## 2 Research Methods for Reservoir Fluid PVT High-Pressure Physical Property Analysis  
### Graph Convolutional Network and Pearson Correlation Analysis  
#### 2.1.1 Graph Theory  
Graph theory is a branch of mathematics that studies graphs, which consist of a set of nodes (vertices) and edges and can be used to represent relationships in various practical problems. In graph theory, common graphs include directed graphs and undirected graphs. An undirected graph is a type of graph where the edges have no direction, meaning that the edges connecting vertices are bidirectional. When constructing an undirected graph to represent the relationships between high-pressure physical properties, each high-pressure physical property can be regarded as a vertex in the graph, and the relationships (such as correlation, similarity, etc.) between high-pressure physical properties can be regarded as edges. Directed graphs and undirected graphs are distinguished by whether the edges have a direction. An undirected graph \(G=\{V,E\}\) is defined, where the vertex set of the graph is \(V=\{v_{1},v_{2},v_{3},\ldots,v_{n}\}\) to represent the vertices of the graph, and the edge set of the graph is \(E=\{e_{1},e_{2},e_{3},\ldots,e_{m}\}\)[23; 24]. In this paper, an undirected graph \(G=\{V,E,A\}\) was constructed, indicating that there are a total of n vertices, and vi is used to represent the \(i\)-\(th\) vertex. A represents the adjacency matrix of graph \(G\), and \(A_{j}^{i}\) represents the relationship between vertex \(v_{1}\) and \(v_{j}\). For an undirected graph, it is obvious that \(A_{j}^{i}=\ A_{j}^{i}\), and the adjacency matrix A is a symmetric matrix, and its value is defined as shown in Equation (1). \[A_{j}^{i}=\left\{ \begin{array}{ll}1,&v_{i}\ is\ connected\ to\ v_{j}\\ 0,&otherwise\end{array} \right. \tag{1}\]  
The Laplacian matrix L of the graph is defined as shown in Equation (2) [25]. \[\rm{L}=D-A \tag{2}\]  
In the above equation, A represents the adjacency matrix of \(G\), and \(D\) represents the degree matrix of \(G\). Therefore, an undirected graph can be described by the adjacency matrix, degree matrix, and Laplacian matrix, which can be represented graphically as shown in Figure 1.  
Figure 1: Representation of the undirected graph G.  
The degree matrix of the undirected graph G can be obtained from Figure 1, as shown in Equation (3). \[ \begin{bmatrix}3&0&0&0&0\\ 0&2&0&0&0\\ 0&0&2&0&0\\ 0&0&0&4&0\\ 0&0&0&0&1\end{bmatrix}  \tag{3}\]  
The adjacency matrix representation of the undirected graph G is shown in Equation (4). \[ \begin{bmatrix}0&1&1&1&0\\ 1&0&0&1&0\\ 1&0&0&1&0\\ 1&1&1&0&1\\ 0&0&0&1&0\end{bmatrix}  \tag{4}\]  
The Laplacian matrix representation of the undirected graph G is shown in Equation (5). \[ \begin{bmatrix}3&-1&-1&-1&0\\ -1&2&0&-1&0\\ -1&0&2&-1&0\\ -1&-1&-1&4&-1\\ 0&0&0&-1&1\end{bmatrix}  \tag{5}\]  
The following are the steps to establish an undirected graph by analyzing the relationships between the high-pressure physical properties of reservoir fluid PVT [26]:  
Step 1: Determine the vertices: Regard each high-pressure physical property as a vertex in the graph, and different high-pressure physical properties correspond to different vertices. Step 2: Determine the edges: Determine the connection relationship of the edges according to the relationships between the high-pressure physical properties. For example, if there is a certain correlation between two high-pressure physical properties, connect an edge between them. Step 3: Draw the graph: According to the determined vertices and edges, draw each vertex and the edges between them in the graph to form an intuitive undirected graph representing the relationships between the high-pressure physical properties. Step 4: Analyze the graph: By analyzing the constructed undirected graph, the relationships between the high-pressure physical properties can be studied, and the highly correlated high-pressure physical properties or the potential patterns and structures between the high-pressure physical properties can be found.