# A Community-Based Construct Method for an Inter-Satellite Communication Network of Satellite Swarm
## 3 Community-Based Construct Method of an ICNS
### Method of Selecting a CCLC According to Structural Entropy  
Now that \(\mathcal{M}_{k}\) has been obtained by Algorithm 1, the next step is to select a suitable CCLC from \(\mathcal{M}_{k}\). We will introduce a method of selecting CCLC according to structural entropy in this section.  
The structural entropy is a type of network entropy [19], which is a measure of the network's heterogeneity [20]. The weaker the heterogeneity, the greater the entropy value [21]. A weaker heterogeneity means there are fewer differences between nodes of the network [22]. Nowadays, there are many kinds of structural entropies in the field of complex networks [19]. We borrowed the idea from the authors in [23], that is, considering both nodes and edges to define a new structural entropy.  
We treat each CLC of \(M_{k}(\varepsilon)\) as a node. We state that if \(\widetilde{e}_{ij}\) is the \(\iota\)-th CLC in \(M_{k}(\varepsilon)\), \(\widetilde{e}_{ij}\) can be denoted by \(c_{\iota}\). We define the degree \(d_{\varsigma}^{[\iota]}\) of \(c_{\iota}\) as the normalized number of links incident to nodes in \(c_{\iota}\):  
\[d_{\varsigma}^{[\iota]}=\frac{d_{\iota}^{[\iota]}+d_{\iota}^{[\iota]}-2}{( \sum\limits_{k=1}^{\Theta}d_{\iota\lambda}^{[\iota]})-2\theta} \tag{8}\]We define the weight \(w_{\text{c}}^{[i]}\) of \(c_{i}\) as the weight of \(\widetilde{e}_{ij}\), i.e., \(w_{\text{c}}^{[i]}=w_{ij}^{[k]}\). Then, we define the importance coefficient \(U_{t}\) of \(c_{i}\) as the weighted sum of \(d_{\text{c}}^{[i]}\) and \(w_{\text{c}}^{[i]}\):  
\[U_{t}=\alpha d_{\text{c}}^{[i]}+\beta w_{\text{c}}^{[i]} \tag{9}\]  
where \(\alpha\) and \(\beta\) are weights and \(\alpha+\beta=1\). \(U_{t}\) needs to be normalized as follows:  
\[\overline{U}_{t}=\frac{U_{t}}{\sum\limits_{v=1}^{\theta}U_{v}}=\frac{\alpha d _{\text{c}}^{[i]}+\beta w_{\text{c}}^{[i]}}{\sum\limits_{v=1}^{\theta}\left( \alpha d_{\text{c}}^{[v]}+\beta w_{\text{c}}^{[v]}\right)}=\frac{\frac{\alpha} {\beta}d_{\text{c}}^{[i]}+w_{\text{c}}^{[i]}}{\sum\limits_{v=1}^{\theta}\left( \frac{\alpha}{\beta}d_{\text{c}}^{[v]}+w_{\text{c}}^{[v]}\right)} \tag{10}\]  
We believe that \(d_{\text{c}}^{[i]}\) is as important as \(w_{\text{c}}^{[i]}\), i.e., \(\alpha=\beta\). Thus, (10) is transformed into  
\[\overline{U}_{t}=\frac{d_{\text{c}}^{[i]}+w_{\text{c}}^{[i]}}{\sum\limits_{v= 1}^{\theta}\left(d_{\text{c}}^{[v]}+w_{\text{c}}^{[v]}\right)} \tag{11}\]  
Finally, we give the formula of the structural entropy \(H_{\text{c}}^{[k]}\) of \(M_{k}(\epsilon)\) with \(\theta\) CLCs:  
\[H_{\text{c}}^{[k]}=-\sum\limits_{i=1}^{\theta}\overline{U}_{t}\ln\overline{U} _{t} \tag{12}\]  
ICNS should have lower heterogeneity to help satellites make full use of their autonomy, which will promote the decentralization and swarm intelligence of a satellite swarm. Therefore, the maximum structural-entropy CCLC (MSE-CCLC) should be selected to become \(E_{k}\), as shown in Equation (13):  
\[E_{k}=\{e_{ij}|\widetilde{e}_{ij}\in M_{k}(\underset{\epsilon}{\text{argmax}}H_ {\text{c}}^{[k]})\} \tag{13}\]