## 3 Cluster Formulation  
In this section, we discuss the downlink transmission of the proposed architecture and design a SC formation method.  
In the downlink network considered in this paper, each GU is connected to a set of nearby satellites. The backhaul link connects the LEO satellites to the GEO satellites, which serve as a CPU to pass the necessary control signals between them. A specified minimum elevation angle determines the GU's candidate service satellite. To ensure a flexible and scalable configuration, each GU is assigned a Master Satellite AP (MSAP) based on the given criteria. In the following, we propose an algorithm for SC formation:* According to the given minimum elevation angle \(\theta_{\text{min}}\), the candidate SCs are selected for each GU. The \(l\)th satellite is added to the \(k\)th GU's set of candidate satellites \(C_{k}\) when the elevation angle of the \(l\)th satellite relative to the \(k\)th GU satisfies \(\theta_{k,l}>\theta_{\text{min}}\).
* For the \(k\)th GU, sequentially traverse the satellites in the set \(C_{k}\), and select the satellite corresponding to the maximum value of \(\alpha_{k,l}\) as the candidate MSAP. Determine whether the satellite is idle or not. If it is idle, then it is selected as the MSAP for the \(k\)th GU and denoted as MSAP\({}_{k}\). If it is not idle, traverse \(\alpha_{k,l}\) in order of value until MSAP\({}_{k}\) is selected among the idle satellites. The K MSAPs of the K GUs are selected sequentially by this step, and the MSAPs are removed from the sets of candidate satellites and added to the sets of serving clusters.
* Choose other accessible satellites to serve the GUs together with MSAPs. In order to ensure rational allocation of resources, the GUs are traversed to select the service satellites sequentially. Each round of traversal sequentially adds a service satellite to the candidate set of K GUs to the service cluster of the GU. The criteria for adding is that the large-scale fading coefficient between the satellite and the GU is maximum and the satellite is idle. The selected satellite is removed from sets of candidate satellites after successful addition until all the satellites in the candidate cluster are selected for all GUs.  
The SC formation strategy is summarized as Algorithm 1.  
As shown in Figure 3, the candidate set \(C_{k}\) for the \(k\)th GU is dependent on the elevation angle. Considering the large-scale fading between the \(k\)th GU and the satellites, an optimal satellite is first selected as the MSAP\({}_{k}\) for the \(k\)th GU among the available satellites, and then the others are selected sequentially.  
For simulation efficiency, our design incorporates a relatively small number of satellites and GUs. When a satellite is tasked with serving a GU, we configure all antennas on that satellite to transmit data exclusively for that GU. In contrast, systems employing a hybrid precoding scheme based on a partially connected architecture typically employ a subarray of antennas connected to the same RF chain to serve a single GU. The number of GUs served by a satellite simultaneously should not exceed the number of RF chains on the satellite. It is worth noting that our proposed service clustering algorithm can be readily extended to accommodate this configuration by merely adjusting the criteria for determining the idle status of a satellite. For instance, when assessing whether the \(l\)th satellite is idle for the \(k\)th GU, it is necessary to examine the state of each RF chain on the satellite. Specifically, the matrix \(\mathbf{A}_{i,l}\) is treated as a diagonal matrix that comprises \(N_{RF}\) block matrices. By traversing the diagonal elements of each block matrix, we can determine the availability of the RF chains. If there exists a block matrix whose diagonal elements are all zero for \(\forall i\in\{1,\dots,K\}\), this indicates that the RF chain on the \(l\)th satellite is idle and capable of serving the \(k\)th GU.  
Figure 3: SC formation for the \(k\)th GU.