# Design and Evaluation of Dynamic Topology for Mega Constellation Networks
## 3 Problem Analysis
### Time Slice Division Principle  
Due to the substantial heterogeneity of the mega constellation network, frequent link breakage and reconstruction of the topology, and the variability of the ISL distance, it is necessary to continuously improve and optimize the dynamic topology of the constellation to obtain a network with better communication performance index. Assuming a Walker constellation with 24 orbits and 66 satellites per orbit, the calculation of the nearest non-adjacent orbiting satellite at different moments requires a total of \(5742\times(24-3)\times 66\) calculations for a single satellite in the constellation. If the nearest satellite is searched in the network of the entire constellation, the number of calculations will be as high as \(5742\times(24-3)\times 66\times 1584\). Therefore, without dividing the time slice, the complexity of the algorithm will be high due to the large data magnitude, which is not suitable for use in mega constellation networks with a large number of nodes, and the irregularity of the topology will also significantly increase the difficulty of prediction for links, which is not conducive to the development of network routing algorithms. Meanwhile, the maximum connectivity of each satellite in the constellation is four. The traditional strategy of building a link based on the shortest distance has not yet taken into account the specified maximum connectivity, which will make the topology less resistant to destruction if the link is disconnected after exceeding the range [11]. The mega constellation is a network of satellites with a constantly changing topology, periodic and predictable satellite movements, and a regular constellation topology so that the time slices can be divided into equal intervals by constructing a virtual topology, thus ultimately achieving dynamic topology management [16]. The [PERSON] constellation assumed in this paper has an operation period equal to 5742 s, and 1584 satellites need to be launched and deployed in the first phase. Suppose only the dynamic links established in the first 33 time slices need to be calculated. In that case, this reduces the computational effort to a great extent and facilitates the further use of routing algorithms to filter dynamic links and generate and analyze regular topologies [17]. Based on the variation in interstellar distance, a time slice division method is constructed using the number of satellites per orbit, corresponding to Figure 1. According to the theory of satellite numbering within the constellation [18], definitions are established for the orbits and inner satellites of the [PERSON] constellation. The named satellite nodes are denoted as \(S(i,j)\), representing the satellite \(j\) within the orbital plane \(i\), where \(i\) ranges from \(1\) to \(24\), and \(j\) ranges from \(1\) to \(66\). \(T_{n}\) represents the time slice \(n\); since there are \(66\) satellites in each orbit, this implies that \(n\in[1,66]\). The core idea is summarized in the following figure. For example, consider \(S(2,x)\), representing the satellite \(x\) in the \(OP_{2}\). Assuming that in the \(T_{1}\), the satellite closest to \(S(2,x)\) is \(S(y,z)\), representing the satellite \(\mathsf{z}\) on the \(OP_{y}\). Both satellites comply with the numbering principle, i.e., \(x\in[1,66]\), \(y\in[1,24]\), and \(z\in[1,66]\). The nearest satellite to \(S(2,x-(n-1))\) in the \(T_{n}\) is \(S(y,z-(n-1))\) when \(x\ -\ (n\ -\ 1)\ \geq\ 0\) and \(z-(n-1)\geq 0\); the nearest satellite to \(S(2,x\ -\ (n\ -\ 1))\) is \(S(y,z-(n-1)+66)\) when \(x-(n-1)\ \geq\ 0\) and \(z-(n-1)<0\); the nearest satellite to \(S(2,x-(n-1)+66)\) is \(S(y,z-(n-1))\) when \(x-(n-1)<0\) and \(z\ -\ (n\ -\ 1)<0\); and the nearest satellite to \(S(2,x-(n-1)+66)\) is \(S(y,z-(n-1)+66)\) when \(x\ -\ (n\ -\ 1)<\ 0\) and \(z-(n-1)<0\).