## 3 System Model  
### Network Model  
The following properties are used to simplify the network model, with a few reasonable assumptions. * The sensor nodes and the base station are assumed to be stationary once they are deployed in the environment. * The base station is aware of each sensor node's location. * A sensor node determines its neighbor node(s) within a specific distance. * All sensor nodes are homogeneous in terms of their energy and processing capabilities. * The basic component of intra- and inter-cluster communications is the single hop. * Communication is symmetric, and a sensor node can compute the approximate distance based on the received signal strength if transmission power is given. * Two nodes can communicate with each other via wireless link if they are within range. * The base station is not limited in terms of energy, memory, and computing power. * The sensor nodes are eligible to determine their own power levels via the standard system call [40]. Considering a set of \(n\) network nodes, each node is independently and randomly placed in a two-dimensional simulation area (\(A\)). A uniform random distribution is used such that for a large network (\(n\)) and a large area (\(A\)), we can define a constant node density, \(\rho=\frac{n}{A}\), which denotes the expected number of nodes per unit area. A radio link model is assumed in which each node has a specific transmission range to represent wireless communications between the nodes. If two nodes are within range of one another, they can directly communicate through a wireless link. To establish connectivity in the whole network, a necessary condition is that each node has at least one neighbor node (\(d_{min}>0\)) [9]. If the node degree is defined by \(d(n)\), then the minimum node degree of a network is denoted as:  
\[d_{min}(\mathrm{N})=\min\{d(n)\} \tag{1}\]  
Therefore, a node is considered to be isolated if \(d=0\). From the definition of a k-connected network, each node pair has at least k mutually independent path(s) (k = 1, 2, 3... ), and the probability of that network is indicated with P(k-con). In our analysis, we consider k = 1 to represent the probability of a 1-connected network, P(1-con), so a network is steadily connected if P(1-con) \(\geq\) 0.95 [41]. Because we are interested in finding the optimum transmission power for the whole network, it is dependent on the distributed node density (\(\rho\)) and the contemplated node degree (\(d_{min}>0\)). If no node is isolated, then the transmission range (r\({}_{0}\)) can be represented as a function of node density and probable node degree, as proposed by [PERSON] [22]. \[\mathrm{P}\left(d_{min}>0\right)=P_{non-iso}=\prod_{u=1}^{n}P_{non-iso}=\left(1 -e^{-\rho\pi r^{2}}\right)^{n} \tag{2}\]  
The transmission range of each node is denoted with \(r_{tx}\). By adding \(ln\) to both sides of Equation (2), we can determine the transmission range as follows [22]:  
\[r_{tx}=\sqrt{\frac{-\ln\left(1-P_{non-iso}\frac{1}{n}\right)}{\rho\pi}} \tag{3}\]  
The significance of this equation is demonstrated by the following example. **Example 1:**_Consider a network of totally deployed sensor nodes, \(n=100\), in a square area, \(A=100\)\(m\)\(\times\)\(100\)\(m\), which yields a node density of \(\rho=\dfrac{n}{A}=\dfrac{100}{100\ \times\ 100}=0.01\)\(m^{-2}\). If the probability of connectivity is \(P=0.94\), then the transmission range\({}_{tx}\) can be calculated with Equation (3):_  
\[r_{tx}=\sqrt{\frac{-\ln\left(1-0.94\dfrac{1}{100}\right)}{0.01*\pi}}=15.0\ \mathrm{m}\]  
If the transmission range \(r_{tx}=15\) m, then Figure 2a depicts two nodes that are isolated. In order to connect at least one of those nodes, the transmission range needs to be modified. For this reason, the probability of connectivity needs to be \(p\geq\left(0.95\right)\) to reconnect the nodes. Here, we consider \(p=0.99\).