## 4 Network Interference Probability Analysis
### Homogenous Node Deployment Probability Analysis
In this situation, the feasible region is \(DBA\) in Figure 9. The whole feasible region is triangle \(ABC\) which can be found in the shade area of Figure 9.  
In Figure 9, when \(r_{1}\) and \(d_{1}\) equal to the values in shade area, the network interference will decrease; otherwise, the interference will increase. In addition, the coordinate of \(B\) is (\(0.85r\), \(0.85r\)), so when \(0<r_{1}<0.85r\), the value range of \(d_{1}\) is (\(r-r_{1}\), \(0.19(\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)\)); when \(0.85r<r_{1}<r\), the value range is (\(r-r_{1}\), \(2.63(\frac{r^{2}}{r_{1}}-r_{1})\)), where \(AC\) represents the line \(r-r_{1}\) in Figure 9.  
In [27], the authors have proved that if the nodes are uniformly distributed in the network, the probability density function (PDF) of the neighbor distance is Rayleigh Distribution:  
\[f(d)=2\pi\rho d\cdot e^{-\pi\rho d^{2}}(d>0), \tag{18}\]  
where \(d\) is the distance between two nodes, and \(\rho\) is the node density. Thus the probability that the neighbors' distances are larger than \(d_{1}\) and smaller than \(d_{2}\) can be calculated as:  
\[P(d_{1}\leq r\leq d_{2})=\int_{d_{1}}^{d_{2}}f(x)dx=e^{-\pi\rho d_{1}^{2}}-e^{ -\pi\rho d_{2}^{2}}, \tag{19}\]  
As shown in Figure 9, the probability that the distance between two nodes, \(r_{1}\), can be calculated by Equation (18). The probability that \(d_{1}\) satisfies the constraints shown in Figure 9 can be calculated by Equation (19). Then the probability that the network is interference-optimal by adjusting the transmission range is:  
\[ \begin{split} p_{\text{hom}}&=\int_{0}^{0.85r}p(r- r_{1}\leq d_{1}\leq 0.19\cdot\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)+\int_{0.85r}^{r}p(r- r_{1}\leq d_{1}\leq 2.63(\frac{r^{2}}{r_{1}}-r_{1}))\\ &=\int_{0}^{0.85r}{(e^{-\pi\rho(r-r_{1})^{2}}-e^{-\pi\rho(0.19 \cdot\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)^{2}})}dr_{1}+\int_{0.85r}^{r}{(e^{-\pi \rho(r-r_{1})^{2}}-e^{-\pi\rho(\frac{263}{r_{1}}r^{2}-2.63r_{1})^{2}})}dr_{1} \end{split} \, \tag{20}\]  
The Equation (20) indicates that the probability relates to both the original transmission range \(r\) and the node density \(\rho\). The properties of this probability under different \(r\) and \(\rho\) will be investigated in the simulation part in detail. However, the probability shown in Equation (20) is just a general probability. This probability measures the possibility that, for the different original transmission ranges, when the transmission ranges are reduced, there exist optimal solutions which can guarantee low network interference. However, the probability of interference-optimality under specific transmission range is different with that in Equation (20). **Theorem 3**. : _In homogeneous networks, the probability that the network is interference-optimal under specific transmission range can be calculated by:_  
\[P_{\text{homs}}=\left\{ \begin{array}{l}\int_{r-r_{1}}^{0.19(\sqrt{{r_{1}}^ {2}+27.67r^{2}}-r)}2\pi\rho d_{1}\cdot e^{-\pi\rho d_{1}^{2}}dd_{1},(0<r_{1}<0. 85r)\\ \int_{r-r_{1}}^{2.63(\frac{r^{2}}{r_{1}}-r_{1})}2\pi\rho d_{1}\cdot e^{-\pi\rho d _{1}^{2}}dd_{1},(0.85r<r_{1}<r)\end{array} \right.\]  
**Proof. ** As shown in Figure 9, when, \(0<r_{1}<0.85r\), the value range of \(d_{1}\) is (\(r-r_{1}\), \(0.19(\sqrt{{r_{1}}^{2}+27.67r^{2}}-r)\)); when \(0.85r<r_{1}<r\), the value range is (\(r-r_{1}\), \(2.63(\frac{r^{2}}{r_{1}}-r_{1})\)). According to the probability density function (PDF) (18), the integral of the PDF shown in (18) on the value range of \(d_{1}\) represents the probability of interference-optimality under specific transmission range, which can be calculated as:  
\[P_{\text{homs}}=\left\{ \begin{array}{l}\int_{r-r_{1}}^{0.19(\sqrt{r_{1}^{2}+27. 67r^{2}}-r)}2\pi pd_{1}\cdot e^{-\pi pd_{1}{}^{2}}dd_{1},(0<r_{1}<0.85r)\\ \int_{r-r_{1}}^{2.63(\frac{r}{r_{1}}-r_{1})}2\pi pd_{1}\cdot e^{-\pi pd_{1}{}^ {2}}dd_{1},(0.85r<r_{1}<r)\end{array} \right., \tag{21}\]  
This probability means that under specific transmission range, the possibility that the network is interference-optimal. **Corollary 1.