# Temporal Continuity Expression for Network Topology of Space Information Systems
## 4 Comparative Analysis  
Taking 300 s as a snapshot length, during which the topology is stable, the network topological dynamic characteristic of the Walker Delta \(32/4/1\) constellation was described as a series through multiple static network topologies in ref. [20], which is shown in Figure 5.  
Obviously, each row vector of the adjacency matrix corresponding to the network topology of the Walker Delta \(32/4/1\) constellation in the first snapshot (\(t=[0\,\mathrm{s},300\,\mathrm{s})\)) is obtained from the figure, as shown in Table 7. \begin{table}
\begin{tabular}{c c} \hline
**Satellite Pairs** & **Expression** \\ \hline \(S_{28}\) and \(S_{11}\) & \(\delta_{S_{11}S_{34}}(t)=\left\{\begin{array}{l}0,t\in(1308\sim 3081,5208\sim 6980)\\ \frac{1+\frac{40888-\delta_{S_{11}S_{34}}(t)}{12084}}{3}+\frac{040175-\delta_{S _{11}S_{34}}(t)}{10844},t\in(0\sim 1308,3081\sim 5208,6980\sim 7798)\\ \\ S_{47}\) and \(S_{11}\) & \(\delta_{S_{11}S_{\sigma}}(t)=\left\{\begin{array}{l}0,t\in(0\sim 2257,2863\sim 6 157,6762\sim 7798)\\ \frac{1+\frac{15120-\delta_{S_{11}S_{\sigma}}(t)}{42084}}{3}+\frac{004075-\delta_ {S_{11}S_{\sigma}}(t)}{42084},t\in(2257\sim 2863,6157\sim 6762)\\ \hline\end{array}\right.\) \\ \hline \end{tabular}
\end{table} 
Table 6: The three elements of the adjacency matrix of \(S_{11}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Satellite Pairs** & \(a_{0}\) & \(a_{1}\) & \(b_{1}\) & \(a_{2}\) & \(b_{2}\) & \(a_{3}\) & \(b_{3}\) & \(a_{4}\) & \(b_{4}\) & \(\omega\) \\ \hline \(S_{28}\) and \(S_{11}\) & \(7.94\times 10^{-6}\) & \(-0.0134\) & \(0.0197\) & \(0.0054\) & \(-0.0021\) & \(-0.003\) & \(-6.49\times 10^{-4}\) & \(0.0011\) & \(0.0012\) & \(0.0016\) \\ \(S_{34}\) and \(S_{11}\) & \(-2.87\times 10^{-5}\) & \(0.0159\) & \(-0.0377\) & \(0.0054\) & \(-0.0053\) & \(0.0068\) & \(-0.0027\) & \(0.0039\) & \(5.46\times 10^{-5}\) & \(0.0016\) \\ \(S_{47}\) and \(S_{11}\) & \(2.6\times 10^{-8}\) & \(-0.0151\) & \(0.01\) & \(-7.5\times 10^{-4}\) & \(-3.1\times 10^{-4}\) & \(-7.99\times 10^{-5}\) & \(-4.1\times 10^{-4}\) & \(3.78\times 10^{-5}\) & \(-3.76\times 10^{-5}\) & \(0.0016\) \\ \hline \end{tabular}
\end{table} 
Table 4: The fourth-order fitting Fourier series coefficients. \begin{table}
\begin{tabular}{c c} \hline
**Satellite Pairs** & **Expression** \\ \hline \(S_{28}\) and \(S_{11}\) & \(\sigma_{3}(t)=\frac{0.025-\delta_{S_{11}S_{28}}(t)}{0.05}\) \\ \(S_{34}\) and \(S_{11}\) & \(\sigma_{3}(t)=\frac{0.04175-\delta_{S_{11}S_{34}}(t)}{0.0835}\) \\ \(S_{47}\) and \(S_{11}\) & \(\sigma_{3}(t)=\frac{0.01775-\delta_{S_{11}S_{2}}(t)}{0.0350}\) \\ \hline \end{tabular}
\end{table} 
Table 5: The relative angular velocity relationship of \(S_{11}\) with the three nodes. Due to space constraints, the first-row vector of the adjacency matrix in the first snapshot (\(t=[0\) s,\(300\) s)) is selected for comparison, as shown in Table 8. \begin{table}
\begin{tabular}{c c} \hline \hline  & \(S_{11}S_{12}S_{13}\cdots S_{17}S_{18}S_{21}\cdots S_{27}S_{28}S_{31}\cdots S_{38} S_{41}S_{42}\cdots S_{48}\) \\ \(S_{11}\) & \([0\ 1\ 0\cdots 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0]\) \\ \(S_{11}S_{12}S_{13}S_{14}\cdots S_{18}S_{21}S_{22}\cdots S_{41}S_{24}S_{43} \cdots S_{48}\) \\ \(S_{12}\) & \([1\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0]\) \\ \(\vdots\) & \\ \(S_{11}\cdots S_{17}S_{18}S_{21}\cdots S_{28}S_{31}S_{32}\cdots S_{38}S_{41}S_{42 }\cdots S_{46}S_{47}S_{48}\) \\ \(S_{48}\) & \([0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0\ \cdots\ 0\ 1\ 0]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The row vectors of the adjacency matrix obtained from the figure. Figure 5: The network topology of the Walker Delta \(32/4/1\) constellation in ref. [20]. Reproduced with permission from [PERSON], et al. [Electronics]; published by [MDPI], [2023]. From Table 8, the comparison of the two results shows the following:  
1. Both results of \(\delta_{S_{\rm Si_{1}}S_{\rm 2}}(t)\) and \(\delta_{S_{\rm 11}S_{\rm 58}}(t)\) are 1.