## 3 A flexible Approach to a Reachable Network Topology with a Minimum Connection Degree
### A Full-Reachable WSN Based on an Adaptive Transmission Range with Minimum Node Degree
The probability is:  
\[P_{0}(d=n_{0})=\frac{(\rho\pi\,r_{0}^{2})^{n_{0}}}{n_{0}!}\cdot e^{-\rho\pi\,r _{0}^{2}}, \tag{9}\]  
where \(P_{0}(d=n_{0})\) is the probability of node with \(n_{0}\) degree  
If \(n_{0}=0\):  
\[P_{0}(d=0)=e^{-\rho\pi\,r_{0}^{2}} \tag{10}\]  
From Equation (10), we have the following equation:  
\[\rho\pi r_{0}^{2}-2n_{0}\,ln\,r_{0}-C=0, \tag{11}\]  
where \(C{=}n_{0}\,ln\,\rho+n_{0}\,ln\,\pi+n_{0}\,ln\,n_{0}!-ln\,P_{0}\)  
Clearly, we have the transmission range (\(r_{0}\)) as a function of node density, expected node degree and expected probability of k-degree nodes as follows:  
\[r_{0}=f(\rho,n_{0},P_{0}) \tag{12}\]The adjustment model as depicted in Equation (12) will be used to determine the optimal transmission range which guarantees the minimum node degree. In implementation, we develop a numerical method to approximate the solution of Equation (12). For the target network in this procedure, the node density, expected node degree and expected probability of k-degree nodes are required to derive a transmission range through the adjustment mode of the transmission range. Next, the network topology will be established by this transmission range, which is less than or equal to the maximum transmission range. Afterward, connectivity and reachability will be calculated by Equations (1) and (3), respectively. The optimal network is established if the energy savings of a node and the reachability of the network have satisfied minimum requirements. This transmission range is considered the optimal transmission range. Otherwise, the expected node degree will be adjusted to obtain a transmission range capable of satisfying the conditions of an optimal network. Step 4: Construction of the transmission route  
An intelligent routing algorithm is employed to discover and construct the transmission range based on the swarm intelligence technique after a full-reachability network topology is established. Each node will select one of degrees of node as delivery path. All paths create a delivery route form source node to data sink in a network. A full-reachability is requirement that ensure existent route between source node and data sink. Therefore, in this approach, the transmission range with required node degree is constructed a full-reachability network topology. And based on this network topology, an intelligent routing algorithm is adopted to discover path between nodes to create at least one delivery route from sensor nodes to data sink. Then, other degrees will be backup delivery path. Consequently, this network not only ensures network reachability  
A network with high node density only requires a 28.5-m transmission range to establish full reachability. The energy savings are more than 98%. Accordingly, high node density saves more energy than low node density for a network with full reachability. The error of expected probability and actual percentage of k-degree nodes for different node degrees in the node densities are shown in the four \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Node density [0.01]**} \\ \hline
**Node** & **Tx\_Range** & **k-Degree** & **Energy** & **Connectivity** & **Reachability** \\
**Degree** & **(m)** & **Percentage** & **Savings** & **Connectivity** & **Reachability** \\ \hline
1 & 6.50 & 73.8\% & 99.9\% & 0.1\% & 0.3\% \\
2 & 11.50 & 90.4\% & 99.8\% & 0.4\% & 3.1\% \\
3 & 20.50 & 99.7\% & 99.4\% & 1.4\% & 80.6\% \\
4 & 28.50 & 100.0\% & 98.9\% & 2.6\% & 100.0\% \\
5 & 36.50 & 100.0\% & 98.1\% & 4.1\% & 100.0\% \\
6 & 45.50 & 100.0\% & 97.1\% & 6.3\% & 100.0\% \\
7 & 53.50 & 100.0\% & 96.0\% & 8.4\% & 100.0\% \\
8 & 62.50 & 100.0\% & 94.6\% & 11.2\% & 100.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The numerical evaluation results for a network with a node density of 0.01 nodes/m\({}^{2}\).