# An Enhanced Multi-Constraint Optimization Algorithm for Efficient Network Topology Generation
## 3 System Model and Problem Description
### Problem Description  
For each satellite in the constellation network, a \(n\times n\) matrix \(E^{t}\) is constructed, starting from the neighboring nodes. This matrix represents the edge set of links between all pairs of satellites at time slot \(t\), it can be expressed as:  
\[E^{t}=\left[ \begin{array}{ccc}e_{11}^{t}&\cdots&e_{1n}^{t}\\ \vdots&\ddots&\vdots\\ e_{n1}^{t}&\cdots&e_{nm}^{t}\end{array} \right] \tag{1}\]  
When \(e_{ij}^{t}=1\), a link is established between node \(v_{i}\) and node \(v_{j}\), permitting the two neighboring nodes to communicate and exchange information. Conversely, when \(e_{ij}^{t}=0\), nodes \(v_{i}\) and \(v_{j}\) cannot establish a link, meaning the nodes cannot communicate directly. Thus, this edge set reflects the connectivity between satellites within the constellation network, forming the network topology. Under ideal conditions, the visibility criterion between two satellite nodes, \(v_{i}\) and \(v_{j}\), is:  
\[d\leq d_{\mathrm{max}}=\sqrt{R_{i}^{2}-R_{h}^{2}}+\sqrt{R_{j}^{2}-R_{h}^{2}} \tag{2}\]  
\[\delta=\cos^{-1}\left(\frac{R_{i}^{2}+R_{j}^{2}-d^{2}}{2\cdot R_{i}\cdot R_{j} }\right)\leq\delta_{\mathrm{max}}. \tag{3}\]  
In this context, \(d\) represents the link length between satellites; \(\delta\) is the geocentric angle formed between nodes \(v_{i}\) and \(v_{j}\); \(R_{h}\) denotes the distance from the geocenter to the line connecting nodes \(v_{i}\) and \(v_{j}\); and \(R_{i}\) and \(R_{j}\) represent the distances from nodes \(v_{i}\) and \(v_{j}\) to the geocenter, respectively. Proposition 1: _For any two satellite nodes, \(v_{i}\) and \(v_{j}\), in an optical communication constellation network (OCCN) with an established link between them, the relationship between the geocentric angle, \(\delta_{ij}\), and the average link duration, \(d_{ij}\), can be mathematically represented as:_  
\[\frac{d(\delta_{ij})}{dt}<0\Rightarrow\frac{d(d_{ij})}{dt}>0. \tag{4}\]  
_This indicates that a decrease in the geocentric angle \(\delta_{ij}\) over time corresponds to an increase in the average link duration \(d_{ij}\) over the same period._  
Figure 1: The topology time-varying diagram of constellation satellite network. **Proof. **  
\[S=k\cdot d^{-2} \tag{5}\]  
where \(k\) is a constant of proportionality. From the visibility criterion between two satellite nodes, \(v_{i}\) and \(v_{j}\), we have:  
\[d=\sqrt{R_{i}^{2}-R_{h}^{2}}+\sqrt{R_{j}^{2}-R_{h}^{2}}. \tag{6}\]  
Substituting \(d\) into the formula for \(S\) gives:  
\[S=k\cdot\left(\sqrt{R_{i}^{2}-R_{h}^{2}}+\sqrt{R_{j}^{2}-R_{h}^{2}}\right)^{-2}. \tag{7}\]  
We also have the formula for \(\delta\), the geocentric angle, as follows:  
\[\delta=\cos^{-1}\left(\frac{R_{i}^{2}+R_{j}^{2}-d^{2}}{2\cdot R_{i}\cdot R_{j} }\right). \tag{8}\]  
When \(\delta\) decreases, from the cosine function properties, the denominator of the fraction inside the cosine function increases. This results in a decrease in \(d\) according to the formula of \(d\). When \(d\) decreases, from the formula for \(S\), \(S\) increases, which means the stability of the link increases. This completes the proof that a decrease in the geocentric angle, \(\delta\), leads to an increase in the average link duration, \(dl\), thus enhancing the stability of the communication link in the OCCN. Due to the unique properties of satellite networks, including large spatiotemporal scales, prolonged delays, dynamic changes, frequent inter-satellite link switching, limited on-board resources, and communication range constraints imposed by antenna elevation angle [25], traditional network node communication limitations are insufficient for on-board communication environments. In this paper, we consider the following constraints for inter-satellite optical communication:  
1. Link propagation delay: \(t\leq t_{\text{max}}\);
2. Link bandwidth: \(b\geq b_{\text{min}}\);
3. Link survival time: \(lt\geq l_{\text{min}}\);
4. Elevation angle between satellites: \(\delta\leq\delta_{\text{max}}\).