## 2 Topological Analysis of Mega-Constellations
For intra-orbit link connections, the Manhattan configuration is retained, wherein each satellite maintains connections with its two adjacent counterparts. However, for inter-orbit link construction, ensuring a fully connected network across the entire constellation necessitates distinguishing satellites within orbits. The procedure involves sorting the 24 orbits into odd and even numbers, and further sorting the 66 satellites within each orbit likewise into odd and even categories. Each satellite is uniquely identified by a five-digit number, with the second and third digits representing the orbit number, and the final two digits representing its position within the orbit. For instance, satellite 10554 denotes the 54 th satellite in the fifth orbit. In selecting permanent links between orbits, priority is given to connections with relatively small relative distances, angular velocities, and angular accelerations to optimize inter-orbit fixed links. We defined a set of geometric structures of orbital and inter-satellite links, as shown in Figure 3. Specifically, consider any two satellites \(i\) and \(j\) within the constellation. Their position vectors in the Earth-centered J2000 inertial coordinate system are denoted as \(\overset{\rightharpoonup}{S_{i}(t)}=\big{(}x_{i}(t)\quad y_{i}(t)\quad z_{i}(t) \big{)}\) and \(\overset{\rightharpoonup}{S_{j}(t)}=\big{(}x_{j}(t)\quad y_{j}(t)\quad z_{j}(t) \big{)}\), respectively, with corresponding velocity vectors \(\overset{\rightharpoonup}{V_{i}(t)}=\big{(}u_{i}(t)\quad v_{i}(t)\quad w_{i}(t) \big{)}\) and \(\overset{\rightharpoonup}{V_{j}(t)}=\big{(}u_{j}(t)\quad v_{j}(t)\quad w_{j}(t) \big{)}\). The link velocity vector to inter-satellite link pointing from satellite \(i\) to satellite \(j\) is represented as \(\overset{\rightharpoonup}{V_{ij\_link}(t)}\), where  
\[\overset{\rightharpoonup}{V_{ij\_link}(t)}=\overset{\rightharpoonup}{S_{j}( t)}-\overset{\rightharpoonup}{S_{i}(t)} \tag{1}\]  
Figure 2: Traditional four fixed link connections. Let the plane perpendicular to the \(\overset{\rightharpoons}{S_{i}(t)}\) orbital plane and passing through the \(\overset{\rightharpoons}{S_{i}(t)}\) be denoted as \(P\). Then, the projection of \(V_{ij\_link}(t)\) onto plane \(P\) is given by  
\[\overset{\rightharpoons}{V_{P}(t)}=\overset{\rightharpoons}{S_{j}(t)}-\frac{( \overset{\rightharpoons}{S_{i}(t)}\cdot\overset{\rightharpoons}{S_{j}(t)}) \overset{\rightharpoons}{S_{i}(t)}}{\left|\left|\overset{\rightharpoons}{S_{j}(t) }\right|\right|^{2}} \tag{2}\]  
The following Equations (3)-(5) represent the calculation formulas for the distance, relative azimuth angle, and elevation angle of satellite \(j\) with respect to satellite \(i\). The azimuth rate and elevation rate are obtained by taking the first derivative of Equations (4) and (5) with respect to time \(t\), while the angular acceleration is obtained by taking the second derivative. \[dist_{ij}(t)=\sqrt{\left[x_{i}(t)-x_{j}(t)\right]^{2}+\left[y_{i}(t)-y_{j}(t) \right]^{2}+\left[z_{i}(t)-z_{j}(t)\right]^{2}} \tag{3}\]  
\[\beta(t)=\cos^{-1}\frac{\overset{\rightharpoons}{V_{P}(t)}\cdot\overset{ \rightharpoons}{V_{i}(t)}}{\left|\left|\overset{\rightharpoons}{V_{P}(t)} \right|\right|\left|\cdot\left|\left|\overset{\rightharpoons}{V_{i}(t)} \right|\right|} \tag{4}\]  
\[\gamma(t)=\cos^{-1}\frac{\overset{\rightharpoons}{V_{P}(t)}\cdot\overset{ \rightharpoons}{V_{ij\_link}(t)}}{\left|\left|\overset{\rightharpoons}{V_{P}( t)}\right|\left|\cdot\left|\overset{\rightharpoons}{V_{ij\_link}(t)} \right|\right|} \tag{5}\]  
Each satellite within the [PERSON] constellation exhibits strong symmetry and similarity, allowing us to select a satellite at random as a focal point for subsequent analysis. In this study, satellite 10554 was chosen at random to serve as the primary subject of investigation within the constellation. In our investigation, satellite 10554 serves as the focal point for experimentation.