## 2 Topological Analysis of Mega-Constellations  
In the analysis of mega-constellation topologies, commonly employed methods include optimal topology planning, integer linear programming, ant colony algorithms, and random graphs [8; 9]. However, these methods tend to be effective only for smaller constellations, and the topology of the network is not necessarily a priori-defined topology topology. lations; when addressing the topology design challenges inherent in mega-constellations, they encounter significant obstacles, such as substantial computational overhead and susceptibility to becoming trapped in local optima. In the Starlink constellation, satellites are deployed in stages, with currently operational satellites primarily concentrated at an altitude of 550 km in LEO, as detailed in Table 1[10]. This paper will conduct a topology analysis based on the satellite layer at an altitude of 550 km, which consists of a Walker constellation arrangement (1584/24/11, 550 km, 53\({}^{\circ}\)) for Starlink satellites. Each satellite in this layer shares an identical orbital inclination, orbital altitude, and the number of co-planar satellites, as illustrated in Figure 1.  
In publicly filed documents submitted by SpaceX to the U.S. Federal Communications Commission (FCC), it is indicated that Starlink satellites are equipped with four laser links, though the specific allocation strategy for these four laser links is not explicitly detailed. In numerous scholarly studies focusing on Starlink, inter-satellite link topologies predominantly utilize four fixed permanent links for connectivity, akin to a Manhattan-like network configuration. This entails each satellite being connected to the two immediately preceding and succeeding satellites along the same orbital plane, forming permanent in-plane links, and also linking with satellites in adjacent orbits that bear either the same or closely numbered positions, ensuring continuous connectivity between adjacent orbits throughout the entire orbital period. The pattern of these connections is depicted in Figure 2. The permanence of these links between satellites contributes to the reliability of data transmission most of the time. High link stability is maintained, with permanent links remaining unaltered, making them suitable for long-term communication purposes. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Phase** & **Height/km** &
\begin{tabular}{c} **Number of** \\ **Satellite** \\ \end{tabular} & **Inclination/deg** \\ \hline Phase One & 550 & 1584 & 53.0 \\ Phase One & 1110 & 1600 & 53.8 \\ Deployment scheduled for & 1130 & 400 & 70.4 \\ March 2027 & 1275 & 374 & 74.0 \\  & 1325 & 450 & 80.0 \\ \hline Phase Two & 335.9 & 2493 & 42.0 \\ Deployment scheduled for & 340.8 & 2478 & 48.0 \\ November 2027 & 345.6 & 2547 & 53.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Starlink construction plan. Figure 1: Starlink at a 550 km orbit. While the current satellite topology exhibits commendable stability and reliability, it is not without limitations. One such constraint arises when two satellites orbiting in intersecting paths find themselves in close proximity but lack a direct line of communication due to specific operational constraints. This necessitates communication detours, impacting the link flexibility and introducing latency. Moreover, the rigidity of these connections prevents adaptation to dynamically evolving communication requirements. To mitigate the challenges posed by permanent link connections, a novel \"3 + 1\" link construction strategy has been proposed. This strategy entails equipping each satellite with three fixed permanent links and one dynamic temporary link, aiming to enhance the flexibility and efficiency in satellite communication networks. In establishing three fixed permanent links, considerations must be made regarding the relatively constant distances between satellites within the same orbit.