## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.6 Problem Modeling and Optimization Algorithm Design
The Adaptive Ant Colony Optimization Algorithm (AACO) [41], with a time complexity of \(O(T\cdot k\cdot(2n+m))\), where \(T\) is the number of iterations, \(k\) is the number of ants, \(n\) is the number of nodes, and \(m\) is the number of edges, optimizes resource allocation through pheromone-based path selection. The Adaptive Ant Bee Colony Algorithm (AABC) [42] combines ant and bee colony behaviors for global and local optimization, exhibiting a time complexity of \(O(T\cdot N\cdot D)\), where \(N\) is the population size, \(D\) is the problem dimension, and \(T\) is the number of iterations. The Improved Hybrid Particle Swarm Optimization Algorithm (AHPSO) [43], with a time complexity of \(O(n\cdot T)\), where \(n\) is the number of particles and \(T\) is the maximum number of iterations, integrates swarm intelligence and genetic operators for robust optimization. Lastly, the Linear Regression Model (LRM) [44], with a time complexity of \(O(k\cdot 2^{n}\cdot m)\), where \(k\) is the number of iterations, \(n\) is the subproblem size, and \(m\) is the number of constraints, provides a deterministic approach for optimizing resource allocation. These algorithms are well-suited to diverse resource scheduling tasks, balancing complexity with adaptability to dynamic environments. ### Evaluation Indicators

This study employed multiple indicators to comprehensively evaluate the performance of the proposed resource allocation and scheduling algorithms. The evaluation indicators included the objective function value, resource utilization rate, algorithm execution time, task execution time, and objective function value optimization rate. The objective function value served as the core evaluation metric, reflecting the algorithm's overall performance in optimizing network throughput and reducing computational complexity. Specifically, the objective function value measures the overall network efficiency by maximizing network throughput \(T\) and minimizing the difference in resource allocation adjustments \(\sum t_{ij}-t_{ij}^{\prime}\). Task execution time evaluated the time taken for each task from initiation to completion, including the total time for data transmission and processing, represented by the task transmission latency \(d_{j}\). This indicator reflected the system's responsiveness and real-time performance in practical applications. Through comprehensive analysis of these evaluation indicators, the study effectively measured the performance of the proposed algorithms in multi-satellite collaborative computing environments, ensuring their superiority in network throughput, resource utilization efficiency, and task execution efficiency. ### Numerical Analysis

In the four experimental scenarios with different satellite scales (Table 3), CMPT demonstrated a significant advantage in terms of objective function values and growth rate stability, as shown in Figure 4. Across all datasets (D1 to D4), CMPT consistently achieved higher objective function values compared to the other algorithms, including AABC, AACO, AHPSO, and LRM, indicating its effectiveness in enhancing the network throughput and reducing computational complexity in resource allocation and task scheduling. This performance is attributed to its cooperative game-theoretic model and real-time resource allocation strategies, which balance global and local optimization, significantly improving resource utilization efficiency. Moreover, CMPT exhibited superior stability in the growth rate of objective function values across datasets, particularly for D2 and D3, where its growth rates remained consistent. In contrast, algorithms such as LRM experienced significant fluctuations, especially on D3. This stability underscores CMPT's adaptability to diverse resource distributions and topological changes in dynamic satellite networks. By leveraging a Nash bargaining model, CMPT efficiently coordinates resource competition among nodes, ensuring balanced resource allocation and effective load balancing under multiple constraints.