## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.3 Network Coding and Multi-Path Transmission  
While traditional path selection methods can optimize some paths, they still suffer from transmission bottlenecks, especially when link failures or losses occur. Therefore, we introduce _Random Linear Network Coding (RLNC)_ to enhance network redundancy, ensuring data transmission, even when some paths experience packet loss. Network coding allows data to be transmitted over multiple paths by performing linear combination of data packets at intermediate nodes, enhancing the network's robustness. Suppose data packets \(X_{1},X_{2},\ldots,X_{m}\) are encoded into \(Y_{1},Y_{2},\ldots,Y_{n}\):  
\[Y_{i}=\sum_{j=1}^{m}g_{ij}X_{j}\quad\text{for}\quad i=1,2,\ldots,n \tag{35}\]  
where \(g_{ij}\) are randomly generated coding coefficients. **Theorem 6** (Robustness under Non-Ideal Conditions). : _Assume that each individual link \(i\) in a transmission path has a success probability \(p_{i}\). For a given path \(P\) consisting of \(L\) links, the end-to-end success probability is_  
\[P_{\text{path}}=\prod_{i=1}^{L}p_{i}.\]  
_In a multi-path transmission scenario with \(K\) independent paths, let the overall probability that at least one path successfully delivers a coded packet be defined as_  
\[P_{\text{multi}}=1-\prod_{k=1}^{K}\left(1-P_{\text{path}}^{(k)}\right).\]  
_Suppose that the source data consist of \(m\) packets and that \(n\) coded packets are transmitted over the multi-path channels, with \(n\geq m\). With Random Linear Network Coding (RLNC), successful decoding is achieved if at least \(m\) linearly independent coded packets are received. Hence, the decoding condition is satisfied with high probability if_  
\[n\cdot P_{\text{multi}}\geq m,\]  
_or equivalently,_  
\[P_{\text{multi}}\geq\frac{m}{n}.\]  
_Thus, as long as the aggregate multi-path reliability satisfies the above inequality, the proposed multi-path transmission strategy maintains its performance, even under non-ideal conditions._  
Proof. : For each individual path \(P^{(k)}\), let the end-to-end success probability be  
\[P_{\text{path}}^{(k)}=\prod_{i=1}^{L_{k}}p_{i}^{(k)},\]  
where \(L_{k}\) is the number of links in the \(k\)th path and \(p_{i}^{(k)}\) denotes the success probability of the \(i\)th link in that path. For \(K\) independent paths, the probability that none of the paths delivers a successful packet is  
\[\prod_{k=1}^{K}\left(1-P_{\text{path}}^{(k)}\right).\]  
Therefore, the probability that at least one path is successful is  
\[P_{\text{multi}}=1-\prod_{k=1}^{K}\left(1-P_{\text{path}}^{(k)}\right).\]  
Assuming \(n\) coded packets are transmitted, the expected number of successfully received packets is  
\[E[N]=n\cdot P_{\text{multi}}.\]  
Since RLNC allows successful decoding if at least \(m\) independent packets are received, a sufficient condition is  
\[E[N]\geq m\quad\Longleftrightarrow\quad n\cdot P_{\text{multi}}\geq m.\]  
Rearranging this yields the condition  
\[P_{\text{path}}=\prod_{i=1}^{L}p_{i}.\]0Thus, the proposed multi-path transmission strategy maintains its performance under non-ideal conditions if the overall multi-path reliability satisfies the above bound. In this paper, the encoding coefficients \(g_{ij}\) used in the network coding process are generated randomly, to improve the robustness and redundancy of the network. To ensure the rationality and stability of these encoding coefficients, the generation process follows the rules and parameters described below:  
1. The encoding coefficients \(g_{ij}\) are randomly generated within the range of \([0,1]\) using a uniform distribution. The generation method employs a standard pseudorandom number generator to guarantee the independence and randomness of the coefficients. 2. To enhance the encoding efficiency and fault tolerance of the network, the generated encoding coefficients \(g_{ij}\) ensure independence, meaning that there is no correlation between the different encoding coefficients.