# An Enhanced Multi-Constraint Optimization Algorithm for Efficient Network Topology Generation
## 3 System Model and Problem Description
### Problem Description
If these constraints are met, an initial potential link is established between the two satellite nodes. The number of optical transceivers for each satellite, however, is limited [26]. Therefore, each satellite node's degree constraint must also be considered. Investigating how to establish inter-satellite links while adhering to these constraints, in addition to ensuring the stability and high communication efficiency of the satellite network topology, is a key issue in generating the constellation network topology. In this paper, we introduce two metrics, the average link length (\(al\)) and the average link duration (\(dl\)), to evaluate the communication efficiency and stability of the constellation network topology. Average Link Length (\(al\)): This metric is defined as the average length of all the links in the network. Physically, the link length has a direct impact on the quality of a link. The longer the link, the greater the attenuation and diffusion of light, leading to a lower signal quality [27]. Therefore, a shorter average link length usually means better link quality and higher communication efficiency. Additionally, the link length directly affects the propagation delay of signals. The longer the link, the longer the time for signals to travel from the sender to the receiver. Hence, maintaining a shorter average link length can reduce the overall delay in the network, enhancing network performance [28]. It is expressed as follows:  
\[al=\frac{1}{|E|}\sum_{(i,j)\in E}d_{ij} \tag{9}\]where \(d_{ij}\) is the length of the link between node \(v_{i}\) and node \(v_{j}\), and \(|E|\) is the total number of links in the network. A shorter average link length signifies a reduced propagation delay and heightened overall communication efficiency among satellites within the constellation network topology. Contrarily, a longer average link length infers a more extended propagation delay and diminished overall communication efficiency between satellites in the constellation network topology. Average Link Duration (\(dl\)): This metric is defined as the average duration of all the links in the network. The average link duration is a direct indicator of link stability. In dynamic networks, the duration of a link is often influenced by many factors such as node mobility, energy consumption, and communication range [29]. The longer the average link duration, the more stable the links in the network, meaning that communication between nodes is less likely to be interrupted. Therefore, the stability of links directly affects the stability and reliability of the network [30]. By maximizing the average link duration, the stability and reliability of the network can be effectively improved. Therefore, by optimizing the average link length and average link duration, the stability of optical communication network links can be effectively enhanced. It is calculated as follows:  
\[dl=\frac{1}{|E|}\sum_{(i,j)\in E}t_{ij} \tag{10}\]  
where \(t_{ij}\) is the duration of the link between node \(v_{i}\) and node \(v_{j}\), and \(|E|\) is the total number of links in the network. A shorter average link duration results in more frequent link switching and adverse network stability. Conversely, a longer average link duration implies superior network stability. Normalization of average link length (\(al\)) and average link duration (\(dl\)):  
\[\delta=\cos^{-1}\left(\frac{R_{i}^{2}+R_{j}^{2}-d^{2}}{2\cdot R_{i}\cdot R_{j} }\right)\leq\delta_{\mathrm{max}}. \tag{3}\]0  
\[\delta=\cos^{-1}\left(\frac{R_{i}^{2}+R_{j}^{2}-d^{2}}{2\cdot R_{i}\cdot R_{j} }\right)\leq\delta_{\mathrm{max}}. \tag{3}\]1  
Therefore, this paper posits that both \(al\) and \(dl\) are equally significant in influencing the communication efficiency and stability of the optical communication constellation network, with \(al\) having an inverse relationship with communication efficiency and \(dl\) exhibiting a direct relationship with stability.