## 4 Simulation Analysis of 3D Network Deployment Experiments
It means that the deployment range of nodes tends to expand and then shrink at the first 500 steps. After 500 steps, all nodes continue to shrink regularly. After 1700 steps, when the utilization rate and the moving distance begin to stabilize, the coverage rate begin to rise slowly. It suggests that our 3D algorithm still has a small-scale adjustment process to obtain a better network structure. Finally, the coverage rate reaches around 0.6 while the whole network has a high utilization. In order to test the validation of the Monte Carlo algorithm on the 3D network coverage rate, Table 1 gives the final values of the coverage rate at the 3000 th time step, for three independent running cases, based on different number of test points \(p_{t}\). It can be seen that, under a different number of test points, the coverage rate does not change much for all three cases. Those very small differences are due to the difference of initial distribution and the calculation error of the Monte Carlo algorithm itself. We think that these small differences can be ignored and the Monte Carlo algorithm does present a good evaluation for the calculation of the system utilization. Figure 4: The variation of the system utilization depending on the calculation time steps. Figure 5: The variation of the system coverage rate as a function of the calculation time steps. Figure 6 shows the variation of the spatial distribution diversion (SDD) value as a function of the calculation time steps. We can observe that, when the nodes in the initial state are randomly distributed at the beginning, the network structure is very chaotic and the SDD value is large. During the first 200 time steps, the network quickly tends to be regular and the SDD value decreases sharply. During the 200-500 steps, all nodes have a process of reverse acceleration due to the action of the formula. According to the above discussions on the system utilization and the coverage rate, it also confirms that all nodes do tend to expand first and then shrink. After 500 steps, the SDD value stably decreases and the network topology becomes more close to the truncated octahedron stack based on our 3D deployment algorithm. Finally, the convergence state is obtained at 3000 steps and the corresponding SDD value tends to be 0. Parameter Effect of the Debye Length \(\lambda_{D}\) and Sensing Radius \(R_{S}\) on the 3D Deployment  
Based on the basic physics theory of dusty plasma [30], the Debye length \(\lambda_{D}\) has an important impact on the operation of the algorithm, which dramatically determines the calculation scale of each algorithm. In this section, we further concentrate on the parameter effect of the Debye length \(\lambda_{D}\) on the performance evaluations of our 3D WSN deployment algorithm. As shown in Table 2, we conducted a large number of experiments to calculate the average moving distance and the variance of the moving distance under different values of \(\lambda_{D}\). It can be seen that, with the continuous increase of \(\lambda_{D}\), both the average moving distance and the corresponding variances decrease. Moreover, their fluctuation ranges have also been reduced. This significantly suggests that the increase of \(\lambda_{D}\) will save the energy cost of the whole network. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Different Number of Points** & **Case 1** & **Case 2** & **Case 3** \\ \hline \(p_{t}=1\times 10^{4}\) & 0.6134 & 0.6106 & 0.6056 \\ \(p_{t}=1\times 10^{5}\) & 0.6157 & 0.6097 & 0.6086 \\ \(p_{t}=2\times 10^{5}\) & 0.6162 & 0.6129 & 0.6078 \\ \(p_{t}=5\times 10^{5}\) & 0.6140 & 0.6103 & 0.6063 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Coverage rates for different number of test points in Monte Carlo calculation. Figure 6: The variation of the spatial distribution diversion (SDD) value as a function of the calculation time steps.