## 4 Performance Analysis
### Link Available Time  
The link available time distribution of a one-hop connectivity between two sensors is described to investigate the property of the node density change rate in a cluster. At a given time \(t\), consider two mobile sensors, a clusterhead and a cluster member, with the velocity vectors \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2}\), respectively, which implies that the relative velocity vector is \(\mathbf{v}_{r}=\mathbf{v}_{1}-\mathbf{v}_{2}\). Assume the initial distance between the clusterhead and a cluster member is \(r\). Thus, the cluster member moves at a relative velocity \(v_{r}\) from the viewpoint of the clusterhead. After time \(\triangle t\), a cluster member may move out of the transmission range of the clusterhead. Consider the one-hop link between a clusterhead and a cluster member with an angle \(\psi\) between the line connecting these two sensor nodes and the X-axis. Suppose a cluster member moves the distance \(\mu\) straightly within this period. Notice that given the transmission range with the mobility model \(R_{\mathcal{M}}\), \(\mu\) is a function of the initial distance \(r\) between the clusterhead and a cluster member and the angle \(\theta_{r}\) between the relative moving direction of the cluster member and the line connecting these two sensors, \(\mu(r,\theta_{r})\). On the basis of the above settings and referring to the derivation in [50], the probability that the link available time is less than \(\triangle t\) is given by  
\[P_{lat}(r,\psi;\triangle t)=\int_{0}^{2\pi}\int_{v_{\mu}}^{\infty}f_{v_{r}}(v_{r },\theta_{r})dv_{r}d\theta_{r}, \tag{10}\]  
where \(v_{\mu}=\mu(r,\theta_{r})/\triangle t\) and \(f_{v}(v_{r},\theta_{r})\) is the polar form of the relative velocity probability density function. The Lindeberg Theorem  
This subsection reviews the probability for analyzing the performance of the proposed scheme. Readers may refer [49] for a complete discussion and proof of the theorem. Suppose for each \(m\)  
\[(X_{11},X_{12},\ldots,X_{1r_{1}})\] \[(X_{21},X_{22},\ldots,X_{2r_{2}})\] \[\vdots \tag{11}\] \[(X_{m1},X_{m2},\ldots,X_{mr_{m}})\]  
are independent random vectors. The probability space may change with \(m\) and (11) is called a _Triangular Array_ of random variables. In the network application, denote \(m\) as the cluster ID information and \(r_{m}\) represents the number of cluster members in cluster \(m\). Put \(S_{m}=X_{m1}+\cdots+X_{mr_{m}}\) and let \(X_{mi}\) take the values 1 and 0 with probability \(p_{mi}\) and \(q_{mi}=1-p_{mi}\), respectively, where \(i\in\{1,2,\ldots,r_{m}\}\). Thus, we may interpret \(X_{mi}\) as an indicator that the link between clusterhead \(CH_{m}\) and cluster member \(CM_{mi}\) is down after time \(\triangle t\) with probability \(p_{mi}\) and \(S_{m}\) is the number of link down events in cluster \(m\). Denote \(Y_{mi}=X_{mi}-p_{mi}\). Hence,  
\[S_{m}^{Y}\equiv\sum_{i=1}^{r_{m}}Y_{mi}=\sum_{i=1}^{r_{m}}X_{i}-\sum_{i=1}^{r_ {m}}p_{mi}=S_{m}-\sum_{i=1}^{r_{m}}p_{mi}, \tag{12}\]  
\[E[Y_{mi}]=E[X_{mi}]-p_{mi}=0, \tag{13}\]  
\[\sigma_{Y_{mi}}^{2}=\sigma_{X_{mi}}^{2}=p_{mi}(1-p_{mi}), \tag{14}\]  
\[\sigma_{S_{m}}^{2}=\sum_{i=1}^{r_{m}}\sigma_{Y_{mi}}^{2}=\sum_{i=1}^{r_{m}} \sigma_{X_{mi}}^{2}=\sum_{i=1}^{r_{m}}p_{mi}(1-p_{mi}). \tag{15}\]  
For our case, the Lindeberg condition [49] reduces to  
\[\lim_{r_{m}\rightarrow\infty}\sum_{i=1}^{r_{m}}\frac{1}{\sigma_{S_{m}}^{2}} \int_{|Y_{mi}|\geq\epsilon\sigma_{S_{m}}}Y_{mi}^{2}dP\leq\lim_{r_{m} \rightarrow\infty}\sum_{i=1}^{r_{m}}\frac{1}{\sigma_{S_{m}}^{2}}\int_{|Y_{mi }|\geq\epsilon\sigma_{S_{m}}}dP=0, \tag{16}\]  
which holds because all the random variables are bounded by 1 and \([|Y_{mi}|\geq\epsilon\sigma_{S_{m}}]\to 0\) as \(r_{m}\rightarrow\infty\). **Theorem 3**: _Suppose that \(Y_{mi}\) is an independent sequence of random variables and satisfies \(E[Y_{mi}]=0\), \(\sigma_{Y_{mi}}^{2}=E[Y_{mi}^{2}]\), \(S_{m}^{Y}=\sum_{i=1}^{r_{m}}Y_{mi}\), and \(\sigma_{S_{m}}^{2}=\sum_{i=1}^{r_{m}}\sigma_{Y_{mi}}^{2}\).