## 3 Materials and Methods
### The Definition of Relgraph  
**Meta-nodes. ** In this paper, we introduce the concept of _meta-nodes_ to represent the relationship between entities and predicates in KGs. These meta-nodes are divided into two categories: entity meta-nodes and predicate meta-nodes. Entity meta-nodes, denoted as \(emn_{i,a}\), represent the component of entity \(e_{i}\) corresponding to predicate \(r_{a}\). Entity meta-nodes express the influence of entities on predicates. The set \(\mathcal{EMN}\) represents all such meta-nodes in the KG, formally defined as \(\{emn_{i,a}|e_{i}\in\mathcal{E},r_{a}\in\mathcal{R}\}\), and belongs to the cross product \(\mathcal{E}\times\mathcal{R}\). Similarly, \(\mathcal{EMN}_{a}\) represents all entity meta-nodes associated with a specific predicate \(r_{a}\). For the relation \(r_{a}\in\mathcal{R}\), we define the graph \(\mathcal{G}_{a}=\langle\mathcal{V}_{a},r_{a},\mathcal{D}_{a}\rangle\), where the set of edges \(\mathcal{D}_{a}\) contains triples \((h,r_{a},t)\), \(h,t\in\mathcal{V}_{a}\) represent source and target vertices, respectively, and \(r_{a}\) is the unique relation of the graph \(\mathcal{G}_{a}\). For each fact \((e_{m},r_{a},e_{n})\in\mathcal{K}\), let \(h,t\) represent the entity meta-nodes \(emn_{m,a},emn_{n,a}\in\mathcal{EMN}_{a}\), respectively. Then, the triple \((h,r_{a},t)\) corresponds to the fact \((e_{m},r_{a},e_{n})\) in the knowledge graph. For a triple \((emn_{m,a},r_{a},emn_{n,a})\in D_{a}\) associated with entities \(e_{m},e_{n}\in\mathcal{E}\) related to the predicate \(r_{a}\in\mathcal{R}\), we define the logical function:  
\[TF(emn_{m,a},r_{a},emn_{n,a})=\left\{ \begin{array}{ll}True&(e_{m},r_{a},e_{ n})\in\mathcal{K},\\ False&(e_{m},r_{a},e_{n})\
otin\mathcal{K}.\end{array} \right. \tag{1}\]  
Equation (1) is referred as the truth function of graph \(\mathcal{G}_{a}\). Dually, for \(e_{i}\in\mathcal{E}\) and \(r_{a}\in\mathcal{R}\), we set \(pmn_{i,a}\) as the predicate meta-node representing the influence of predicate \(r_{a}\) on entity \(e_{i}\). The set \(\mathcal{PMN}=\{pmn_{i,a}|e_{i}\in\mathcal{E},r_{a}\in\mathcal{R}\}\subseteq \mathcal{R}\times\mathcal{E}\) represents all predicate meta-nodes in the knowledge graph. Similarly, \(\mathcal{PMN}_{i}=\{pmn_{i,a}|r_{a}\in\mathcal{R}\}\) represents all predicate meta-nodes associated with entity \(e_{i}\). **Entity Links. ** Let \(e_{i}\in\mathcal{E}\) be an entity. We define a graph \(\mathcal{RG}_{i}=\langle\mathcal{RV}_{i},e_{i}|,\mathcal{RD}_{i}\rangle\) such that each edge \(rd\in\mathcal{RD}_{i}\) is a triple \((h,el_{i},t)\) where \(h,t\in\mathcal{RV}_{i}\) are source and destination vertices. The only relation in \(\mathcal{RG}_{i}\) is \(el_{i}\). For predicates \(r_{f},r_{g}\in\mathcal{R}\), let \(h,t\) be predicate meta-nodes \(pmn_{i,f},pmn_{i,g}\in\mathcal{PMN}_{i}\), define \(el_{i}\) as an _entity link_. The graph \(\mathcal{RG}_{i}\) is called a relation graph_ about entity link \(el_{i}\). To distinguish it from entity graphs, we refer to \(\mathcal{G}_{n}\) as an _entity graph_. It follows from the definition that for each entity \(e_{i}\in\mathcal{E}\), there exists a corresponding collection of predicate meta-nodes \(\mathcal{PMN}_{i}\) and an entity link \(el_{i}\). The collection of all entity links is denoted by \(\mathcal{EL}\), and \(|\mathcal{EL}|=|\mathcal{E}|\). To reflect the influence of entities on predicate pairs, we extend the truth function Equation (1) to relation graphs. The truth function for relation graph \(\mathcal{RG}_{i}\) is defined as follows:  
\[TF(pmn_{i,f},el_{i},pmn_{i,g})=\left\{ \begin{array}{ll}\textit{True}&\exists e _{x},e_{y}\in\mathcal{E},\\ &(e_{x},r_{f},e_{i})\in\mathcal{K}\\ &and\\ &(e_{i},r_{g},e_{y}),\in\mathcal{K},\\ &\textit{False}&else.\end{array} \right.