## 4 Algorithm Design
### Fast Rerouting Mechanism
From the graph, it can be observed that the convergence characteristic curve is not strictly monotonic, indicating that the simulated annealing algorithm possesses the ability to escape from local optima and converge towards global optimum solutions. ### The Impact of Networking Methods on Algorithm Performance

Network topology methods refer to the schemes by which network nodes form a topology. Common satellite network topology methods include Sparse, +Grid, and Ideal. Sparse indicates a sparse topology where each satellite can maintain only two connections. Specifically, a satellite connects to the satellites immediately before and after it in the same orbit. +Grid represents a grid topology where each satellite can maintain four connections. In addition to the two connections mentioned in Sparse, a satellite can also connect to the nearest satellites in the adjacent left and right orbits. In an ideal topology, each satellite can connect to any other satellite within its visible range, and there is no limit on the number of connections. Figure 5: Comparison of (**a**) path stability and (**b**) latency of different algorithms. Figure 6: The convergence process of multi-objective simulated annealing. Due to the sparsity of the topology in the Sparse case, which may hinder the establishment of a continuous neighbor set, the MOLM algorithm is tested in both +Grid and Ideal scenarios. The results are averaged over 1000 steps and compared with _Shortest Path_ and \(Optimum\). The comparison results for path stability and path latency are shown in Figure 7. In terms of average path duration, MOLM performs similarly in both +Grid and Ideal scenarios and tends to approach the Optimum. In the +Grid scenario, the average path duration obtained by the MOLM algorithm is improved by 104.29 s, approximately 82.36%, compared to the _Shortest Path_. In the Ideal scenario, the average path duration achieved by MOLM is improved by 130.52 s, approximately 162.24%, compared to the _Shortest Path_. Regarding path latency, in the +Grid scenario, MOLM's path latency increases by about 7.66 ms, approximately 10.81%, compared to the _Shortest Path_. In the Ideal scenario, MOLM's path latency increases by about 7.22 ms, approximately 20.36%, compared to the _Shortest Path_. In the Ideal scenario, the path latency for all three strategies is significantly lower than the path latency in the +Grid scenario. This is because the _Shortest Path_ strategy prioritizes the shortest path, and in the Ideal scenario, where the topology has high connectivity and numerous selectable neighbors, it is easier to find paths with lower latency. However, the trade-off is more frequent path switches and worse path stability. Therefore, in the Ideal scenario, the MOLM algorithm exhibits a particularly noticeable improvement in path stability. Through the MOLM algorithm, nodes that can replace congested nodes and maintain longer connections with predecessors and successors are added to the continuous neighbor set. The multi-objective simulated annealing algorithm selects the optimal node from this set to replace the congested node, reducing path switches and significantly improving path stability. As the multi-objective simulated annealing algorithm considers both path latency and path stability, it enhances path stability while maintaining a relatively low latency cost. ### The Impact of Constellation Density on Algorithm Performance

In general, a higher constellation density provides more options for selecting the next-hop neighbor when establishing paths using the shortest path algorithm. This increases the likelihood of finding low-latency paths. However, due to the increased number of choices, path stability may decrease. Figure 8 shows the constellation topologies for \(10\times 10,30\times 30\), and \(50\times 50\), and the +Grid scheme is used for network formation. Figure 7: The impact of networking methods on algorithm performance.