## 3 Problem Definition and Formulation  
### System Model  
The proposed satellite network comprises three primary components: satellite nodes, ground stations, and user terminals. Each satellite node \(i\) is characterized by its resource vector \(R_{i}=(C_{i},B_{i},Q_{i})\), where \(C_{i}\) denotes the computational capacity, \(B_{i}\) represents the available bandwidth, and \(Q_{i}\) signifies the storage capacity. The network topology is encapsulated by the adjacency matrix \(A=[a_{ij}]\), where each element \(a_{ij}\) defines the connection attributes between satellite nodes \(i\) and \(j\). Specifically, \(a_{ij}^{B}\) denotes the bandwidth and \(a_{ij}^{L}\) denotes the latency between node \(i\) and node \(j\), respectively. The satellite network employs a SDN controller to enable centralized management and dynamic resource scheduling. Acting as the global coordination center, the SDN controller continuously monitors the resource status of satellite nodes--including the computing capacity \(C_{i}\), bandwidth \(B_{i}\), and storage \(Q_{i}\)--as well as link parameters such as transmission delay \(a_{ij}^{T}\) and bandwidth \(a_{ij}^{B}\), along with task queue information. Its core functions encompass dynamic domain clustering, multi-path optimization, and resource reallocation. Specifically, each satellite node periodically reports its local resource utilization, link reliability \(R_{ij}\) (Equation (32)), and task execution status to the SDN controller via dedicated control channels. The controller aggregates this global information and updates the real-time network topology graph \(\mathcal{G}\). Based on the current network state, the controller performs dynamic domain clustering by optimizing the utility function \(U_{i}=\frac{1}{1+\alpha\frac{d_{i}^{2}}{d_{i}^{2}}}\cdot\sum_{j\in D_{k}}R_{ij}\) as defined in Equation (30), determining the affiliation of nodes to logical domains \(D_{k}\). The clustering process minimizes the weighted objective \(\min_{D}[\lambda(1-R)+(1-\lambda)\cdot\frac{\eta_{\text{task}}}{\eta_{\text{ max}}}]\) to balance intra-domain link reliability and overall network overhead, ensuring proximity-based cooperation among nodes, while reducing signaling costs. Subsequently, for each task \(j\), the controller dynamically selects the optimal path \(P_{jk}\) from the available path set \(\mathcal{P}\) based on real-time link conditions and node workloads, considering the path reliability (Equation (33)). The path assignment is then disseminated to the corresponding satellite nodes through the binary variable \(x_{jk}\) (Equation (2)). During task execution, if a resource overload (\(\sum t_{ij}>R_{i}\)) or link failures are detected, the controller triggers Nash bargaining models (Algorithm 1) and Lagrangian optimization (Algorithm 2) to adjust the resource allocation \(t_{ij}\) and update the task matrix \(T\), ensuring all system constraints (Equations (3)-(9)) are satisfied. Through this mechanism, the SDN controller achieves real-time adaptation of the network topology and efficient resource scheduling, significantly enhancing the robustness and task execution efficiency of the satellite network in dynamic environments. Data transmission initiates from user terminals, traverses through ground stations, and culminates in task requests directed to the satellite nodes. Task allocation is modeled by the matrix \(T=[t_{ij}]\), where \(t_{ij}\) represents the quantity of resources allocated to task \(j\) by satellite node \(i\). To ensure the physical realizability of resource allocation variables, this paper imposes the non-negativity assumption:  
\[t_{ij}\geq 0,\,\forall i\in\{1,\ldots,n\},\forall j\in\{1,\ldots,m\} \tag{1}\]  
A multi-path scheduling strategy facilitates parallel data transmission across multiple pathways, mitigating packet loss and reducing latency. Formally, for each task \(j\), a possible path \(P_{jk}\in\mathcal{P}_{j}\) represents a potential transmission route that can be used to transmit data from the source to the destination in the network.