## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.3 Network Coding and Multi-Path Transmission
Additionally, the distribution range of the encoding coefficients can be adaptively adjusted using a scaling factor. In particular, this scaling factor can help adjust the distribution characteristics of the encoding coefficients to suit different network topologies. 3. The generation process involves extracting each encoding coefficient \(g_{ij}\) from a uniform distribution \(U(0,1)\) using a pseudorandom number generator, ensuring the randomness and independence of each coefficient, thereby guaranteeing the reliability and effectiveness of the network coding process. Through these control measures, the generated encoding coefficients not only meet the requirements of randomness, but also ensure the stability and reliability of the network coding, thereby helping to enhance the redundancy and recovery capability during data transmission. In multi-path transmission, combining network coding strategies can effectively improve data transmission reliability, especially in scenarios where multiple paths provide redundancy and can mitigate the impact of single link losses on the entire system. Proof. : Network coding performs encoding combination of data packets over multiple paths, such that even if some links fail, data can still be transmitted successfully over other paths. Let data packets \(X_{1},X_{2},\ldots,X_{m}\) be encoded into \(Y_{1},Y_{2},\ldots,Y_{n}\), where each data packet \(Y_{i}\) is a linear combination of the original data packets, and the coding coefficients \(g_{ij}\) are randomly chosen. When some links in a path fail  
Following this, for each data packet \(X_{1},X_{2},\ldots,X_{m}\), the data packets are encoded to generate network-coded data packets \(Y_{i}\) (Lines 7-8), and the encoded data packets are transmitted over multiple paths to achieve redundancy (Line 9). After the iterative process concludes, the computed utility values \(U\), link reliabilities \(\mu_{ij}^{E}\), path reliabilities \(R_{h}\), and encoded data packets \(Y_{i}\) are returned (Line 12). ```
1:Input: Set of satellite nodes \(N\), resource demands \(D\), link reliability \(R_{ij}\), interference coefficients \(\epsilon_{ij}\), optimal power and bandwidth allocation \((P_{n}^{*},W_{n}^{*})\), interference threshold \(I_{\text{th}}\), SNR threshold \(S_{\text{NR}}\). 2:Output: Utility values for each satellite node \(U\), link reliability values \(\mu_{ij}^{E}\), path reliability values \(R_{h}\), network-coded data packets \(Y_{i}\). 3:for each node \(i\in N\), each link \(E_{ij}\in N\), each path \(p_{EI}\in\mathcal{P}\)do
4: Calculate utility as Equation (30);
5: Assign node \(i\) to domain \(D_{k}\) based on \(U_{i}\);
6: Calculate link reliability as Equation (32);
7: Calculate path reliability as Equation (33);
8:endfor
9:for each packet \(X_{1},X_{2},\ldots,X_{m}\)do
10: Encode data packets as Equation (35);
11: Transmit \(Y_{i}\) over multiple paths for redundancy;
12:endfor
13:Return
14:\(U=\{U_{1},U_{2},\ldots,U_{N}\}\), link reliabilities \(\mu_{ij}^{E}\), path reliabilities \(R_{h}\), coded packets \(Y_{i}\)
```
**Algorithm 3** Preprocessing Framework for Resource Allocation and Reliability Optimization (PF-RARO)