# Uncertainty Theory-Based Resilience Analysis for LEO Satellite Communication Systems
## 4 LEO-SCS and Its Uncertain Evolution Model
### Uncertain Satellite Network Evolution Model
#### 4.2.4 Evolution Rule of Dynamic Uncertain Satellite Network Model  
The dynamic uncertain satellite network model describes the operation of an LEO-SCS in the normal state. However, in the actual operation, the satellite network is evolving due to the failure and recovery of satellites. **Definition 11**. : _The **topology evolution of an LEO-SCS** is the topology change of the system caused by satellite entry and exit due to its failure and recovery._  
We present the evolution rule of the dynamic, uncertain satellite network model, including timing and an algorithm, to describe the topology evolution of the network structure. (1) Evolution timing  
According to Definition 11, the topology evolution of the satellite network is caused by satellite failure and recovery. Once a satellite fails, it will de-orbit and a backup satellite will be transferred to the site of failure. The evolution timing of the system is as follows:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Adjacency Matrix** & **Nodes** & **Edges** & **Connection Rules** & **Delays** \\ \hline \multirow{4}{*}{\(S_{m}(t)\)} & \multirow{4}{*}{Satellites} & \multirow{4}{*}{ISLs} & Each satellite connects to its adjacent satellites & \(d_{s}(t)= \begin{cases}l_{s}(t)/c+d_{p,s},\text{(case 1)}\\ l_{s}(t)/c+2d_{p,s},\text{(case 2)}\end{cases} \), where \\  & & & Each satellite connects to \\  & & & & \(l_{s}(t)\) is the length of ISL at time \(t\), \(c\) is the propagation speed of electromagnetic waves, and \(d_{p,s}\) is the processing delay of satellites \\ \hline \multirow{4}{*}{\(G_{m}(t)\)} & \multirow{4}{*}{Satellites and gateways} & \multirow{4}{*}{FLs} & Each satellite connects to a connectable gateway according to the access criteria & \(d_{f}(t)=l_{f}(t)/c+d_{p,s}\), where \(l_{f}(t)\) is the length of FL at time \(t\), and \(d_{p,s}\) is the processing delay of the gateway \\ \hline \multirow{4}{*}{\(U_{m}(t)\)} & \multirow{4}{*}{Satellites and VNs} & \multirow{4}{*}{ULs} & Each satellite connects to \\  & & & & \(d_{u}(t)=l_{u}(t)/c+d_{p,v}\), where \(l_{u}(t)\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & & \(d_{p,v}\) is the \\ \cline{1-1}  & & & &* **Node exit time** depends on the satellite failure time;
* **Node entry time** depends on the backup strategy of the system. Both in-orbit and ground backup are applied in an LEO-SCS [23; 32; 33].