## 4 Network Interference Probability Analysis
### Homogenous Node Deployment Probability Analysis  
In homogenous WSNs, all the nodes have the same transmission ranges and the transmission ranges are circular. **Theorem 1**. : _In homogenous WSNs, the relationship between the overlapping coverage area \(s\) and the node distance \(d\) is approximately linear, which is \(s=\pi\sigma^{2}-1.93r\cdot d\)._  
**Proof**. As shown in Figure 5, the distance between node \(u\) and node \(v\) is \(d\); the intersection points of the overlapping coverage area are \(A\) and \(B\), respectively. Thus, the acree of the shade area in Figure 5 can be calculated as:  
\[s=2\cdot r^{2}\cdot(\arccos\frac{d}{2r})-d\cdot\sqrt{r^{2}-\frac{d^{2}}{4}} \tag{5}\]  
where \(r\) is the transmission radius. Based on the Taylor Formula, the Taylor Expansion of Equation (5) is:  
\[s_{t}=\pi r^{2}-(r+1)\cdot d+o[(\frac{d}{2r})^{3}] \tag{6}\]  
where \(\frac{d}{2r}\in(0,\frac{1}{2})\); since \(o[(\frac{d}{2r})^{3}]\) is small, then the relationship between the overlapping coverage area \(s\) and the node distance \(d\) is approximately linear according to Equation (6). For simplifying the calculation of Equation (5), we assume:  
\[f=\pi r^{2}-kr\cdot d \tag{7}\]  
Additionally, we define the difference between \(s\) and \(f\) is:  
\[y =s-f \tag{8}\] \[=2\cdot r^{2}\cdot(\arccos\frac{d}{2r})-d\cdot\sqrt{r^{2}-\frac{d^ {2}}{4}}-(\pi r^{2}-kr\cdot d)\enspace^{\prime}\]  
The variance of the difference \(y\) is:  
\[D(y)=E(y^{2})-[E(y)]^{2} \tag{9}\]  
\[D(y)|_{k}^{\prime}=E(y^{2})|_{k}^{\prime}-[E(y)]^{2}|_{k}^{\prime} \tag{10}\]  
where \(E(\cdot)\) is the operator of mathematical expectation. In Equation (10), when \(0<k<1.93\), \(D(y)\left|_{k}^{\prime}<0\right.\); when \(k>1.93\), \(D(y)\left|_{k}^{\prime}>0\right.\); therefore, when \(k=1.93\), the \(D(y)\) can get the minimum value. This conclusion also can be found in Figure 6.  
Figure 4: Interference after adjusting the transmission range. From Figure 6, we can conclude that with the increasing of \(k\), the variance will obtain the minimum value when \(k\) is 1.93, which is consistent with the theoretical analysis in Equation (10). Figure 7 indicates that when the value of \(k\) is 1.93, the evaluation error between \(s\) and \(f\) is smaller than 0.015, which is small enough to be ignored. In addition, when \(k=1.93\), the variance and evaluation error are minimum simultaneously. Therefore, the Equation (11) can be used to evaluate two nodes' overlapping coverage area in homogenous WSNs. The simulation results of Equations (5) and (11) are shown in Figure 8. In Figure 8, the Equations (5) and (11) are fitted with small evaluation error. This indicates that in homogenous WSNs, the relationship between the overlapping coverage area and the node distance is linear. Even this conclusion is obtained under the assumption that \(r=100\); however, when the value of \(r\) changes, we can get the same conclusion as Equation (11). **Theorem 2**. : _In homogeneous networks, the probability of interference-optimality by adjusting the transmission range is:_  
\[s_{t}=\pi r^{2}-(r+1)\cdot d+o[(\frac{d}{2r})^{3}] \tag{6}\]0 \[s_{t}=\pi r^{2}-(r+1)\cdot d+o[(\frac{d}{2r})^{3}] \tag{6}\]1  
**Proof. ** Considering the conclusion of Theorem 1, the coverage area before transmission range adjustment (shown in Figure 2a) is:  
\[s_{t}=\pi r^{2}-(r+1)\cdot d+o[(\frac{d}{2r})^{3}] \tag{6}\]2 \[s_{t}=\pi r^{2}-(r+1)\cdot d+o[(\frac{d}{2r})^{3}] \tag{6}\]3  
where \(r\) is the original transmission range, \(d\) is the distance between two nodes. When the transmission ranges are reduced, the network topology is shown in Figure 2b. In Figure 2b, the distance (\(u\), \(n\)) is \(d_{0}\) and the distance (\(n\), \(v\)) is \(d_{1}\).