## 3 System Model
The number of direct neighbors of a given cluster head is defined as the degree of the connectivity. The simplest case is taken into consideration as an initial step in which it is assumed that the graph consists of a large number of scattered sensors in the coverage area and paired up whenever calculating the probability that a pair of sensors \((u,j)\in\kappa\) separated by Euclidean distance \(d_{ij}\) is connected, where \(d_{ij}\) is given by  
\[d_{ij}=\sqrt{(x_{i}-x_{j})^{2}+(y_{j}-y_{j})^{2}} \tag{1}\]  
where \(x_{i}\), \(y_{j}\) are the coordinates of \(k\) connecting sensors. The hop-distance between two connected sensors can be calculated as [37]  
\[r_{o}=\sqrt{d_{ij}}\frac{3\varepsilon f_{s}}{2E_{ec}} \tag{2}\]  
where \(E_{ec}\) is the energy dissipation rate to run the radio, \(\varepsilon f_{s}\) is the one-path model for the transmitter amplifier. Meanwhile \(\varepsilon_{mp}\) is the multipath model for the transmitter amplifier. In other words, any two sensors can directly communicate with each other if and only if their Euclidean distance is smaller than a given threshold \(r_{o}\). Suppose a graph \(\mathbf{G}\) is said it to be connected if and only if at least a path  
Figure 1: Structure of the system model; (**a**) Uniform Linear Array (ULA), (**b**) Uniform Mesh Array (UMA), (**c**) Random Antenna Array (RAA). exists \((x_{tx},y_{rx})\) between any pair of sensors \(\imath\) and \(\jmath\), where \(\{\imath,\ldots,\jmath\}_{\imath\
eq\jmath\in\mathbb{N}\subseteq N}\). When there is no path between any pair of the sensors, the network is said to be disconnected. We express the sensor coordinates as \((x_{t},y_{\jmath}),\imath_{\jmath}=1,\ldots,N\), where \(\imath\
eq\jmath\) and \(\kappa\in\mathbb{N}\subseteq N\). The sensors are chosen as linear, mesh, and randomly distributed in a circle with radius \(R\) and the spacing between sensors is \(\lambda/2\)[40]. Meanwhile, the random deployment is assumed to be Gaussian distributed with zero mean and variance \(\sigma_{0}^{2}\). Initially, one of the collaborative sensors is selected to act as a cluster head, which then serves as the geometrical reference point for all collaborative sensors in the respective cluster. The cluster head broadcasts Hello messages to all its neighbors. Each connected sensor receives the message, and align its signal phase with reference to the cluster head by multiplying the message by a complex weight. This process is performed during information sharing among the collaborative sensors to ensure that they have aligned the signal to be added constructively at the direction of base station. At the end of this stage, the CB stage starts by the source sensor and its connected collaborators which simultaneously transmit the data packets with a field amplitude of \(\xi_{\kappa}\). Due to the phase alignment among sensors during the sharing phase, the data packets add constructively at the intended location. The channel characteristics in various environments such as outdoor, or outdoor-to-indoor IoT applications, i.e., scenarios such as open-loop or closed-loop in which CB is needed are exposed to large-scale fading which is considered to be a dominant factor for the channel among these collaborative sensors and base station. Therefore, the channel coefficient for each \(\kappa\)th collaborative sensor which servers as connectable source-destination pair is multiplied by the corresponding weight to align the phase of the signal. This ensures that signals from all collaborative sensors are in phase toward the direction of the intended base station. Moreover, a closed-loop scenario is considered where the phase alignment is done by compensating the distance between the collaborative sensors and the intended base station, with respect to the cluster head. Assume the location of the intended location is expressed in spherical coordinate system \((r_{o},\theta_{o},\phi_{o})\), where the elevation and azimuth angles correspond to \(\theta\in[0,\pi]\) and \(\phi\in[-\pi,\pi]\), respectively.