## 4 Parameter Effects on VFA-SF and VFA-SF-OPT
### Effect of External Central Force  
The calculation of the external central force in Figure 6 of our previous paper [13] was a theoretical indication. On the other hand, network-deployment experiments may not require a very precise value. Usually, designers can choose a suitable value, slightly greater than the optimal value. However, the choice of external central force does have important impact on VFA-SF-OPT optimizations. Typical simulations for three different values of external central force are shown in Figure 5.  
The top panels show the deployment topology for \(F_{extern}=2\). The left panels are the network topologies in the middle of the deployment process (VFA-SF-OPT) at the 600 th time step. The right panels are the final networks after VFA-SF-OPT optimization. Red circles in the left panels indicate the region involved in the calculation of the PCD curve and the equilibrium distance at the 600 th and 5000 th time step to evaluate the algorithm performance. It is very clear that, if the external virtual force is small, it is still hard to eliminate the coverage hole in the final node distribution (see the orange ellipse in the top-right panel), as well as the twisted structure. The reason is that small external force may not contribute enough to the total force, which drives the sensor to the optimal position within the fixed calculation time. The middle panels illustrate the results corresponding to \(F_{extern}=3.5\) that were presented in a previous work [13]. Bottom panels show the results corresponding to \(F_{extern}=5.5\). Larger external force can usually yield good hexagonal topology in a large-scale WSN. Figure 4: Corresponding possibilities of the final PCD value for different values of \(v_{max}\). However, it should be noted that the gap between the deployed nodes by the VFA-SF-OPT and those nodes outside of the calculation region increases when the external central force becomes larger. The reason is that the VFA-SF-OPT only calculates the positions and the total virtual force to drive the node movements for the sensors involved in the optimization process. Sensor nodes within the optimization region follow the action of the external force while the rest do not temporally move. Centrally preferred deployment can effectively reduce energy consumption for sensors excluded from the VFA-SF-OPT. Furthermore, average equilibrium distance and average moving distance within the region of radius = 10 for the indicated values of the external central force from the VFA-SF-OPT are listed in Table 3. If \(F_{exten}=0\), equilibrium distance \(D_{m}\approx\sqrt{3}\), which is consistent with theoretical analysis of the original VFA-SF. When external central force increases, the average equilibrium distance  
Figure 5: Typical simulation results of node deployment by the VFA-SF-OPT for the indicated external central force \(F_{exten}=2.0\), 3.5 and 5.5, respectively. Left panels, deployment at 600 th time step. Right panels, final network topology at 5000 th time step. between two neighbor nodes becomes smaller. It means that a larger external central force causes the compression of the central area during the process of optimization deployment. It is consistent with the gap variation in Figure 3. In such a case, those nodes that are newly involved in the VFA-SF-OPT process have to move for a longer distance, which consequently leads to more energy consumption. That is also consistent with Column 3 of Table 3. The final topology may also not be regular hexagonal topology, and it is difficult to recover after releasing this external force in the optimization algorithm. Generally, extremely large or small value of external central force positively or negatively affects the energy consumption of the sensor nodes, and impacts system performance. It can be determined based on the requirements of environment scenarios or location accuracy for some real WSN applications in a large target area.