## 2 Network Architecture and Signal Model
### Hybrid Interference Matrix  
The interference matrix \(\mathbf{T}\) is constructed from the following analysis. The antennas on board the satellite receive signals, including the desired signals from SSs intra-beam to be modeled as an equivalent factor 1, interference signals from TCNs intra-beam to be modeled as an equivalent interference factor \(\beta\) and interference signals from SSs inter-beam and TCNs inter-beam respectively to be modeled as equivalent interference factors \(\alpha\) and \(\alpha\beta\), respectively. \(\alpha\) can be used as a substitute for \(\alpha=10^{-4\rho}\), where \(\rho\) is defined as a parameter to vary beam isolation [32]. The \(j\)-th interference signal from the TCNs is assumed to be equivalent to \(\beta_{j}\), and the interference from the TCNs per beam is equivalent to \(\beta=\beta_{1}+...+\beta_{j}\) ( \(1\leq j\leq J\)). From the Annex 4 of International Telecommunications Union-Radio (ITU-R) F.1336-4 [45], the mathematical model of omnidirectional generic average radiation patterns can be approximated and used for terrestrial antenna radiation patterns and are expressed as \(\beta_{j}=\mathrm{G}(\varphi)\). The following average side-lobe patterns should be used for elevation angles that range from \(-90^{\circ}\) to \(90^{\circ}\), given by [45]  
\[G(\varphi)=\left\{ \begin{array}{ll}G_{0}-12\left(\frac{\varphi}{\varphi_{3} }\right)^{2},&0\leq|\varphi|<\varphi_{4}\\ G_{0}-12+10\log\left(t+1\right)+F\left(\varphi\right),&\varphi_{4}\leq|\varphi |<\varphi_{3}\\ G_{0}-12+10\log\left[\left(\frac{|\varphi|}{\varphi_{3}}\right)^{-1.5}+k\right] +F\left(\varphi\right),&\varphi_{3}\leq|\varphi|\leq 90^{\circ}\end{array} \right. \tag{5}\]  
with:  
\[\varphi_{3} =107.6\times 10^{-0.1\mathrm{G}_{0}}\] \[\varphi_{4} =\varphi_{3}\sqrt{1-\frac{1}{12}\log\left(t+1\right)}\] \[G_{0} =2+15\log\left(D/\lambda\right)\] \[F(\varphi) =10\log\left(0.9\mathrm{sin}^{2}\left(\frac{3\pi\varphi}{4 \varphi_{3}}\right)+0.1\right)\]  
where:  
\(G(\varphi)\): gain relative to an isotropic antenna (dBi). \(G_{0}\): the maximum gain in the azimuth plane (dBi). \(\varphi\): elevation angle relative to the angle of the maximum gain (degrees) \((-90^{\circ}\leq\varphi\leq 90^{\circ})\). \(\varphi_{3}\): the 3-dB beamwidth in the elevation plane (degrees). \(t\): in cases involving typical antennas operating in the 1-3 GHz range, the parameter \(t\) should be 0.7; in cases involving antennas with improved side-lobe performance in the 1-3 GHz range and for all antennas operating in the 3-70 GHz range, the parameter \(t\) should be zero. Figure 3 shows the relationships between the substitution \(\alpha\) and beam isolation parameter \(\rho\), the substitution \(\beta\) and elevation angle \(\varphi\). In Figure 3b, we can see that the terrestrial interference for the satellite depends on the terrestrial antenna elevation angle (\(\lambda=0.1\) m, \(D=0.3\) m, where \(\lambda\) is the wavelength and \(D\) is the equivalent antenna diameter); there may be tens of terrestrial communication networks per satellite beam. Whilst \(\alpha\) and \(\beta\) may not reflect the actual characteristics of the beam  
patterns and antenna patterns, they can be utilized to analyze the effects of \(\alpha\) and \(\beta\) on the performance of IWMSNNs. The arrangement of a hybrid 1D beam is \(1\times N\), where \(L=1\) and \(M=N\), as shown in Figure 2. The 1D beam interference matrix can be expressed as \(\mathbf{T}_{\mathrm{1D}}\)[32], while only considering satellite sink interference from the adjacent beams:  
\[\mathbf{T}_{\mathrm{1D}}=\left[ \begin{array}{cccc}1&\alpha&&&\\ \alpha&1&\ddots&\\ &\ddots&\ddots&\alpha\\ &&\alpha&1\end{array} \right] \tag{6}\]  
The 1D terrestrial interference matrix from the TCNs intra-beam and inter-beam can be expressed as \(\mathbf{B}_{\mathrm{1D}}\), while again, only considering the TCNs interference from the adjacent beams:  
\[\mathbf{B}_{\mathrm{1D}}=\beta\left[ \begin{array}{cccc}1&\alpha&&&\\ \alpha&1&\ddots&\\ &\ddots&\ddots&\alpha\\ &&\alpha&1\end{array}