## 6 Simulations and Discussion
### Comparison of Analysis and Simulation  
Link Dynamics (\(N_{out}\))  
The effect of mobility on the link down rate is studied with varying transmission ranges. The various properties investigated in Section 4. characterize the behavior of the links of mobile sensors, which can be used as a basis for analyzing the performance bounds of the proposed SAMCA protocol. Thus, the link dynamics may be used as an indicator of topology change and the distribution of link available time can be used to examine the link stability in the network. Given the network density (\(n=200\)), the transmission range \(R_{\mathcal{M}}\), and the cluster formation parameters (_LB_ = 4, \(z\) = 8, and _UB_ = 12), Figure 17 illustrates the comparison of the theoretical analysis and simulation results for characterizing the average number of link down event \(N_{out}\) in a cluster within \(\triangle t=1\) second during the observation time 200 \(\sim\) 201 second with varying the node speed. As expected, the average number of of link down event \(N_{out}\) in a cluster increases with increasing the node mobility. Moreover, Figure 17 also implies that the theoretical analysis and the simulation result match well and the characteristic of the link down event can be roughly monitored and described by a function of the link lifetime distribution as detailed in Section 4.2.  
Figure 17: The average \(N_{out}\) in a cluster for varying the node speed (left) and the ratio of the average \(N_{out}\) to the average cluster size for varying speed (right); the average cluster sizes for \(R_{\mathcal{M}}\) = 6, \(R_{\mathcal{M}}=10.5\), and \(R_{\mathcal{M}}=15\) are 3.18, 8.82, and 9.93, respectively.  
Simulations vs. Simplified Models  
The network behaviors in the transitional and steady phases are explored when applying the SAMCA approach and the simplified models in a random network of 100 sensors. In each method, the results of 100 typical runs are merged. Given in Figure 18 are the standard deviations of the mean number of clusterheads (left) and gateways (right) with varying the transmission power \(R_{\mathcal{M}}/\ell\), which shows the comparison of the initial cluster formation (without warm-up time) and the Mobility and Density Model (MDM) in the transitional phase and the comparison of the simulated results (with warm-up time) and the Averaged Model (AM) in the steady phase.  
The graph suggests that the MDM method and the AM method well approximate the SAMCA performance. This is reasonable because the MDM retains global connectivity and mobility information for cluster construction in the transitional phase. Due to the uniform convergence of the sensor spatial distribution, the approximation of the AM method may be a way to predict the performance of the SAMCA. Moreover, with an appropriate transmission range (\(R_{\mathcal{M}}\) = 15.0), Figure 18 also shows that both the network behaviors during the transitional phase and the steady phase are close, which may imply that the effect of the system warm-up operation on the network performance is not significant since the whole network is reasonably connected. On the other hand, without a proper transmission range (\(R_{\mathcal{M}}\) = 10.5), experiencing the system warm-up time may help stabilize the number of clusters under the control of the proposed self-organization scheme.