## 5 Connectivity Model
Each cluster of connected collaborative sensors can be labeled by a set of \(\aleph\) coordinates \(\vec{x}_{i}=\{(x_{\iota},y_{\iota 1}),(x_{\iota 2},y_{\iota 2}),\ldots,(x_{\iota r },y_{\iota r})\}\), whereas \((x_{\iota j},y_{\iota j})\) defines the components of one cluster denoted as \(\vec{x}_{i}\) of collaborative connected sensors. The Latin hypercube sample that is denoted as \(h(\vec{x})\) is used to obtain a random selection of \(\aleph\) of clusters \(\vec{x}_{1},\vec{x}_{2},\ldots,\vec{x}_{\aleph}\) conditioned by the probability of sensor connection for each component of a cluster. The set \(\{\vec{x}_{\iota}\}_{i=1}^{\aleph}\) is defined as a permutation of the integers \(1,\ldots,N\). Moreover, the density function of \(N\) given as [44]  
\[N\
i(x_{\iota j},y_{\iota j})= \begin{cases}1,&\text{if}\quad\aleph^{\kappa}f( \vec{x})\longleftrightarrow(x_{\iota j},y_{\iota})\
i\vec{x}_{\aleph}\\ 0,&\text{otherwise}\end{cases}  \tag{17}\]  
The distribution of connected sensors is seen to be the measurement of the local network connectivity (i.e., the connectivity of collaborative sensors). **Definition 1**. : _The probability of connectivity \(\mathsf{P}(conn)\) is the probability that for each node in a network, there is connection to any other node._  
The probability of a sensor connects with any of its neighbors is an important metric to evaluate the local network connectivity. Moreover, the distribution of connected sensors \(\mathsf{P}_{\vec{x}}(conn)\leq d_{max}\) follows Gaussian process is given as  
\[\mathsf{P}_{\vec{x}}(conn)\leq d_{max} =\sum_{\forall\{\vec{x}_{\iota}\}_{i=1}^{\aleph}}\mathsf{P}_{ \vec{x}}(conn|N\
i\{\vec{x}_{\iota}\}_{i=1}^{\aleph})\mathsf{P}(N\
i\{\vec{x} _{\iota}\}_{i=1}^{\aleph}) \tag{18a}\] \[=\sum\int_{\{\vec{x}_{\iota}\}_{i=1}^{\aleph}(\vec{x})\leq\exp^{- \mathbb{E}[\wp]}}f(\vec{x})d\vec{x}(\frac{1}{\aleph^{\kappa}})\] (18b) \[=\int_{h(\vec{x})\leq\exp^{-\mathbb{E}[\wp]}}f(\vec{x})d(\vec{x}) \tag{18c}\]  
Where \(\wp\) denotes the node degree of connectivity, which is defined as the number of sensors such that any given sensor can connect with directly, and \(\mathbb{E}[\cdot]\) denotes the expectation operator. Thus, the average sensor degree can be expressed as \(\mathbb{E}[\wp]=\rho\pi[d_{max}]\). To present the variance of the Latin hypercube sampling, it is usually not possible to obtain closed-form expressions for Equation (17), an approximation is thus required. We establish this by means of Latin hypercube sampling in the coverage area where the collaborative sensors are deployed. We use an indictor denoted as \(\varepsilon\) with  
\[\varepsilon_{\vec{x}_{\iota j}}= \begin{cases}1,&\text{if}\quad\{(x_{\iota 1},y_{\iota 1 }),(x_{\iota 2},y_{\iota 2}),\ldots,(x_{\iota r},y_{\iota r})\}\leq d_{max}\\ 0,&\text{otherwise}\end{cases}  \tag{19}\]It is shown that the probability of connectivity is influenced by shadow fading and antenna gain, which depends on the position and distance among the selected sensors for each cluster with boundary defined as \(\mathbb{I}=[\vec{x}-f_{N},\vec{x}+f_{N}]\), where \(\mathbb{I}\) depends on the point \(\vec{x}\) of interest and limits \(f_{N}\), which defines the boundaries of the coverage area. Therefore, we considered the randomly directed antenna for the cluster of sensors selection scheme. Each sensor can randomly select its main beamforming direction and then progressively change its direction based on robustness of particle swarm optimization by acquisition of the pre-knowledge of locations of all its neighbors. In other words, starting with initial random weights, the selection of sensors can refine the weights based on sharing the information among them. Figure 4 shows the relative positions of collaborative sensors, where \(d\), and \(\phi_{0}\), \(\theta_{0}\) denote the distance and angle between two connected sensors, respectively. All sensors with angles are Gaussian distributed in \((0,2\pi]\). Figure 4: Relative Positions of the Collaborative Beamforming.