## 3 The Proposed DRL + GNN-Based Handover Scheme
### DQN Framework
The state is defined as the combination of the inherent state characteristics of the satellite handover graph with zero-padding vectors, which autonomously determines the size of the state space \(N\). These characteristics include propagation delay, remaining service time, and user data rate. The specific description is as follows:  
\[S(t)=\{\{d_{1},r_{1},t_{1},padding\},\{d_{2},r_{2},t_{2},padding\},\ldots,\{d_{k },r_{k},t_{k},padding\}\} \tag{19}\]  
The size of the state space is configured to exceed the number of features, with vectors padded with zeros, enabling each edge to retain its own information along with aggregated information from all neighboring edges. Nevertheless, it should not be excessively large to avoid the state space becoming bigger, which could lead to the GNN model becoming overly large and thus prone to overfitting [40]. The feature description is shown in Table 1:  
* Action space During the path selection for satellite handover graphs, we calculate the number of paths and the set of nodes for each path (i.e., all nodes traversed by each path) from the virtual start node, on a hop-by-hop basis. We take advantage of the generalization capability of GNNs to introduce actions to intelligent agents in the form of graphs. Successfully trained GNNs can comprehend actions on various graphical structures (i.e., different network states and topologies). \[a(t)\in\{1,2,\ldots,k\}\] (20) where k denotes the number of paths from the virtual start node. * Reward function The reward function of the system should be related to the optimization objective, based on the handover decision problem studied in this paper, the reward function is defined with respect to the quality of the user's service, thus the reward function is defined as:  
\[R(s,a)\ =\ w_{1}N(r_{i})\ -\ w_{2}N(d_{i})+\ w_{3}N(t_{i}) \tag{21}\]  
where \(N(d_{i})\) is a normalized function of the transmission delay on the candidate path; \(N(r_{i})\) is a normalized function of the user data rate on the candidate path; and \(N(t_{i})\) is a normalized function of the remaining service time on the candidate path. We used an Analytic Hierarchy Process (AHP) analysis to calculate the weights of different handover factors in the reward function with the following steps. Step 1: Constructing the judgment matrix. We employ the consistent matrix method to construct a judgment matrix to compare the handover factors with each other. Among all handover factors, transmission delay holds the highest significance, as real-time monitoring of services by the power system is crucial for timely reaction to faults. Service time ranks as the second-most critical handover factor, as it dictates the handover frequency to mitigate power data loss caused by frequent handover, subsequently impacting the system's real-time monitoring and other functionalities. User data rate stands as the third-most significant handover factor, as it influences the efficient transmission of power data information, albeit with a lesser impact on real-time services. Based on the importance to different handover factors, a judgment matrix A is constructed, where each row from top to bottom and each \begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Description** \\ \hline \(d_{k}\) & Transmission delay \\ \(r_{k}\) & Data Rate \\ \(t_{k}\) & remaining service time \\ \(x_{4}-x_{n}\) & Zero padding \\ \hline \end{tabular}
\end{table} 
Table 1: Description of characteristics. column from left to right represents a different handover factor, which are transmission delay, remaining service time, and data rate in that order. \[\text{A}= \begin{bmatrix}1&1/3&1/5\\ 3&1&1/2\\ 5&2&1\end{bmatrix}  \tag{22}\]  
Step 2: Calculating the weights of the handover factors.