# Measuring the Inferential Values of Relations in Knowledge Graphs
## 4 The Proposed Approach
### Some Preliminaries of RIMIE  
**Definition 1**.: _Information Table. In a knowledge base, an information table is a tuple \((U,A)\), where \(U=\{x_{1},x_{2},\ldots,x_{n}\}\) represents a non-empty finite sample set of inference chains based on knowledge information, and \(A=\{a_{1},a_{2},\ldots,a_{n}\}\) is a non-empty finite attribute set on inference chains._  
Figure 2: High-inference value relations. In a burglary case, there are clues pointing to two suspects (\(A\) and \(B\)), in which the surveillance camera caught \(A\) entering the store one hour before the burglary, and \(B\) entering the store the day before. Meanwhile, the neighbor saw \(A\) entering the store, and frequently appeared around the store in the previous days, and found \(A\)’s fingerprint in the store; here, surveillance, witness and fingerprint are all important relations with high inferential value. Changing the relation value of “2024-6-22 13:14:00” will reduce the suspicion of \(A\) and may lead to different results.  
There exists a mapping from the sample space to the attribute value space, denoted as \(f_{a}:U\to V_{a}\). For each sample \(x\in U\), there exists a value \(f_{a}(x)\in V_{a}\), where \(V_{a}=\{f_{a}(x)\mid x\in U\}\) is called the codomain of \(a\).  
**Remark 1**.: _According to Definition 1, in a knowledge graph \(G\), an inference chain can be viewed as a subgraph or chain composed of a series of fact triplets or 4-tuples, for example, (head, relation: value, tail) or (head, relation, tail, value). The relations can be regarded as the attributes of an inference chain._  
According to Definition 1 and Remark 1, in Figure 3, there are three inference chains, \(x_{1}\), \(x_{2}\) and \(x_{3}\), the shaded part, where \"conflict\" is a relation in the inference chains.  
**Definition 2**.: _Equivalence Class. The sample set \(U\) can be divided by an attribute \(a\) into some disjoint subsets, which are called a group of equivalence classes. Every equivalence class on \(a\) is expressed by a set as the following_  
\[[x]_{a}=\{y|f_{a}(y)=f_{a}(x),y\in U\}, \tag{1}\]  
where \([x]_{a}\) is a set of samples, the elements in which have equal values on attribute \(a\). \([x]_{a}\) is also called the equivalence class obtained by the partitioning with attribute \(a\). Different equivalence classes are disjoint, and the union of all equivalence classes forms a partition of the sample set \(U\).  
**Definition 3**.: _Knowledge Structure ([PERSON] and [PERSON], 2015). A partition of the sample set \(U\) on attribute \(a\), which is composed of the equivalence classes such as \([x]_{a}\), is called a knowledge structure \(\chi\) on attribute \(a\), defined as the following_  
\[\chi(a)=\frac{U}{a}=\{[x]_{a}|x\in U,a\in A\}. \tag{2}\]  
For example in Figure 3, a knowledge structure on the relation attribute \"conflict\" generated by partitioning the sample set \(U\) can be expressed as  
\[\chi(r)=\frac{U}{r}=\{(x_{1},x_{2}),(x_{3})\}, \tag{3}\]  
where \(r\) represents the relation \"conflict\", and \((x_{1},x_{2}),(x_{3})\) are two equivalence classes.  
Figure 3: The inference chain in a knowledge graph. In a case, according to the clues, the police locks up three suspects \(A\), \(B\) and \(C\); there is a conflict between them and the victim \(S\); the severity of the conflict between \(A\) and \(B\) and \(S\) is 1, and the severity of the conflict between \(C\) and \(S\) is 2; there are three reasoning chains in the figure, that is, the part covered by the shadows.