## 4 Reliability Assurance Topology Control Strategy  
In the previous section, this paper proposed some possible methods (e.g., increasing the number of redundant paths) to make the network more reliable and improve the link cost metric's reliability. In this section, this paper tries to construct a topology control strategy for a satellite cluster with EO missions by using the results discussed above. Considering that a satellite's approximate position can be predicted with an orbit dynamic model, this paper uses the centralized topology control strategy in general. This means one satellite in the cluster periodically calculates all of the possible network links and then distributes the results to others. The process of the topology control strategy in this paper includes two main parts: topology matrix calculation and its distribution. Topology matrix calculation means one of the DRS analyzes the network links' connections and obtains the reliability-optimized topology. Then, the procedure of distribution is that the satellite with the links' relations sends it to the others. After that, in one topology updating period, the network changes the link connections according to the topology matrix. The topology control strategy with reliability assurance in the satellite cluster can be regarded as the following three steps:  
1. Topology Matrix Calculation  
In this step, we present a centralized algorithm for one satellite cluster with EO missions. One of the DRS will calculate the links for all. This step mainly includes three sub-steps: (1) calculating the paths for sending image data from any of the EOS \(s_{i}\) to the GSC \(\left(g_{1},g_{2},\cdots,g_{m}\right)\) to obtain \(p_{i}=\left\{s_{i}\rightarrow\left(g_{1},g_{2},\cdots,g_{m}\right)\right\}\), including the abundant paths; (2) repeating Sub-step (1) to obtain sub-topology \(G_{sg}=\left\{p_{1},\cdots,p_{m}\right\}\) for all EOS to GSC; and (3) connecting the isolated satellites to the nearest and connected satellite to obtain the final topology \(G_{t}\). The centralized topology control algorithm is described in Algorithm 1, where \(s\), \(g\) and \(r\) represent the position matrix of EOS, GSC, and DRS at time \(t\), respectively. ```
0:\(s\), \(g\), \(r\) and \([\chi_{ij}]\)
0:\(G_{t}=(V_{t},E_{t})\)
1:% Sub-step (1,2)
2:\(V_{t}=(s,g,r)\), \(E_{t}=\emptyset\), \(n=length(s)\), \(m=length(g)\)
3:for\(i=1\) to \(n\)do
4:for\(j=1\) to \(m\)do
5:\(p_{s_{i}g_{j}}=ShortestPath(s_{i},g_{j})\) in \(s_{i},g_{j}\), \(r\)
6:endfor
7:\(p_{s_{i}g}=\{p_{s_{1}g_{1}},\cdots,p_{s_{1}g_{m}}\}\)
8: Sort \(p_{s_{i}g}\) into non-decreasing order by the path cost with \([\chi_{ij}]\)
9: Let \(r_{1},\cdots,r_{m}\) be the sets of DRS on the paths \(p_{1},\cdots,p_{m}\), respectively
10:for\(k=2\) to \(m\)do
11:if\(r_{k}\cap(r_{1}\cup\cdots\cup r_{k-1})\
eq\emptyset\)then
12:\(p^{{}^{\prime}}_{k}=ShortestPath(s_{i},g_{k})\) in \(s_{i},g_{k},r/(r_{1}\cup\cdots\cup r_{k-1})\)
13:if\(p^{{}^{\prime}}_{k}=\emptyset\)then
14:\(p_{k}=p_{k}\)
15:endif
16:else
17:\(p^{{}^{\prime}}_{k}=p_{k}\)
18:endif
19:endfor
20:\(p^{{}^{\prime}}_{s_{i}g}=\{p^{{}^{\prime}}_{1},\cdots,p^{{}^{\prime}}_{m}\}\)
21:endfor
22:\(G_{t}=\{p^{{}^{\prime}}_{s_{1}g},\cdots,p^{{}^{\prime}}_{s_{t}g}\}\)
23:%Sub-step (3)
24:if\(G_{t}\) exists isolated satellite then
25:for any isolated satellite do
26: Connect the isolated satellite to the nearest and connected satellite and get \(G_{t}\)
27:endfor
28:endif
29:return\(G_{t}\)
```  
**Algorithm 1** Reliability-Assured Topology (RAT)  
Theorem 1: _If the SCN's initial topology without any topology control is connected, the topology after using the RAT algorithm is also connected._  
Proof of Theorem 1.: Let us assume the SCN's topology \(G_{t}\) with RAT is not connected, i.e., the network can be divided into several unconnected sub-topologies, like \(G_{t}=G_{0}+\cdots+G_{s}\). As the initial topology is connected, this means the RAT algorithm of Sub-step (3) cannot change the connectivity by just connecting the isolated satellite to the nearest and connected satellite.