# An Accelerated Maximum Flow Algorithm with Prediction Enhancement in Dynamic LEO Networks
## 3 System Model and Problem Formulation
### Visibility Constraints of Satellite Networks  
In LEO satellite networks, communication between any pair of nodes, that is, satellites, or ground stations, is governed by geometric visibility conditions. A data transmission link can only be established if two nodes have an unobstructed line-of-sight (LoS) connection during a given time slot [35]. Since LEO satellites move rapidly along their orbits, the availability of communication links varies over time, requiring a time-dependent representation of network connectivity.  
For a direct communication link between two nodes \(v_{i}\) and \(v_{j}\) to be feasible, the minimum elevation angle \(\theta_{min}\) defines the threshold for communication. The maximum allowable elevation angle \(\delta_{ij}\) for establishing a communication link between two nodes is determined by the Earth's curvature and the altitude of the satellites. This angle can be calculated as follows for satellite-to-ground and satellite-to-satellite links:  
\[\delta_{ij}=90^{\circ}-\theta_{\min}-\arcsin\left(\frac{r_{E}+h_{i}}{r_{E}+h_ {j}}\cdot\sin(90^{\circ}+\theta_{\min})\right) \tag{5}\]  
where \(r_{E}\) is the Earth's radius, and \(h_{i}\) and \(h_{j}\) are the altitudes of satellites \(v_{i}\) and \(v_{j}\), respectively, assuming that \(v_{i}\) is at a relatively higher altitude than \(v_{j}\).  
For satellite-to-satellite communication, the visibility condition is satisfied when the angular \(\angle OPv_{j}\) does not exceed the maximum allowable link angle \(\delta{ij}\). The angle \(\angle OPv_{j}\) is defined as the angular separation between the center of the Earth \(O\), an intermediate intersection point, \(P\), and the receiving satellite, \(v_{j}\). The intersection point, \(P\), is determined along the line connecting \(O\) and \(v_{j}\) such that it satisfies the minimum elevation angle constraint required for satellite visibility. Using trigonometric relationships, this angle is computed as follows:  
\[\angle OPv_{j}=2\arcsin\left(\frac{|Pv_{j}|}{2(r_{E}+h_{j})}\right)\leq\delta_ {ij} \tag{6}\]  
where \(\left|Pv_{j}\right|\) is the Euclidean distance between the intersection point \(P\) and the satellite \(v_{j}\)For both satellite-to-satellite and satellite-to-ground links, communication is only possible if the angular separation between the two nodes is less than or equal to \(\delta_{ij}\), ensuring LoS. This condition is mathematically represented as follows:  
\[(v_{i},v_{j})\in\mathcal{E}^{h}\text{ iff }\theta_{ij}^{h}\leq\delta_{ij} \tag{7}\]  
where \(\mathcal{E}^{h}\) represents the set of feasible links at time slot \(\tau_{h}\), and \(\theta_{ij}^{h}\) denotes the angular separation between the two nodes \(v_{i}\) and \(v_{j}\) at time slot \(\tau_{h}\).  
Thus, these visibility constraints are fundamental for determining which links are available for data transmission at any given time, ensuring that only valid communication paths are used. These constraints are crucial when calculating the maximum flow in the network, as they directly limit the set of feasible paths through which data can be transmitted.