# Jamming Analysis between Non-Cooperative Mega-Constellations Based on Satellite Network Capacity
## 3 Satellite Network Capacity Calculation Method
### Average Hop Count  
The topology of an inclined orbit constellation has symmetry. Take the center node as the source node (e.g., \(\left(\frac{N}{2},\frac{M}{2}\right)\) if \(N\) and \(M\) are even). Either of the other nodes is equivalent to \(\left(\frac{N}{2},\frac{M}{2}\right)\) because either node can be rotated to the center by rotation. Take the example that both \(N\) and \(M\) are even. When \(\left(\frac{N}{2},\frac{M}{2}\right)\) sends data to other nodes, the total hop count is  
\[\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}\left(\left|i-\frac{N}{2}\right| +\left|j-\frac{M}{2}\right|\right) \tag{18}\]  
The topology of \(\Psi_{A}\) can be divided into four regions, i.e., the lower left, lower right, upper left and upper right, as shown in Figure 3. The hop counts for each region are \(H_{ll}\), \(H_{lr}\), \(H_{ul}\) and \(H_{ur}\), respectively. \[\left\{ \begin{array}{l}H_{ll}=\sum\limits_{i=0}^{N/2}\sum\limits_{j=0}^{M/2 }\left(\left|i-\frac{N}{2}\right|+\left|j-\frac{M}{2}\right|\right)\\ H_{lr}=\sum\limits_{i=N/2}^{N-1}\sum\limits_{j=0}^{M/2}\left(\left|i-\frac{N} {2}\right|+\left|j-\frac{M}{2}\right|\right)\\ H_{ul}=\sum\limits_{i=0}^{N/2}\sum\limits_{j=M/2}^{M-1}\left(\left|i-\frac{N} {2}\right|+\left|j-\frac{M}{2}\right|\right)\\ H_{ur}=\sum\limits_{i=N/2}^{N-1}\sum\limits_{j=M/2}^{M-1}\left(\left|i-\frac{N} {2}\right|+\left|j-\frac{M}{2}\right|\right)\end{array} \right. \tag{19}\]  
Therefore, the total hop count \(H\) is  
\[ \begin{array}{l}H=H_{ll}+H_{lr}+H_{ul}+H_{ur}\\ =\sum\limits_{i=0}^{N-1}\sum\limits_{j=0}^{M-1}\left(\left|i-\frac{N}{2}\right| +\left|j-\frac{M}{2}\right|\right)\\ =\sum\limits_{i=0}^{N-1}\left|i-\frac{N}{2}\right|+\sum\limits_{j=0}^{M-1} \left|j-\frac{M}{2}\right|\\ =\frac{NM^{2}+N^{2}M}{4}\end{array}  \tag{20}\]Thus, the average hop count \(E(H)\) of each ISL is  
\[E(H)=\frac{NM^{2}+N^{2}M}{4(NM-1)} \tag{21}\]  
The above is the case when both \(N\) and \(M\) are even. Other cases are similar. Thus, the average hop count of each ISL is, respectively,  
\[E(H)=\left\{ \begin{array}{l}\frac{NM^{2}+N^{2}M}{4(NM-1)},N=2n,M=2n\\ \frac{N(M^{2}-1)+N^{2}M}{4(NM-1)},N=2n,M=2n+1\\ \frac{NM^{2}+(N^{2}-1)M}{4(NM-1)},N=2n+1,M=2n\\ \frac{N(M^{2}-1)+(N^{2}-1)M}{4(NM-1)},N=2n+1,M=2n+1\end{array} \right. \tag{22}\]  
Therefore, we obtain the network capacity \(C_{A}\) of \(\Psi_{A}\) by dividing the transmission rate \(R\) by the average hop count \(E(H)\). \[C_{A}=\frac{R}{E(H)} \tag{23}\]  
Thus, when the link rates in \(\Psi_{A}\) change dynamically, the network capacity \(C_{A}\) is defined as follows:  
\[C_{A}=\left\{ \begin{array}{l}\frac{4}{N+M}(R_{o}+R_{h}),N=2n+1,M=2n+1\\ \frac{4(NM-1)}{NM^{2}+NM}(R_{o}+R_{h}),N=2n,M=2n\\ \frac{4(NM-1)}{NM^{2}+(N^{2}-1)M}(R_{o}+R_{h}),N=2n+1,M=2n\\ \frac{4(NM-1)}{N(M^{2}-1)+N^{2}M}(R_{o}+R_{h}),N=2n,M=2n+1\end{array} \right. \tag{24}\]  
where \(R_{o}\) and \(R_{h}\) denote the sums of the actual transmission rates of intra-plane and inter-plane ISLs, respectively. \[R_{o}=\sum_{i=0}^{N-1}\sum_{j=0}^{M-1}\sum_{k=0}^{\frac{M-1}{2}}(\frac{M-1}{2} -k)(P+Q) \tag{25}\]  
\[R_{h}=\sum_{i=0}^{N-1}\sum_{j=0}^{M-1}\sum_{k=0}^{\frac{N-1}{2}}(\frac{N-1}{2} -k)(U+V) \tag{26}\]  
where \(P=\tilde{R}^{\
earrow}_{i,(j-k)\bmod M,t^{\prime}}\), \(Q=\tilde{R}^{\
earrow}_{i,(j+k)\bmod M,t^{\prime}}\), \(U=\tilde{R}^{\searrow}_{(i-k)\bmod N,j,t^{\prime}}\), \(V=\tilde{R}^{\searrow}_{(i+k)\bmod N,j,t^{\prime}}\). In summary, network capacity \(C_{A}\) is a function of the satellite transmission rate. Denote the network capacity function as \(\Gamma\); then, \(C_{A}=\Gamma(R^{\
earrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t^{\prime}},R^{ \searrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t})\). Therefore, the network capacities under jamming-free conditions and non-cooperative jamming conditions are \(C_{A}=\Gamma(R^{\
earrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t^{\prime}},R^{ \searrow}_{i,j,t})\) and \(\tilde{C}_{A}=\Gamma(R^{\
earrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t^{\prime}},R^{\searrow}_{i,j,t^{\prime}})\), respectively.