# Adaptive Network Routing Technology for Near-Moon Space Cross-Domain Transmission
## 2 Hybrid Link Resource Prediction and Scheduling Model
### Communication Networking Model  
In this design, we consider that all communication nodes are equal and propose a communication data transmission method based on probability, as well as an optimal routing search method based on this model. The notations and definitions are listed in Abbreviations. The entire networking system is divided into four communication domains, namely, the lunar node domain \(U_{m}\), circumlunar node domain \(U_{a}\), near-Earth node domain \(U_{c}\), and Earth-Moon transfer node domain \(U_{t}\), with \(U=\{U_{a},U_{c},U_{t},U_{m}\}\). The system contains \(M=M_{m}+M_{a}+M_{e}+M_{t}\) satellites, where \(M_{m}\), \(M_{a}\), \(M_{e}\) and \(M_{t}\) represent the number of communication nodes in different domains. The near-Earth node domain \(U_{e}\), circumlunar node domain \(U_{a}\), and Earth-Moon transfer node domain \(U_{t}\) maintain relatively stable movement, which could be modeled as the two clusters of fixed topology sub-networks \(T_{e,l}(i\in[0,Q_{e,i}-1],i\in Z^{+})\), \(T_{a,j}(j\in[0,Q_{a,i}-1],j\in Z^{+})\), and \(T_{s,k}(k\in[0,Q_{t,k}-1],k\in Z^{+})\), where \(\sum Q_{e,i}=M_{e}\), \(\sum Q_{a,i}=M_{a}\), \(\sum Q_{t,k}=M_{t}\), and \(Q\) represents the node number of a single network. Then, the fixed topology sub-network group can be denoted as \(T_{F}=\{\widetilde{T}_{e},\widetilde{T}_{a},\widetilde{T}_{s}\}=\{T_{c,i},T_ {a,j},T_{s,k}\}\), in which the data transfer in the three domains depends on the scheduled routing principle, and the inter-data transfer of domains relies on the fixed relay nodes. It is difficult to maintain a fixed relative position in lunar node domain \(U_{m}\) for a long time, and it belongs to the time-varying topology structure \(T_{m,p}(p\in[0,Q_{m,p}-1],p\in Z^{+})\). Therefore, the time-varying topological sub-network set is labeled as \(T_{V}=\{\widetilde{T}_{m}\}=\{T_{m,p}\}\). The interaction between the time-varying topology structure and the fixed topology sub-network set also requires interaction through fixed relay communication nodes, which may not be a single node. It is necessary to clarify the data path relationship between two clusters. The communication interaction between two different sub-networks \(\widetilde{T}_{i}\) and \(\widetilde{T}_{j}\) is denoted as \(I_{(\widetilde{T}_{i},\widetilde{T}_{j})}\). At any time t, the sub-network interaction communication relationship is as follows:  
\[I_{(\widetilde{T}_{i},\widetilde{T}_{j})}(t)=\bigcup C_{(q_{0},q_{1},q_{0}^{ \prime},q_{1}})(t) \tag{1}\]  
where \(q_{0},q_{1}\in\widetilde{\widetilde{T}_{i}},q_{0},q_{1}\in\widetilde{T}_{j}\). \(\cup\) represents the aggregation operation, and the communication channel between any two links is \(C_{(q_{0},q_{1},q_{0}^{\prime},q_{1}})(t)\). The communication rate function is \(f(q_{0},q_{1},q_{0},q_{1},t)\), which represents the communication rate between the \(q_{1}\)-th channel (\(n\in[0,N_{m}-1]\)) of the \(q_{0}\)-th node and the \(q_{1}\)'-th channel of the \(q_{0}\)'-th node at time \(t\), where \(q_{0}\
eq q_{0}\)'. The complete description of the optimal communication topology architecture for cross-domain networks is as follows: There is data interaction and communication between each node domain, and the total communication efficiency \(E_{C}\) is proportional to the reciprocal of the communication rate variance \(\xi/\delta_{f}\) between each node. The input output balance of communication rate is guaranteed by a single node and planned in advance, where \(\xi\) represents the efficiency coefficient and \(\delta_{f}\) represents the communication rate variance of each node:  
\[\delta_{f}=\text{var}\{f(q_{\Psi},q_{\
u},q_{\psi}{}^{\prime},q_{\
u}{}^{ \prime},t)\},\psi\in[0,2,\cdots,10],\upsilon\in[1,3,\cdots,11] \tag{2}\]  
Therefore, under the optimal communication topology architecture conditions, it is hoped that each node will achieve a rate balance as much as possible to reduce the additional increase in individual node power and storage costs caused by rate differences.