## 2 Network Description
### EO Missions on Satellite Cluster Network  
According to the usual workflow in Earth observation [32; 33], after the step of mission planning, the next is taking images of targets using the payloads. Once the target area has been detected and covered, the satellite begins the procedure of transmitting target images to the ground stations. Even though, in some cases, satellites can work in both states of image-taking and ground communication, the download bandwidth is very low due to the limited energy onboard and vast distance between the ground and satellite. Thus, it may take a long time from gathering images to  
Figure 1: Illustration of satellite cluster networks (SCNs) with multi-clusters.  
Figure 2: An example of topology changing varying with time.  
accomplishing image-sending. To resolve this problem, this paper puts forward a cooperative work pattern in SCNs.  
According to different payload requirements when observing the Earth, let satellites in SCNs be divided into three categories: Earth observation satellites (EOS), ground-station communication satellites (GSC), and data routing satellites (DRS), i.e., \(V_{[N,3]}=V_{EOS}+V_{GSC}+V_{DRS}\). EOS are the satellites having EO cameras and play the role of taking images of the ground targets; GSC should have the ability of high-speed data transmission and are responsible for sending these data to ground stations via satellite-ground links of high bandwidth; and the rest satellites are categorized as DRS, which undertake the task of image data relaying, to transmit the image data from EOS to GSC. For example, as shown in Figure 3, two EOS work cooperatively to accomplish the target imaging and, at the same time, send these data to the ground stations via several DRS and GSC. In this above cooperative work pattern, it not only greatly extends the functionality of the SCN by working as a team, but also makes possible the transmission of large amounts of image data to the ground, which is usually strongly demanded in some emergency cases [34].  
In SCNs, this paper respectively indicates any satellite in EOS, DRS, and GSC by using \(s_{i}\), \(r_{i}\) and \(g_{i}\), i.e., \(s_{i}\in V_{EOS}\), \(r_{i}\in V_{DRS}\) and \(g_{i}\in V_{GSC}\). \(n\) and \(m\) denote the number of satellites in EOS and GSC separately. Additionally, \(l\) is the number of DRS with data routing requirements in one specific topology snapshot, and \(l^{{}^{\prime}}\) shows the rest of the DRS not getting involved in routing tasks during one snapshot. Based on the states of whether participating in cooperative work, the topology of SCNs can be indicated with the gathering of the working sub-graph \(G^{{}^{\prime}}_{t}\) and non-working sub-graph \(G^{{}^{\prime\prime}}_{t}\), i.e., \(G_{t}=G^{{}^{\prime}}_{t}+G^{{}^{\prime\prime}}_{t}=\left(V^{{}^{\prime}}_{t}, E^{{}^{\prime}}_{t}\right)+\left(V^{{}^{\prime\prime}}_{t},E^{{}^{\prime\prime}}_{t}\right)\). The position vertex of satellites in the cluster satisfy the following relations:  
\[\begin{array}{rcl}V_{t}&=&V^{{}^{\prime}}_{t}+V^{{}^{\prime\prime}}_{t}\\ &=&V\left(s_{1},\cdots,s_{n},r_{1},\cdots,r_{l},g_{1},\cdots g_{m}\right)\\ &&+V\left(r_{l+1},\cdots,r_{l+l^{{}^{\prime}}}\right)\end{array} \tag{1}\]  
where:  
\[\left\{\begin{array}{l}n+m+l+\dot{l}^{{}^{\prime}}=N\\ V_{EOS}=\left\{s_{1},\cdots,s_{n}\right\}\\ V_{DRS}=\left\{r_{1},\cdots,r_{l},r_{l+1},\cdots,r_{l+l^{{}^{\prime}}}\right\} \\ V_{GSC}=\left\{g_{1},\cdots g_{m}\right\}\end{array}\right. \tag{2}\]  
Figure 3: Illustration of cooperative work in SCNs.  
The equations above describe the position relations \(V_{t}\) of one topology snapshot, and the subsequent content will discuss the edge relations \(E_{t}\) in detail, with consideration of orbits' prediction and periodicity.