## 4 Performance Analysis
### Analysis of Neighboring Sensor Properties  
The sensor distribution and neighboring sensor properties in mobile ad-hoc networks are crucial metrics for describing the network performance such as connectivity and topology management. This subsection derives the statistic properties of the critical neighbor number (CNN) in a typical network topology and proposes a sensible way to select the key parameters for the SAMCA algorithm such as the _UB_, _LB_, and the reference cluster size. The following theorem investigates the condition and convergence of node distribution with respect to the mobility models.  
**Theorem 1**: _If the sensor velocities are independent of time and the sensor positions, the mobile sensors will asymptotically uniformly distribute over the whole given region._  
According to the random walk mobility model and Theorem 1[48], the steady-state sensor distribution should be able to maintain the uniform distribution property. Therefore, under this circumstances, the critical neighbor number may be determined by  
\[CNN\simeq E[N_{i}]=\frac{\pi R^{2}}{A}\cdot(n-1), \tag{5}\]  
where \(n\) is the number of sensors, \(A\) is the bounded region \(A=[0,\ell]^{2}\), \(\ell\) is the side length of a square, and \(R\) is the transmission range of a sensor. Thus, for the network operation, the selection of the reference cluster size \(z\) is suggested to be  
\[z=\alpha(CNN+1)\simeq\alpha(E[N_{i}]+1), \tag{6}\]  
where \(1\leq\alpha\) since the CNN may be considered as the possible number of cluster members in a cluster. In order to reduce the frequency of topology change (i.e., merging and splitting clusters), appropriate upper and lower bounds of the cluster size are necessary. Hence, _UB_ may be defined by  
\[UB=\beta(CNN+1)\simeq\beta(E[N_{i}]+1), \tag{7}\]  
where \(\alpha<\beta\). Similarly, _LB_ may be expressed by  
\[LB=\gamma(CNN+1)\simeq\gamma(E[N_{i}]+1), \tag{8}\]  
where \(0<\gamma<\alpha\).  
Observe that the above analysis is suitable for any transmitting range. However, overly small transmission ranges may result in isolated clusters whereas overly large transmission ranges may result in a single cluster. Therefore, in order to optimize energy consumption and encourage linking between clusters, it is sensible to consider the critical transmission power which will result in a reasonably connected network. This _range assignment_ problem is investigated in [9], which proves that every bounded and obstacle-free type of node mobility is detrimental for network connectivity. Assume that the sensors are distributed in \(A\) with probability density function \(\mathcal{F}_{\mathcal{M}}\) where \(\mathcal{F}_{\mathcal{M}}\) is the asymptotic sensor spatial distribution based on the mobility model \(\mathcal{M}\) with initial deployment \(\mathcal{F}\). The result is represented as the following theorem.  
**Theorem 2**: _Let \(\mathcal{M}\) be an arbitrary mobility model that is bounded within \(A=[0,\ell]^{2}\) and obstacle-free. Furthermore, assume that the asymptotic sensor spatial distribution \(\mathcal{F}_{\mathcal{M}}\) is continuous on the boundary \(\partial A\), and \(\min_{A}\mathcal{F}_{\mathcal{M}}>0\). The critical transmission range of an ad-hoc network with the mobility model \(\mathcal{M}\) is_  
\[R_{\mathcal{M}}=c\cdot\ell\sqrt{\frac{\log n}{\pi n}} \tag{9}\]  
_for constant \(c\geq 1\)._  
Given the random walk mobility model, Theorem 2 may suggest a good initial value for the search of optimized range assignment strategies to provide a high probability of connectivity. The performance of the SAMCA with different selections of \(R_{\mathcal{M}}\), the reference cluster size \(z\), and possible values of _UB_ and _LB_ (i.e the values of \(\alpha\), \(\beta\), and \(\gamma\)), is examined via simulation.