## 5 Connectivity Model  
Let the collaborative sensors be considered to be input variables \(\chi=[x_{1},x_{1},x_{1},\dots,x_{\kappa}]\) assumed to be Gaussian distributed with density \(\rho\) over a given area in a \(2D\) plane. Each sensor first calculates the approximate number neighbors after the deployment by exchanging the information among the sensors in the network. The expected number of sensors in an area \(\mathbb{I}\) can be calculated as \(S=\rho\pi d_{max}\). However, the output function of the area as \(f_{N}(\aleph)=\frac{\rho}{\kappa}\), in the range \(\kappa\in 0\leq\aleph\leq N\) of each of the \(\aleph\) of \(N\) grouped by clusters of collaborative sensors of probability of size \(\frac{1}{N}\). Therefore, the expected number of collaborative sensors in each cluster is given by \(\aleph=\frac{S}{N}\)  
Linear beamforming technique implies that every sensor directs its mainbeam towards the geometric center of the network, where the connection can be easily established if any two sensors are aligned within each other's mainbeam. Hence, we can say that every two sensors are facing each other if their main beams are pointing towards each other or, more precisely, if they are deployed within the width of each other's main beam. We expect that the sensors near the base station have high probability of being connected together via their mainbeams. The size of the connectivity area is proportional to the gains of the mainbeam. Thus, the radiated power is fixed at each sensor, where is an ideal distributed beamforming with fixed number of collaborative sensors results in maximum gain which is approximately equal to \(N^{2}\) fold whereas the expected value of \(P_{r}\) at the base station increases as \(N\) increases [43]. On the other hand, the probability of two sensors facing each other in the random deployment is independent of the distance from the base station. Therefore, the connectivity performance of random beamforming is almost the same throughout the network. The gain of mainbeam is independent of the boresight direction; it is always equal to the number of sensors \(N\). As \(N\) increases, the mainbeam of each sensor can reach a greater distance from the neighbor, while its width stays the same. Collaborative beamforming concentrates the radiated power in a certain direction toward the base station. Moreover, CB distributes power consumption overall sensors when the collaboration involves \(\aleph^{\kappa}\) sensors up to an \(N^{\kappa}\)[41]. The gain \(G_{r}\) is defined as the maximum in mainlobe depending on the network topology as well as the minimum value of an \(\frac{1}{N^{\kappa}}\) fold with the reduction in the received power everywhere else [41]. An example of IoT application is monitoring rural areas by deploying sensors. The locations are considered random due to wind, releasing mechanism, speed, and height of the local environment. The array factor is written in terms of the weighted sum of the individual elements as follows  
\[A{F}^{\kappa}(\phi/r,\phi)\triangleq\int_{\kappa=0}^{\aleph-1}\omega_{\kappa}( \vec{x})\exp^{\beta\kappa}\exp^{-j\frac{2\pi}{\lambda}d(\vec{x})_{\kappa}(\phi)} \tag{15}\]  
This describes the spatial response of \(\aleph\) element array factor. The positions are denoted as \(\vec{x}=\{x_{\kappa},y_{\kappa}\}\), \(\kappa\) ranges from \(0\) to \(\aleph-1\), and each element has a weight \(\omega_{\kappa}(\vec{x})\) defined as  
\[\omega=\int_{\kappa=0}^{\aleph-1}\omega_{\kappa}(\vec{x})\exp^{-j\kappa(\vec{x })} \tag{16}\]  
The randomness of positions and the number of sensors cause variations in the respective beampatterns, whereas both the sidelobes' positions and magnitude are affected by the network topology. To ensure the connectivity among these collaborative sensors, the Cartesian product for each cluster grouped as \(\aleph^{\kappa}\) of probability size \(\aleph^{-\kappa}\) is used.