## 2 Relate Work
### GCN and Graph SAGE  
The GCN is a network that represents data in the form of graph data structures for learning. It can simultaneously extract the topological structure and vertex attributes of data. Then, it will continue to learn the information of target vertices in the global space. The GCN belongs to the field of transductive learning.  
The graph is usually represented by \(G(V,E)\), where \(V\) represents the node set and \(E\) represents the edge set. Under this definition, nodes are usually represented by pixels in HSIs, and adjacent nodes have a connection relationship, which is represented in Equation (1):  
\[A_{ij}=\begin{cases}1&if\ \big{\{}u_{i},u_{j}\big{\}}\in E\ and\ i\
eq j\\ 0&otherwise\end{cases} \tag{1}\]  
when there is an edge relationship between node \(u_{i}\) and \(u_{j}\), the value between the two nodes is 1, otherwise it is 0. A key aspect of graph processing is the appropriate construction of the Laplacian matrix [39]. After determining \(A\), we can calculate the Laplace matrix \(L\) of the graph as Equation (2):  
\[L_{ij}=\begin{cases}d(u_{i})&if\ i=j\\ -1&if\big{\{}u_{i},u_{j}\big{\}}\in E\ and\ i\
eq j\\ 0&otherwise\end{cases} \tag{2}\]  
where \(d(u_{i})\) is the number of edges associated with the node [40]. Based on the above definition, we can derive the connection between the data.  
Graph SAGE [41], on the other hand, belongs to a variant form of the GCN. It extends the GCN to trainable aggregation functions and uses the attributes of vertices to efficiently generate embedding of unknown vertices. Thus, unlike the GCN, SAGE belongs to inductive learning. SAGE makes up for the disadvantage of the GCN that it cannot be generalized directly. In a recent study, [PERSON] et al. investigated a center-weighted convolution and GraphSAGE (CW-SAGE) for HSI classification, exploiting the ability of SAGE to capture contextual relationships [42]. The SAGE Algorithm 1 is as follows:  
The graph GCA first randomly samples the neighboring nodes of each node in the graph to reduce the computational complexity. Then, it generates the target node embedding based on the information embedded in the aggregated neighboring nodes. During the propagation process, the information of the node is extended to the \(k_{th}\) order neighbors after \(k\) times of aggregation. The embedding, as an input to the fully connected layer, will be used to predict the target label of the node.  
GraphSAGE is a spatial domain-based graph convolution whose features include both neighbor sampling and feature aggregation. The neighbor sampling setting enables the GCN to attempt large-scale data processing. Similarly, SAGE has some issues that need to be addressed. The randomness in SAGE sampling leads to feature instability in its inference process. Meanwhile, the sampling process may lead to the loss of important information on the nodes. SAGE also relies on the construction of graph data structures.  
To address the problems exhibited by SAGE, we introduce the features of neighboring pixels as a reference during sampling. The input from the original image is shifted to the input of the features. The local convolution in the previous stage effectively compensates for SAGE's lack of information. In order to ensure the similarity relationship between nodes, we built an edge between the nodes. By iterating, the connection relationship of the model will be optimized and strengthened.