p}\in\mathbb{R}^{d}\) are the embeddings of relational predicates. \(\|\cdot\|_{2}\) represents the two-norm.
2. TransR [14]: \[f_{p}(s,o)=\left\|\mathbf{s}_{p}+\mathbf{p}-\mathbf{o}_{p}\right\|_{2}^{2}\] (2) where \(\mathbf{s}_{p}=\mathbf{s}\mathbf{M}_{p}\), \(\mathbf{o}_{p}=\mathbf{o}\mathbf{M}_{p}\), and at the same time, the embedding and mapping matrices must be constrained, that is, \(\left\|\mathbf{s}\right\|_{2}\leq 1\), \(\left\|\mathbf{o}\right\|_{2}\leq 1\), \(\left\|\mathbf{p}\right\|_{2}\leq 1\), \(\left\|\mathbf{s}_{p}\right\|_{2}\leq 1\), \(\left\|\mathbf{o}_{p}\right\|_{2}\leq 1\). These scoring functions do not consider temporal information.  
To introduce the temporal information of relational predicates, TBRm uses the projection matrix \(V_{t}\) to define the projection of relational predicates as:  
\[\mathbf{p}_{t}^{s}=\mathbf{p}\mathbf{V}_{t}\mathbf{t}\,,\;\mathbf{p}_{t}^{o}= \mathbf{p}\mathbf{V}_{t}\mathbf{t} \tag{3}\]  
where \(\mathbf{p}_{t}^{s}\) represents the projection of the entity (subject) set at the beginning of the relation predicate on the time dimension, and \(\mathbf{p}_{t}^{o}\) represents the projection of the entity (object) set at the end of the relation predicate on the time dimension.  
The scoring function is correspondingly defined as:  
\[f_{t}(s,p,o)=\|\mathbf{s}_{p}^{t}+\mathbf{p}_{t}^{o}-\mathbf{p}_{t}^{s}- \mathbf{o}_{p}^{t}\|_{2}^{2} \tag{4}\]  
where \(\mathbf{s}_{p}^{t}=\mathbf{s}_{p}\mathbf{V}_{t}\,,\;\mathbf{o}_{p}^{t}= \mathbf{o}_{p}\mathbf{V}_{t}\). At the same time, the constraints of the projection matrix are satisfied, that is, \(\|\mathbf{s}_{p}^{t}\|_{2}\leq 1\,,\;\|\mathbf{o}_{p}^{t}\|_{2}\leq 1\).  
Figure 4: The overall structure of the TBRm model.