# Resource Mapping Allocation Scheme in 6G Satellite Twin Network
## 3 System Model of 6G Satellite Twin Network
### Function-Resource Mapping Model  
The essence of the communication-twin-computing integration network mapping is to map the communication network topology to the twin network topology and then to the computation network topology. At the same time, the resources are used to abstractly describe the functions. The function \(F\) deployed to the resource \(R\) can be expressed as \(\mathrm{H}(\mathrm{F})\to R\) in the domain. The twin network is taken as an example for analysis.  
The twin function \(F^{S}\) deployed to the twin resource \(R^{S}\) can be expressed as \(\mathrm{H}(F^{S})\to R^{S}\). We design a flexible coefficient autoregressive model based on the basis function matrix to describe the mapping relationship among the networks.  
\[R^{S}\leftarrow\mathrm{H}(F^{S})=B^{S}\times(A\circ F^{S}), \tag{8}\]where \(\text{A}=(a_{1},\cdots a_{I})\) is the matrix containing all linear parameters, \(F^{S}\) indicates a function, and \(B^{S}=(b_{\text{cap}f}^{S},b_{\text{memf}}^{S},b_{\text{band}}^{S})\) is denoted as the parameter of the Gaussian basis function.  
\[A\circ F^{S}=(a_{1},\cdots a_{I})\circ(f_{1}^{S},\cdots,f_{I}^{S})=\sum_{j=1}^{I }a_{j}\cdot f_{1}^{S}+,\cdots,+\sum_{j=1}^{I}a_{j}\cdot f_{j}^{S}, \tag{9}\]  
\[b_{f}^{S}=\exp{(-\frac{\|NUM-c_{f}\|^{2}}{2\sigma_{f}^{2}\cdot\sum_{z=1}^{Z}p_ {z}^{S}})}, \tag{10}\]  
In Equation (14), \(NUM\) is the number of events required by the twin function, \(c_{f}\) is denoted as the number of resources required to complete a unit event, and \(\sigma_{f}\) is denoted as the variance of resources required to complete a unit event. Then, the parameter of the Gaussian basis function can be expressed as follows:  
\[b_{\text{capf}}^{S}=\exp{(-\frac{\|NUM-c_{capf}\|^{2}}{2\sigma_{expf}^{2} \cdot\sum_{z=1}^{Z}p_{z}^{S}})}, \tag{11}\]  
\[b_{\text{memf}}^{S}=\exp{(-\frac{\|NUM-c_{memf}\|^{2}}{2\sigma_{memf}^{2} \cdot\sum_{z=1}^{Z}p_{z}^{S}})}, \tag{12}\]  
\[b_{\text{bandf}}^{S}=\exp{(-\frac{\|NUM-c_{hmf}\|^{2}}{2\sigma_{hmf}^{2} \cdot\sum_{z=1}^{Z}p_{z}^{S}})}, \tag{13}\]  
The mapping relationship of the twin resources and functions can be denoted as  
\[R^{s}=B^{s}\times(A\circ F)= \left(\begin{array}{c}b_{\text{capf}}^{S}\\ b_{\text{memf}}^{S}\end{array}\right)\times\left((a_{1},\cdots a_{I})\cdot f _{1}^{S},\cdots,(a_{1},\cdots a_{I})\cdot f_{I}\right)\] \[=\left(\begin{array}{ccc}\sum\limits_{j=1}^{I}a_{j}\cdot f_{1} ^{S}b_{\text{capf}}^{S}&\cdots&\sum\limits_{j=1}^{I}a_{j}\cdot f_{j}^{S}b_{ \text{capf}}^{S}\\ \sum\limits_{j=1}^{I}a_{j}\cdot f_{1}^{S}b_{\text{memf}}^{S}&\ddots&\sum\limits _{j=1}^{I}a_{j}\cdot f_{j}^{S}b_{\text{memf}}^{S}\\ \sum\limits_{j=1}^{I}a_{j}\cdot f_{1}^{S}b_{\text{bandf}}^{S}&\cdots&\sum\limits _{j=1}^{I}a_{j}\cdot f_{j}^{S}b_{\text{bandf}}^{S}\end{array}\right)=(r_{1}^{ S},\cdots,r_{I}^{S}) \tag{14}\] \[=\left(\begin{array}{ccc}\text{capf}_{1}^{S}&\cdots&\text{capf} _{1}^{S}\\ \text{memf}_{1}^{S}&\cdots&\text{memf}_{1}^{S}\\ b_{\text{bandf}_{1}}^{S}&\cdots&\text{bmf}_{1}^{S}\end{array}\right),\]  
where the row is the CPU, memory, and bandwidth required by the function \(f_{j}^{S}\), and \(a_{j}\) is the mapping factor of the function \(f_{j}^{S}\), with values in [0, 1].