# A Community-Based Construct Method for an Inter-Satellite Communication Network of Satellite Swarm
## 3 Community-Based Construct Method of an ICNS  
### Link Community  
Communities in networks are groups of nodes. There are dense connections in the communities and sparser connections between the communities [16]. As a satellite is only allowed to establish one ISL with another satellite within a slot, an ICNS is composed of multiple satellite pairs, within which there are ISLs yet between which there are no ISLs. These satellite pairs can be treated as communities, which are called link communities. The link community has two essential features: (1) one link community contains two and only two nodes, and (2) there is no edge between any two link communities. We propose that the link community is equivalent to the ISL, so the link community that consists of \(v_{i}\) and \(v_{j}\) is also denoted by \(e_{ij}\). The edge \(\widetilde{e}_{ij}\) in the PLN represents a candidate for the link community (CLC) \(e_{ij}\) accordingly.  
In fact, the ICNS is a network that is composed of \(\theta\) link communities if there are \(\theta\) satellite pairs that have established ISLs (\(\theta\leq\frac{1}{2}\Theta\)). The combination of candidates for link communities (CCLC) is defined as a special subset of the edge set of a PLN in which no two members, i.e., candidates for link communities, have a node in common. Hence, \(G_{k}\) can be constructed by collecting CCLCs based on \(\widetilde{G}_{k}\) and by choosing one of them to become \(E_{k}\)We then introduced three fundamental matrices of \(\widetilde{G}_{k}\), which are vital to construct ICNS. They are the adjacency matrix \(\mathbf{A}_{k}=\left(a_{ij}^{[k]}\right)_{\Theta\times\Theta^{\prime}}\) degree matrix \(\mathbf{D}_{k}=\left(a_{ij}^{[k]}\right)_{\Theta\times\Theta^{\prime}}\) and weight matrix \(\mathbf{W}_{k}=\left(w_{ij}^{[k]}\right)_{\Theta\times\Theta}\). The element \(a_{ij}^{[k]}\) of \(\mathbf{A}_{k}\) is defined by Equation (3).  
\[a_{ij}^{[k]}=\begin{cases}1,&\widetilde{e}_{ij}\in\widetilde{E}_{k}\\ 0,&\widetilde{e}_{ij}\
otin\widetilde{E}_{k}\text{ or }i=j\end{cases} \tag{3}\]  
The element \(d_{ij}^{[k]}\) of \(\mathbf{D}_{k}\) is defined by Equation (4).  
\[d_{ij}^{[k]}=\begin{cases}\text{degree of }v_{i}\text{ in }\widetilde{G}_{k},&i=j\\ 0,&\text{others}\end{cases} \tag{4}\]  
The element \(w_{ij}^{[k]}\) of \(\mathbf{W}_{k}\) is defined by Equation (5).  
\[w_{ij}^{[k]}=\begin{cases}\text{weight of }\widetilde{e}_{ij},&a_{ij}^{[k]}=1\\ 0,&\text{others}\end{cases} \tag{5}\]  
There are two steps to obtain \(\mathbf{W}_{k}\). Firstly, we establish a distance matrix \(\mathbf{R}=\left(r_{ij}^{[k]}\right)_{\Theta\times\Theta}\) whose element \(r_{ij}^{[k]}\) is defined by (6).  
\[r_{ij}^{[k]}=\begin{cases}\text{distance between }v_{i}\text{ and }v_{j},&a_{ij}^{[k]}=1\\ 0,&\text{others}\end{cases} \tag{6}\]  
Secondly, we normalizes \(r_{ij}^{[k]}\) as (7), so that \(w_{ij}^{[k]}\) is obtained.  
\[w_{ij}^{[k]}=\begin{cases}\frac{\min\limits_{\lambda_{jk}^{[k]}=1}^{[k]}}{r_{ ij}^{[k]}},&a_{ij}^{[k]}=1\\ 0,&\text{others}\end{cases} \tag{7}\]  
In this paper, \(d_{ii}^{[k]}\) denotes the degree of \(v_{i}\) in \(\widetilde{G}_{k}\) and \(w_{ij}^{[k]}\) denotes the weight of \(\widetilde{e}_{ij}\) in \(\widetilde{G}_{k}\).