## 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
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Node density [0.001]**} \\ \hline
**Node** & **Tx\_Range** & **k-Degree** & **Energy** & **Connectivity** & **Reachability** \\
**Degree** & **(m)** & **Percentage** & **Savings** & **Connectivity** & **Reachability** \\ \hline
1 & 18.50 & 64.0\% & 99.6\% & 1.1\% & 1.8\% \\
2 & 37.50 & 89.1\% & 98.4\% & 4.1\% & 23.7\% \\
3 & 64.50 & 99.6\% & 95.4\% & 11.7\% & 100.0\% \\
4 & 90.50 & 99.8\% & 90.9\% & 21.4\% & 100.0\% \\
5 & 116.50 & 100.0\% & 84.9\% & 32.9\% & 100.0\% \\
6 & 143.50 & 100.0\% & 77.1\% & 45.2\% & 100.0\% \\
7 & 170.50 & 100.0\% & 67.7\% & 57.5\% & 100.0\% \\
8 & 198.50 & 100.0\% & 56.2\% & 69.7\% & 100.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The numerical evaluation results for a network with a node density of 0.001 nodes/m\({}^{2}\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Node Density [0.005]**} \\ \hline
**Node** & **Tx\_Range** & **k-Degree** & **Energy** & **Connectivity** & **Reachability** \\
**Degree** & **(m)** & **Percentage** & **Savings** & **Connectivity** & **Reachability** \\ \hline
1 & 8.50 & 64.9\% & 99.9\% & 0.2\% & 0.4\% \\
2 & 16.50 & 90.3\% & 99.7\% & 0.9\% & 6.6\% \\
3 & 28.50 & 99.5\% & 99.1\% & 2.7\% & 99.0\% \\
4 & 40.50 & 100.0\% & 98.2\% & 5.1\% & 100.0\% \\
5 & 52.50 & 100.0\% & 96.9\% & 8.3\% & 100.0\% \\
6 & 64.50 & 100.0\% & 95.4\% & 12.0\% & 100.0\% \\
7 & 76.50 & 100.0\% & 93.5\% & 16.2\% & 100.0\% \\
8 & 88.50 & 100.0\% & 91.3\% & 20.8\% & 100.0\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The numerical evaluation results for a network with a node density of 0.005 nodes/m\({}^{2}\). tables. The error decreases with increasing node degree. If the node degree is greater than five, the error decreases to zero. This result indicates that the proposed approach is reliable and feasible. The performance index curves for low and high node densities are displayed in Figure 4 and 5, respectively. This result implies that the transmission range in low node density is more easily influenced by node degree than that in high node density. A long transmission range is required to connect more adjacent nodes for node. In a uniform node distribution, node density produces different distances between nodes. And the required transmission range is also different because of node density for identical node degree. This result shows that node degree and node density are important factors in the adjustment of the transmission range. And these factors seriously affect energy saving of nodes. Figure 4: The corresponding relationship of network evaluations with a node density of 0.0005 nodes/m\({}^{2}\). Figure 5: The corresponding relationship of network evaluations with a node density of 0.005 nodes/m\({}^{2}\). The performance index curves for the connection quality of a network with different node densities are shown in Figure 6. The two curves imply the relationship of reachability and connectivity of network. The difference between reachability and connectivity becomes apparent by comparing the different node densities. In low nodes density, the two estimates are closer to present that a full-reachability network easily approach to a full-connectivity network. This condition is obvious when the node degree is large. The difference between connectivity and reachability is influenced by node density. Regarding the stability of the evaluation, network reachability maintains stability under different node densities. Therefore, generally, network reachability can serve as a useful indicator of connection quality for a network. The quality of network connectivity with different node densities is shown in Figure 7. This performance index is easily influenced by node density. A network with full connectivity is difficult to construct with high node density. Accordingly, except for special applications, network connectivity does not serve as a useful indicator of connection quality for a network.