# A Community-Based Construct Method for an Inter-Satellite Communication Network of Satellite Swarm
## 4 Optimized Constructions of ICNS
### Dynamic-Connected and Dynamic Paths  
PLNs for different slots may have the same adjacency matrices and degree matrices, which may lead to the same sets of CCLCs. Although different slots can result in different weight matrices, it is possible to select the same CCLC according to structural entropy, that is, the MSE-CCLCs for different slots may be the same. In this situation, each satellite would keep establishing the ISL with the same satellite. As there is no edge between any two link communities, an ICNS-OSS must be non-connected. Consequently, if all satellites do not change their link objects over time, the ICNS-OSP will remain disconnected throughout the state period. In order to ensure effective communication in the satellite swarm, the ICNS-OSP must be dynamically connected, which requires that the dynamic paths between any two satellites are always available. A dynamic path in ICNS is a space-time sequence of nodes that begins with the source node and ends with the destination node. The space-time sequence indicates that a dynamic path has both temporal length and spatial length. The temporal length is the number of slots that a dynamic path spans. The spatial length is the number of edges that are constituted of two neighboring nodes in a dynamic path. We define an S&D (the abbreviation of \"source and destination\") as the ordered pair \((v_{i},v_{j})\) that contains the source node \(v_{i}\) and the destination node \(v_{j}\). The dynamic path for an S&D \((v_{i},v_{j})\) is written as follows:  
\[DP(v_{i},v_{j},k)=\{v_{i}^{[k]},\cdots,v_{j}^{[k+p]}\} \tag{15}\]  
where \(v_{i}^{[k]}\) represents \(v_{i}\) during slot \(k\), and so on. There are two important parameters in (15): \(k\) is the slot in which the dynamic path is planned and it is called the plan slot (\(1\leq k\leq\Gamma\)), and \(k+p\) is the slot in which the dynamic path finishes and it is called the finish slot (\(p\geq 0\)). Consequently, \(p+1\) represents the temporal length of \(DP(v_{i},v_{j},k)\). A dynamic path can be planned by the method of finding the earliest route from a source satellite to all the other satellites [24]. Given the slot \(k\), if every S&D \((v_{i},v_{j})\) has an existent \(DP(v_{i},v_{j},k)\), the ICNS is believed to be dynamically connected from slot \(k\) to slot \(k+p\). Under the circumstances, the maximum temporal length of all dynamic paths can be defined by dynamic connect duration (DCD), which means how long it takes an ICNS to achieve dynamic connectivity from slot \(k\). However, if some S&D does not have a dynamic path, the ICNS will never be dynamically connected until the state period ends, so it is necessary to optimize the construction of the ICNS. We can prove that there is no plan to make ICNS dynamically connected during any slot. The binary tree in Figure 3 shows all dynamic paths that are from \(v_{i}\). Nodes of this binary tree represent satellites. A parent node's left child node is itself, and its right child node is the satellite with which it establishes an ISL during the corresponding slot. Satellites in different black rectangles are also different (their numbers are fictitious), which means each satellite cannot establish an ISL with satellites that it has connected with and their link objects. Under the circumstances, the binary tree can cover all \(|V|\) satellites when it has had \(\log_{2}|V|\) levels, which means the maximum temporal length of dynamic paths from \(v_{i}\) is \(\log_{2}|V|\). Nevertheless, the link plan in Figure 3 is highly impracticable because it does not consider those dynamic paths from another satellite. If some satellites established an ISL with one of the link objects of its link objects, the maximum temporal length of dynamic paths from \(v_{i}\) may increase to at least \(1+\log_{2}|V|\). Therefore, \(\log_{2}|V|\) is the ideal minimum of the maximum temporal length of dynamic paths. Hence, the ICNS will spend at least \(\log_{2}|V|\) slots dynamically connected to slot \(k\), no matter what \(k\) is.