# Research on the Coverage of Ultraviolet Communication Network in the Arbitrary Polygon Area
## 3 k-Connectivity
### The Connectivity Properties  
Five sets of elevation angle combinations are selected in the simulation process based on three working methods. This not only guarantees the diversity of work modes but also avoids work redundancy caused by too many angle combinations.  
Without the loss of generality, a regular quadrilateral is selected as the simulation area, and the number of simulations is 100,000.  
Figure 7 indicates the changing trend of _P(k-connected)_ with the node density in different working modes (\(R_{b}\) = 10 kbps, \(P_{t}\) = 0.5 W). The results illustrate that the k-connection probability increases with the increase of node density, and there is a similar growth trend under different transceiver elevation angles.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline wavelength (\(\lambda\)) & 250 nm \\ quantum efficiency of the filter and photodetector (\(\eta\)) & 0.045 \\ speed of light (\(c\)) & 3.0 \(\times\) 10\({}^{8}\) m/s \\ bit error rate (\(P_{e}\)) & 10\({}^{-6}\) \\ Planck constant (\(h\)) & 6.62 \(\times\) 10\({}^{-34}\) J-s \\ transmitted power (\(P_{t}\)) & 0.5 W \\ data rate (\(R_{b}\)) & 10 kbps \\ \hline \hline \end{tabular}
\end{table}
Table 1: System model parameters.  
Figure 6: The relationship between node coverage range and the transmitting elevation angle.  
If the UV network needs to reach 2-connectivity, that is, the network connection rate reaches more than 95%, the node density under NLOS-a working mode is at least \(1.71\times 10^{-3}\) m\({}^{-2}\). The node density under NLOS-b (\(\theta_{1}\) is chosen to be 50\({}^{\circ}\) and \(\theta_{2}\) is unchanged) is at least \(2.19\times 10^{-4}\) m\({}^{-2}\), which is 12.8% of NLOS-a mode.  
A horizontal comparison illustrates that the conditions for achieving a 3-connected network under the same communication parameters is more stringent. In NLOS-a mode, the node density needs to increase by 54.0% if the 1-connected network needs to reach the 2-connected network, and the node density increases by 116.9% to reach 3-connectivity.  
Figure 8 shows the changing trend of _P(k-connected)_ with the data rate in different working modes (\(\rho=5.0\times 10^{-4}\) m\({}^{-2}\), \(P_{t}=0.5\) W). We know that the k-connection probability decreases with the increase of the data rate from Figure 8. When the working mode is NLOS-c, the data rate of the network from 1-connected state to 2-connected state needs to be reduced by 26%, and the data rate needs to be reduced by 39.4% to reach a 3-connected state, as shown in Figure 8e.  
From Figure 9, the changing trend of _P(k-connected)_ with the transmission power (\(\rho\) = 5.0 \(\times\) 10\({}^{-4}\) m\({}^{-2}\), \(R_{b}\) = 10 kbps) can be found. That is, the _P(k-connected)_ increases as the transmission power increases. When the working mode is NLOS-c, the transmission power of the network from 1-connected state to 2-connected state needs to increase by 27.0%, and the transmission power needs to increase by 53.7% to reach a 3-connected state, as shown in Figure 9e. By comparing Figures 7-9, we notice that the smaller the transceiver elevation angles are, the better the network connectivity performance is.