## 5 The Relationship between Energy-Efficiency and Interference-Optimality
### Homogenous Node Deployment Mode  
**Theorem 7**: _In the homogeneous networks, when the network is energy-efficient, the probabilities that the network is also interference-optimal are from 0.972 to 0.998._  
**Proof.** As discussed in Section 4.1, in homogeneous networks, the feasible region of interference-optimality is shown in Figure 9. Considering the Inequations (40) and (17), the feasible region of both energy-efficiency and interference-optimality can be found in Figure 16. The boundary conditions of energy-efficiency and interference-optimality in Figure 16 are shown in Equations (41) and (42), respectively:  
\[d_{e}=(r^{\gamma}-r_{1}{}^{\gamma})^{\frac{1}{\gamma}}\,, \tag{41}\]  
\[d_{i}=\left\{\begin{array}{l}0.19(\sqrt{r_{1}{}^{2}+27.67r^{2}}-r),(0<r_{1}< 0.85r)\\ 2.63\cdot(\frac{r^{2}}{r_{1}}-r_{1}),(0.85r<r_{1}<r)\\ \end{array}\right., \tag{42}\]  
Figure 15: Transmission range adjustment in energy-efficiency analysis.  
Figure 17 illustrates the relationship between energy-efficiency and interference-optimality. From Figures 16 and 17, we can find that with the increasing of the distance-power gradient \(\gamma\), the feasible region of energy-efficiency increases; moreover, the probability that the requirements of interference-optimality and energy-efficiency cannot be satisfied simultaneous increases, too.  
The coordinates of \(E\), \(F\) and \(H\) in Figure 16 are shown in Table 3.  
Therefore, as shown in Figure 16, when the network is energy-efficient, the value ranges that the network is not interference-optimal at the same time are the region (\(AE\), \(\widetilde{AE}\)) and region (\(FC\), \(\widetilde{FC}\)).  
Figure 16: The feasible region of energy-efficiency and interference-optimality in homogeneous networks.  
Figure 17: The boundary of energy-efficiency and interference-optimality in homogeneous networks.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Coordinates** & \(\mathbf{F}\) & \(\mathbf{E}\) & \(\mathbf{H}\) \\ \hline \(\gamma=2\) & \((0.94r,\,0.36r)\) & \((0.36r,\,0.94r)\) & \((0.71r,\,0.71r)\) \\ \(\gamma=3\) & \((0.88r,\,0.68r)\) & \((0.68r,\,0.88r)\) & \((0.79r,\,0.79r)\) \\ \(\gamma=4\) & \((0.86r,\,0.83r)\) & \((0.83r,\,0.86r)\) & \((0.84r,\,0.84r)\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The coordinates of \(E\), \(F\) and \(H\).  
Moreover, these two regions are equal since they are symmetrical to line \(OB\). Therefore, the probabilities can be calculated as:  
(1) The minimum probability can be calculated when \(\gamma=2\):  
\[\begin{array}{l}p_{\min}=2\cdot\int_{0}^{0.36r}\left(e^{-\pi\rho(r^{2}-{r_{1}} ^{2})}-e^{-\pi\rho(0.19\cdot\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)^{2}}\right)\cdot 2 \pi\rho r_{1}\cdot e^{-\pi\rho{r_{1}}^{2}}dr_{1}\\ =0.002\end{array}\, \tag{43}\]  
(2) The maximum probability can be calculated when \(\gamma=4\):  
\[\begin{array}{l}p_{\max}=2\cdot\int_{0}^{0.83r}\left(e^{-\pi\rho(r^{4}-{r_{1 }}^{4})\frac{1}{2}}-e^{-\pi\rho(0.19\cdot\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)^{2}} \right)\cdot 2\pi\rho r_{1}\cdot e^{-\pi\rho{r_{1}}^{2}}dr_{1}\\ =0.028\end{array}. \tag{44}\]  
According to Equations (43) and (44), when the network is energy-efficient, the probabilities that the network is also interference-optimal are from 0.972 to 0.998, which are pretty high. So in most cases, we can observe that the energy-efficient network is approximately an interference-optimal network.