## 4 Parameter Effects on VFA-SF and VFA-SF-OPT
### Effect of Node Velocity  
Node velocity is calculated at every time step, while node locations and spring forces are determined from the VFA-SF or VFA-SF-OPT. A series of theoretical exercises were carried out \begin{table}
\begin{tabular}{l l} \hline \hline
**Category** & **Parameter Values** \\ \hline Basic parameters & \(N=500,\ m=1,\ R_{g}=1,\ R_{c}=3,\ D_{m}=\sqrt{3}\) \\  & \(F_{enti}=0.05,\ dt=0.08\) \\ Spring coefficient & \(\kappa=15\) \\ Damping coefficient & \(\gamma_{critical}=\sqrt{4 km}=7.75\) \\  & \(\gamma=1.08\times\gamma_{critical}\) (0-300 time steps) \\  & \(\gamma=\gamma_{critical}\) ( after 300 time steps) \\ External central force & \(F_{exten}=2.873\), the theoretical value in the reference [13] \\  & \(F_{exten}=2.35,5.5\) in Figure 6 \\ Radius to calculate PCD & \(\tau_{T}=\)18, 15, 10, respectively in Figure 1 of Section 4.1 \\  & \(\tau_{T}=\)18 used in Section 4.2 \\  & \(\tau_{T}=\)10 used in Section 4.3 \\  & red dashed circles in Section 4.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Initial parameters used in simulation experiments of following subsections. Note: PCD, pair correlation diversion. Figure 1: Final node distributions by the VFA-SF for indicated node equilibrium distance \(D_{m}=1.732\), 1.5, and 1.3, respectively. Corresponding coverage area by these 500 nodes is 1299, 974.1 and 732, respectively. to prove the validation of two algorithms, but accuracy is very important to optimal coverage. How node velocity can potentially affect deployment performance needs to be further discussed. Here, the experiments and statistical analysis were implemented to show the effect of node velocity on the VFA-SF (external central force \(F_{extern}=0\)). Sensor networks used to have coverage holes or twisted structures during the deployment process, especially at the early stage. The distance between two nodes in such a region is possibly much larger than the average distance. The corresponding node velocity, determined by spring force, can be large. Therefore, maximum node velocity has to be limited in the virtual-spring-force algorithms in case the sensor node has an unreasonable speed. In the simulations, maximum node velocity \(v_{max}\) is given as a constant. If the determined node velocity is larger than the fixed value of \(v_{max}\) at one time step, it would be reset to be \(v_{max}\). Figure 2 presents the pair correlation diversion as a time-step function for indicated maximum node velocity \(v_{max}=0.05,0.1,0.2,0.4,0.6,1.0\), and 2.0, respectively. The comparison is based on the identical initial node distribution. When \(v_{max}\) is very small, like \(v_{max}=0.05\), the PCD curve fluctuates at the early stage of the VFA-SF process. It decreases to 0.3 at the 2500 th time step, and finally stays around 0.2. The reason is that a large number of sensor nodes can easily reach the limitation of a small \(v_{max}\), which significantly responds to a short moving distance for each sensor at any time step. In such a case, sensor nodes have to spend more time and consume more energy in the deployment process to form quasihexagonal topology. When \(0.1\leq v_{max}\leq 0.4\), the PCD curves can finally stay around 0-0.2, which means that a medium \(v_{max}\) helps improve the effect of node deployment in some extent. However, this improvement is still unstable. When \(v_{max}\geq 0.6\), the final WSN deployment successfully forms a perfect hexagonal topology (PCD value goes to 0). Most importantly, it should be noted that the PCD curves converge quicker as the \(v_{max}\) limit increases. Figure 3 shows the variation of convergence time as a function of \(v_{max}\) for the case of PCD = 0.4. Based on statistical analysis, 0.4 is a typical value for the PCD performance, in which the network nearly forms a quasihexagonal topology but still has holes or twisted structures in some small regions. After that, designers can use the VFA-SF-OPT to further optimize deployment.