# A Community-Based Construct Method for an Inter-Satellite Communication Network of Satellite Swarm
## 2 Complex Network Models of the Satellite Swarm
### Introduction of ICNS  
The members of a satellite swarm are denoted by \(V=\{v_{i}\}\), where \(v_{i}\) represents the \(i\)-th satellite. The cardinality of \(V\) is designated by \(\Theta\) (i.e., \(|V|=\Theta\)). If \(v_{i}\) and \(v_{j}\) need to establish ISL within slot \(k\), two conditions must be met throughout this slot:  
1. The distance between the Earth's core and the line from \(v_{i}\) to \(v_{j}\) (\(|OD|\) in Figure 2) is greater than the sum of the Earth's radius \(v_{j}\) (\(|OE|\) in Figure 2) and the height of the ionosphere(\(|EF|\) in Figure 2), as is shown in Equation (1): \[|OD|>|OE|+|EF|\] (1)
2. The elevation angle from \(v_{i}\) to \(v_{j}\) (\(\angle OI\)) in Figure 2) is in the onboard antenna scanning range of \(v_{i}\) (\(\gamma_{i}\) in Figure 2); meanwhile, the elevation angle from \(v_{j}\) to \(v_{i}\) (\(\angle OII\) in Figure 2) is in the onboard antenna scanning range of \(v_{j}\) (\(\gamma_{j}\) in Figure 2), as is shown in Equation (2): \[\begin{cases}\angle OIJ<\gamma_{i}\\ \angle OJI<\gamma_{j}\end{cases}\] (2)  
Once ISLs are established in a satellite swarm, an ICNS is created. According to the notation of the graph theory, the ICNS that is created during slot \(k\) can be modeled as a graph \(G_{k}=(V_{k},E_{k})\), where \(V_{k}=\{v_{i}\}\) is the node set and \(E_{k}=\left\{e_{ij}|v_{i},v_{j}\in V,i<j\right\}\) is the edge set. Herein, nodes are equivalent to satellites and edges are equivalent to ISLs. Thus,  
Figure 1: Division of a task time interval through FSA.  
Figure 2: Conditions of establishing an ISL.  
\(v_{i}\) is not only the \(i\)-th satellite but also the \(i\)-th node. Furthermore, \(e_{ij}\) is not only the edge connecting the \(i\)-th node and the \(j\)-th node but also the ISL between the \(i\)-th satellite and the \(j\)-th satellite.  
In this paper, \(e_{ij}\) does not indicate the direction of ISL but only means that there is an ISL between \(v_{i}\) and \(v_{j}\). The satellite with a smaller number \(v_{i}\) is called the left satellite while the satellite with a bigger number \(v_{j}\) is called the right satellite. An ICNS is not an undirected graph, and it is not a simple directed graph. In fact, there is two-way alternate communication between two satellites connected with an ISL, that is, the direction of ISL is from the left satellite to the right satellite during the former half slot and from right the satellite to the left satellite during the former half slot. Thus, the ICNS is a special directed graph whose edges have changeable directions.  
The edge set \(E_{k}\) is also a set of ISLs that are established during slot \(k\). As a satellite is only allowed to establish one ISL with another satellite within a slot [6], \(E_{k}\) has \(\frac{1}{2}\Theta\) elements at most. The difference between these ICNSs for various slots only concerns ISLs rather than satellites, according to [12]. Therefore, the ICNS for slot \(k\) can be further denoted by \(G_{k}=(V,E_{k})\). As a result, the construction of ICNS can be reducible to the construction of the edge set of the ICNS for each slot. Since each satellite establishes at most one ISL in any slot, no two edges in an ICNS have a vertex in common. Hence, the ICNS is a disconnected graph at any time.