## 3 Analysis of Energy Efficiency and Network Connectivity
### The Network Connectivity  
As aforementioned, minimizing the energy-distance ratio \(\epsilon(r)\) in Equation (9) will lead to an optimal transmission range \(r\) that achieves minimal energy consumption. However, the optimal transmission range might cause some connectivity problem, since sensors are assumed to be placed uniformly and randomly in a fixed area. Thus, the selection of the transmission range needs to consider the connectivity requirement. In this section, we analyze the network connectivity given a network radius \(a\), a node density \(\rho\) and a transmission range \(r\). Here the network **connectivity** is defined as the probability that a sensor node can find at least one path to reach the sink (_i.e._, node \(D\)).  
Let \(u_{k}\), \(v_{k}\) and \(x_{k}\), \(k=1,2,\dots\), be the distance between the \(k^{th}\) forwarding node and sink \(D\), the distance between its one-hop router and sink \(D\), and the \(k^{th}\) distance progress, respectively. Accordingly, denote r.v.s \(\mathbb{U}_{k}\), \(\mathbb{V}_{k}\) and \(\mathbb{X}_{k}\), respectively, correspond to \(u_{k}\), \(v_{k}\) and \(x_{k}\). It is easy to see that \(u_{k+1}=u_{k}-x_{k}\) for \(k=1,2,\dots\).  
Based on the definitions and Figure 2, we can derive the following conditional probability distribution function  
\[\begin{split} F(x_{k}|u_{k})&\doteq\text{Pr}\{ \mathbb{X}_{k}\leq x_{k}|\mathcal{G}^{k}\cap(\mathbb{U}_{k}=u_{k})\}\\ &=\text{Pr}\{u_{k}-\mathbb{V}_{k}\leq x_{k}|\mathcal{G}^{k}\cap( \mathbb{U}_{k}=u_{k})\}\\ &=\text{Pr}\{\mathbb{V}_{k}\geq u_{k}-x_{k}|\mathcal{G}^{k}\cap( \mathbb{U}_{k}=u_{k})\}\\ &=e^{-\rho A_{\alpha}^{k}(u_{k}-x_{k},u_{k},r)}-e^{-\rho A_{s}^{ k}(u_{k},r)}\end{split} \tag{15}\]  
where \(A_{s}^{k}\), \(A_{\alpha}^{k}\) and \(\mathcal{G}^{k}\) are the counterparts in the \(k^{th}\) forwarding of \(A_{s}\), \(A_{\alpha}\) and \(\mathcal{G}\) defined in previous subsection. Here we would like to stress that the triplet (\(A_{s}^{k}\), \(A_{\alpha}^{k}\), \(\mathcal{G}^{k}\)) is indeed correlated to the triplet (\(A_{s}^{k-1}\), \(A_{\alpha}^{k-1}\), \(\mathcal{G}^{k-1}\)). They, however, can be assumed to be independent of each other because UWSNs are generally deployed in a sparse way.  
Further, we define \(g_{k}(x_{k}|u_{k})=\frac{dF(x_{k}|u_{k})}{dx_{k}}\). Let \(p_{k}(\rho,u_{1})\) be the conditional probability that the source node \(S\) can be connected to the destination node \(D\) through \(k\) times forwarding with the initial distance between \(S\) and \(D\) being \(u_{1}\). By [18], assuming each forwarding is independent, \(p_{k}(\rho,u_{1})\) can be approximately computed as:  
\[p_{k}(\rho,u_{1})=\int_{0}^{r}g_{1}(x_{1}|u_{1})\int_{0}^{r}g_{1}(x_{2}|u_{2} )\cdots\int_{0}^{r}g_{k}(x_{k}|u_{k})f(u_{k})dx_{1}\cdots dx_{k} \tag{16}\]  
where  
\[f(u_{k})=\left\{\begin{aligned} & 1,&\text{if}\;\;u_{k}\leq r \\ & 0,&\text{otherwise}\end{aligned}\right. \tag{17}\]  
Thus, the connectivity of the network (denoted by \(p_{c}\)) can be obtained as:  
\[p_{c}=\sum_{k}\int_{0}^{a}p_{k}(\rho,u_{1})\frac{1}{a}du_{1} \tag{18}\]Note that r.v. \(\mathbb{U}_{1}\) here is assumed to be uniformly distributed in the interval [0, \(a\)].