## 2 Time Synchronization Scheme
### Sum of Weighted Clock Hierarchy Calculation  
The weight value of the satellite is the same as the time stratum of itself. For example, the weight value of the time source satellite of the 0 clock layer is 0. The weight values of all satellites are summed to calculate the weighted time level, and the minimum value of the sum of the weighted clock stratum is considered as the optimization target. Using the bionic optimization algorithm, time sources are initialized and randomly selected with the number of \(D\). The time layer of each satellite is estimated according to the above measurement, and then the sum of the weighted clock levels is obtained. The \(D\) time sources are consistently selected iteratively until the minimum sum of the weighted time cascades is obtained through optimization. At this time, when the \(D\) time sources are the optimal solution, the clock stratum and the parent node of each satellite are recorded. To complete the setting of time hierarchy of all satellites in the LEO satellite networks, the synchronization information is broadcasted to other satellites.  
In this paper, the particle swarm optimization (PSO) algorithm is used [24; 25; 26; 27; 28], which is one kind of the bionic algorithms. The low-orbit constellation takes the Iridium constellation as an example, which has 66 low-orbit satellites divided into six orbital planes. For the Iridium constellation, six satellites are selected as the time references, as every plane has one time source. The traditional traversal approach for obtaining the six time references with the smallest weighted clock cascade needs to calculate all combinations of time references, i.e., \(11^{6}\), in order to determine the optimal combination of time references. In contrast, the proposed means, with the number of particles set to 20 in the PSO algorithm, only requires a minimum of 10 iterations to determine the optimal time source combination.  
\[s_{k}=\sum_{i=1}^{N}a_{i,k} \tag{1}\]  
In Equation (1), \(s_{k}\) represents the sum of the weighted time stratum of the \(k\)-th clock source satellite combination in the time slice, \(N\) represents the number of all the satellites in the constellation; in this article, it is 66. \(a_{i,k}\) represents the time hierarchy of the \(i\)-th satellite in the \(k\)-th clock source combination in the time slice.  
In order to obtain a better global time synchronization effect, the sum of the clock weighted layer should be as small as possible, so as to minimize the time level of each satellite. There are 11 selections of clock sources for each orbital plane with 11 satellites, and a total of \(11^{6}\) combinations for six orbital planes, namely 1,771,561.  
In order to make a comparison with the method proposed in this paper, the time layers calculations have also been performed with the traditional traversal method [29; 30]. The traditional traversal measure mainly differs from the PSO algorithm in the time source selection, while the process of calculating the clock transfer trees and the sum of the satellite weighted time cascade is the same as described earlier. Traditional traversal methods calculate the weighted clock stratum for each combination in each time slice. That is, for the first time slice, \(s_{1}\)-\(s_{1771561}\) are calculated. Firstly, six time sources in six orbital planes are selected, such as the first combination of the satellites with the numbers of 1, 12, 23, 34, 45, 56. Secondly, the connection relationship between satellites is obtained based on the satellite network topology of the time slice, and the clock transfer trees are obtained based on this connection relationship and the time source satellite number. Finally, \(s_{k}\) is calculated according to the clock transfer trees.