## 3 Results
Figure 3: The curves of the number of iterations of the PSO algorithm and the traversal algorithm with respect to the minimum sum of the weighted time in the first time slice. in Equations (4) and (5); the parameters \(v_{j}\) and \(x_{j}\) of the previous iteration are computed to cover the current \(v_{j}\) and \(x_{j}\), i.e., the parameters of the current iteration are updated using the parameters of the previous iteration, so that the results of the previous computation has a contributing effect on the next computational results. In addition, in Equation (3), PSO updates \(NP\) particles at one time simultaneously, i.e., \(NP\) kinds of combinations and the computational results of these \(NP\) particles update \(\mathbf{pbest}_{j}\) and \(\mathbf{gbest}\), and then update \(\mathbf{v}_{j}\) and \(\mathbf{x}_{j}\) of each particle according to Equations (4) and (5) so that the iteration of each particle can benefit from the iterative process of other \(NP\)-1 particles. In summary, instead of enumerating sequentially as the traversal algorithm does, the PSO algorithm significantly improves computational efficiency through these beneficial update mechanisms. Four time source combinations can be generated after the program has run for several times, as shown in Table 3.  
In order to study the time synchronization effect of the method in the case of network topology changes, we obtained another time slice of the constellation in one operation cycle by simulation, and the constructed network topology is shown in Figure 5. In the network topology, the numbers represent the satellites' numbers. The difference with the  
We also calculated the optimal clock source combinations using the PSO algorithm for the network topology shown in Figure 5, obtaining the fourth group clock source combination in Table 4. Clock delivery trees are constructed for the fourth group of clock source combinations, as shown in Figure 6. And, the sum of weighted clock levels is calculated as 103 based on the clock transfer trees in Figure 6. The value is larger than the sum of weighted clock levels 102 in the first time slice, which indicates that the clock synchronization effect is degraded. This is because the network topology has changed in this time slice, the links between satellites have been reduced, the available clock transfer paths have been reduced, and the matrix connection table \(A\) has been changed, which affects the simulation results according to Equation (7), so the simulation results become worse accordingly. The iteration times of the two algorithms are compared in Figure 7. The traversal algorithm iterates to the 94,277 th time to obtain the optimal result, but still needs to calculate all \(11^{6}\) times to determine, while the PSO algorithm only needs to iterate 43 times to obtain the optimal result. Using a time monitoring program, it takes 190.96 s for the traversal algorithm to calculate \(11^{6}\) combinations, and 0.039 s for the PSO algorithm to calculate 43 times. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Group**} & \multicolumn{3}{c}{**Optimal Clock Source**} & \multicolumn{3}{c}{**Satellite Numbers**} \\  & **Combination Serial Numbers** & & & & & & \\ \hline
1 & 94,277 & 1 & 18 & 27 & 43 & 46 & 62 \\
2 & 778,085 & 5 & 21 & 24 & 40 & 49 & 66 \\
3 & 993,477 & 7 & 13 & 32 & 38 & 51 & 56 \\
4 & 1,677,285 & 11 & 16 & 29 & 35 & 54 & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal clock source combination table. Figure 6: Time transfer trees of the fourth group in Table 4.  
Figure 7: The curves of the number of iterations of the PSO algorithm and the traversal algorithm with respect to the minimum sum of the weighted time in another time slice.