## 3 System Model  
In conventional beamforming, the elements of an antenna array are arranged in a predefined regular configuration such as linear or circular. Meanwhile, in CB where each cluster contains randomly distributed sensors acting as a virtual antenna array in an Ad-hoc manner with no common controller, i.e., the weights are synchronized at each sensor. Then, the whole clusters over a distributed area simultaneously transmit a weighted signal which constructively combines in the direction of the intended base station. We consider a system model where a group of randomly deployed sensors are co-located in the \((x,y)\) horizontal plane. Each sensor has a transmission radius \(r_{o}\) as shown in Figure 1 which illustrates the cluster head, collaborating sensors and intended base station. All sensors jointly form a topology of such a network which is dynamically changing in an IoT environment due to the signal variations. However, from the topology point of view, a WSN can be represented as a graph \(G(\mathbf{V},\mathbf{E})\), where the sensors correspond to the vertices and the paths between theses sensors correspond to the edges of the graph. Table 1 lists the symbols used throughout the paper along with their definitions. \begin{table}
\begin{tabular}{l l} \hline \hline
**Symbol** & **Definition** \\ \hline \((x,y)\) & coordinates of deployment sensor over plane \\ \(G(\mathbf{V,E})\) & Graph representation of WSN \\ \((t,j)\) & pair of connected sensors \\ \((x_{nx},y_{\pi K})\) & connected path between any pair of sensors \(t\) and \(j\) \\ \(\aleph\) & subset of a group of deployed sensors \\ \(\kappa\) & number of selected of deployed sensors \\ \(N\) & Total number of deployed sensors \\ \(R\) & Radius \\ \(\lambda\) & wavelength \\ \(\varphi_{\kappa}\) & initial phase of the \(\kappa\)th sensor carrier frequency \\ \((r_{o},\varphi_{\kappa}),\theta_{o},\phi_{o}\) & corresponding coordinate of paired sensors, elevation, and azimuth angle respectively \\ \(d_{\kappa}\) & Euclidean distance between \(\kappa\)th sensor and any point located in \((x,y)\) \\ \((A,\phi_{o})\) & incident coordinate over (x,y) plane \\ \(AF\) & Array Factor of sensors \\ \(\omega\) & weight of transmission signal \\ \(\eta\) & the efficiency factor \\ \(\varepsilon\) & define as a function of the phase distribution (the connectivity factor \\ \(\alpha_{\kappa}\) & initial phase of \(\kappa\)th collaborating sensors \\ \(\zeta_{\kappa}\) & complex amplitude of transmission signal \\ \(P_{r}\) & receipting power \\ \(P_{t}\) & transmit power \\ \(G_{r},G_{t}\) & Gain of transmitter and receiver \\ \(\alpha\) & path-loss \(2\leq\alpha\leq 6\) \\ \(\beta\) & Power attenuation \\ \(d_{\max}\) & maximum transmission range between two collaborative sensors \\ \(S\) & Density of the deployment sensors \\ \(\mathbb{I}\) & the dependent area of interest \\ \(Z\) & Objective function \\ \(particle\) & Sensor \\ \(\vec{x}(r_{t,\phi_{t}})\) & Reference of the active cluster \\ \(\vec{x}(r_{t,\phi_{t}})\) & Reference of the active cluster \\ \(v_{t}\) & Velocity toward selecting optimal solution \\ \(P_{t}\) & Personal-best position for \(sensor_{t}\) \\ \(G_{t}\) & Global-best position for \(sensor_{t}\) \\ \(p_{best}\) & Personal-best \\ \(\beta_{best}\) & Global-best \\ \(\phi_{1}\) & Personal-best coefficient \\ \(\phi_{2}\) & Neighbor best coefficient \\ \(\zeta\) & Constriction coefficient \\ \(n\) & The number of samples that must be taken out of area of interest \(\mathbb{I}\) \\ \(E_{elec}\) & The energy dissipation rate to run the radio \\ \(\varepsilon f_{s}\) & The one-path model for the transmitter amplifier \\ \(\varepsilon_{mp}\) & The multipath model for the transmitter amplifier \\ \(\rho\) & The density of distributed sensors over a given area in a 2D \\ \(\wp\) & The node degree of connectivity \\ \(t\) & A certain amount of iterations \\ \(h(\vec{x})\) & The Latin hypercube sample \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation.