## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.6 Problem Modeling and Optimization Algorithm Design
Figure 4: Objective function values and growth rates of all algorithms. Figure 5 and Table 4 illustrates the variation in objective function values for the different algorithms over the time steps. Throughout the entire time period, CMPT consistently maintained a significant performance advantage, with its objective function value steadily improving over time and surpassing those of AABC, AACO, AHPSO, and LRM. This superiority stems from the integration of Lagrangian dual optimization and multi-path scheduling strategies within CMPT, which enable fine-grained resource allocation and real-time feedback mechanisms in dynamic networks. This ensures system reliability and efficiency even in complex environments. In contrast, the other algorithms failed to simultaneously optimize resource utilization and scheduling performance under multiple constraints, resulting in inferior performance. In this study, the growth rate of the task execution time was calculated based on the variation in task execution time across consecutive time steps. Specifically, the growth rate \(G_{i}\) between two consecutive time steps \(t_{i}\) and \(t_{i+1}\) was computed as the relative change in task execution time. The growth rate is defined by the following formula:

\[G_{i}=\frac{T_{i+1}-T_{i}}{T_{i}}\times 100\% \tag{49}\]

where \(T_{i}\) and \(T_{i+1}\) represent the task execution times at time steps \(t_{i}\) and \(t_{i+1}\), respectively. This formula quantifies the relative change in task execution time between two consecutive time steps and is used to assess the rate of performance change over time. As shown in Figure 6 and Table 5, CMPT exhibited clear advantages in task execution time across the four datasets (D1 to D4). Compared to AABC, AACO, AHPSO, and LRM, CMPT consistently achieved lower execution times, especially for datasets D3 and D4, where it demonstrated better scalability and stability. This performance was due to CMPT's combination of Nash bargaining and Lagrangian dual optimization, which enables efficient resource scheduling and conflict mitigation, significantly reducing task execution time. In contrast, the other algorithms, particularly LRM, showed a substantial increase in execution

Figure 5: Objective function values in 10 time steps. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dataset** & **Satellite Nodes** & **Tasks** & **Paths** \\ \hline D1 & 5 & 10 & 10 \\ D2 & 10 & 25 & 25 \\ D3 & 100 & 1000 & 1000 \\ D4 & 200 & 2000 & 2000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Configuration of selected datasets. time as the dataset size grew, highlighting their limited adaptability to large-scale and complex scenarios. Figure 7 illustrates the growth rate of task execution time, where CMPT also demonstrated exceptional stability. While AHPSO achieved lower growth rates for D1 and D2, it exhibited a sharp increase in D3, reflecting significant fluctuations. In contrast, CMPT maintained consistently low and stable growth rates across all datasets, showcasing its strong adaptability to dynamic network environments. This stability is enabled by CMPT's domain clustering management and multi-path transmission strategies, which optimize path selection and task allocation in real time, effectively reducing the latency caused by link failures or resource contention. On the other hand, AABC and LRM exhibited greater fluctuations, struggling to maintain consistency and control over growth rates.