## 6 Simulations and Discussion
### The Number of Clusters  
The network topology management is investigated from three perspectives: (1) with varying the \(R_{\mathcal{M}}/\ell\) ratio given the network density, the node mobility, and the cluster formation parameters, (2) with varying the network density given the node mobility and the cluster formation parameters, and (3) with varying the degree of mobility given the network density and the cluster formation parameters. From the first perspective, as shown in Figure 7, load balancing may not be achieved without an appropriate transmission range since this may lead to either too large or too small cluster sizes. Hence, the cluster formation is examined with respect to the \(R_{\mathcal{M}}/\ell\) ratio and network density suggested in (9) when applying the SAMCA. With random waiting time parameters \(C=500,\lambda=20\), and \(\rho=0.5\), Figure 7 depicts typical runs of the algorithm with different \(R_{\mathcal{M}}/\ell\) ratios. The results show that each cluster is a collection of sensors which are up to 2 hops away from a clusterhead. Figure 7 (left) also shows that when the transmission range is small, the network with a lower sensor density will have a larger percentage of isolated sensors which eventually become clusterheads in their own right. This is because the network is only weakly connected with these values. On the other hand, given in Figure 7 (right), when the transmission power is large enough, the network is connected with high probability. Figure 8 shows the relationship between the average number of clusters and the \(R_{\mathcal{M}}/\ell\) ratio with varying the number of sensors. The average number of clusters in each case is the sample mean of the  
Figure 7: The typical runs of random networks of 200 sensors with a given mean node speed (2.5 m/sec), different \(R_{\mathcal{M}}/\ell\) ratios, \(R_{\mathcal{M}}=5.25\) (left) and \(R_{\mathcal{M}}=10.5\) (right), and the cluster parameters (_LB_ = 4, \(z\) = 8, and _UB_ = 12). Note that a red \(\
abla\) represent a clusterhead in the network and a red dashed line (right) represents a connection between a pair of distributed gateways. results of 100 typical runs. Observe that the average number of clusters decreases as the ratio \(R_{\mathcal{M}}/\ell\) increases (i.e., the transmission power increases). Since larger transmission power allows larger radio coverage, a cluster has more cluster members, which reduces the number of clusters in the network. However, observe that because of the cluster size constraints, the average number of clusters stabilizes as the number of sensors increases. From the second perspective, Figure 9 depicts that the average number of the SAMCA clusterheads increases nearly linearly with increasing network density, which demonstrates that a good selection of transmission range may lead to a minimal variation of the cluster size with increasing network density. Though the cluster maintenance operations stabilize the network topology with a variety of cluster size parameters, balancing the load among clusters with an appropriate setting for the desired number of neighbors is essential. Notice that the average cluster sizes with the parameters \([LB,z,UB]\) = [4,8,12], [6,12,18], and [8,16,24] are close to the corresponding reference cluster sizes. However, the cluster formation with the parameter [10,20,30] may not spread the load among clusterheads effectively due to the wide range between the two bounds of the cluster size. The approach for determining appropriate cluster parameters is further discussed in Section 6.3. Considering the third perspective, Figure 10 depicts the average numbers of clusters in the network and the average cluster size for different speeds over 200 seconds, which implies that the average numbers of clusters and cluster size remain approximately stable regardless of the node moving speed.