# A Practical Neighbor Discovery Framework for Wireless Sensor Networks
## 7 Evaluations
### Increasing Discovery Rate  
In the network with multiple nodes, communication collisions could affect the discovery results. We evaluate the performance of the proposed collision reducing methods (PPR and DPR in Section 6), and compare them with the bare (not running in Spear) algorithms.  
**Number of neighbors**. In a star network, the central node \(u_{c}\) has \(|N_{c}|\) neighbors and it selects duty cycle \(\theta_{c}=0.3\). We select Hedis as the example and set \(p_{1}=p_{2}=0.5\) for PPR and DPR. Figure 4 shows the discovery rate (\(y\)-axis) of \(u_{c}\) within \(100,000\) time slots when \(|N_{c}|\) (\(x\)-axis) increases from 1 to 100. From the figure, (bare) Hedis cannot discover all neighbors when \(|N_{c}|\geq 7\), and it cannot find even one neighbor when \(|N_{c}|\geq 18\). Modified by PPR and DPR, all neighbors can be discovered when \(|N_{c}|\leq 43\) and \(|N_{c}|\leq 33\), respectively. We also evaluate the performance of PPR and DPR at \(p_{1}=0.4\) and \(p_{2}=0.2\), and they outperform the methods when \(p_{1}=p_{2}=0.5\).  
**Sensitivity of \(p_{1}\)**. In a star network, the central node \(u_{c}\) has \(|N_{c}|=20\) neighbors (we set \(|N_{c}|=20\) since the bare algorithms fail to find any neighbor) and we evaluate PPR's performance under different values of \(p_{1}\). As shown in Figure 5a, \(\theta_{c}\) is set to 0.3 and the average discovery latency (\(y\)-axis) of different algorithms is changed by different values of \(p_{1}\) (\(x\)-axis). When \(p_{1}\in[0.3,0.4]\), the performance is better. In Figure 5b, we select U-Connect as the example and set \(\theta_{c}\) as \(0.1,0.2,0.3,0.4,0.5\), respectively. The average discovery latency is changed by different values of \(p_{1}\) (\(x\)-axis), and the performance is also better when \(p_{1}\) approaches \([0.3,0.4]\). Overall, our discovery latency is stable when \(p_{1}\leq 0.6\).  
Figure 4: Spear increases discovery rate by incorporating PPR and DPR.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Figures** & \multicolumn{4}{c}{**Parameters**} \\ \hline All figures & \(d_{c}=50\) m & \(P_{max}=100,000\) & \(p_{n}=1\) & \(t_{0}=20\) ms \\ Figure 4 & \(|N_{c}|\in[1,100]\) & \(\theta_{c}=0.3\) & \(p_{1}=0.5\) & \(p_{2}=0.5\) \\ Figure 5 & \(|N_{c}|=20\) & \(\theta_{c}=0.3\) & \(p_{1}\in[0.1,0.9]\)  
**Sensitivity of \(p_{2}\).** In a star network, the central node \(u_{c}\) has \(|N_{c}|=20\) neighbors and the DPR's performance is evaluated under different values of \(p_{2}\). As shown in Figure 6a, \(\theta_{c}\) is set to 0.3 and the average discovery latency (\(y\)-axis) of different algorithms are changed by different values of \(p_{2}\) (\(x\)-axis). When \(p_{2}\) is close to 0.2, the performance is better. In Figure 6b, we select U-Connect for different \(\theta_{c}\) (\(0.1,0.2,0.3,0.4,0.5\), respectively). The average discovery latency (\(y\)-axis) is changed by different values of \(p_{2}\) (\(x\)-axis), and the performance is better when \(p_{2}\approx 0.2\). Overall, our discovery latency is stable when \(p_{2}\leq 0.7\).  
**1000 nodes**. In a randomly generated network with \(N=1000\) nodes, we evaluate the performances of different algorithms. As shown in Figure 1, modified by PPR (\(p_{1}=0.4\)) and DPR (\(p_{2}=0.2\)), the discovery rates (\(y\)-axis) are much larger than those of the bare algorithms. Especially for Hello, the discovery rate is only 33.0%, while PPR and DPR could greatly increase the rate to 99.2%, 95.5%, respectively.