## 3 Network Connectivity Analysis
### Link Probability Analysis
As the transmission power increases, the average transmission range also increases accordingly, thus a lower node density is sufficient to  
\begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Settings**} & \multirow{2}{*}{\(\mathbf{\rho(P(G)\)=90\(\pm\).05\%)} } & \multirow{2}{*}{\(\mathbf{\rho(P(C)\)=90\(\pm\).05\%)} } & \multirow{2}{*}{\(\mathbf{\rho(P(\bar{I})\)=90\(\pm\).05\%)} } & \multirow{2}{*}{\(\mathbf{\rho(P(\bar{I})\)=90\(\pm\).05\%)} } \\  & (\(\mathbf{m^{-2}}\)) simulation & (\(\mathbf{m^{-2}}\)) simulation & (\(\mathbf{m^{-2}}\)) simulation & (\(\mathbf{m^{-2}}\)) analysis  
make the network connected with the same high probability, as shown in \[\varphi(\gamma)=(1-\beta_{M}(\gamma))^{8(2/-h)} \tag{15}\]4 The giant component probability shows consistent tendency. * Figure 5(d) shows the impact of a different encoding scheme. The connectivity is better using SECDED encoding than Manchester encoding. This result is due to the error correction capabilities of SECDED, which comes at a cost of energy efficiency (encoding ratio 1:3) while Manchester does not provide error correction and the encoding ratio is 1:2. Simulated results for different encoding schemes agree with the expected analytical behavior. * Figure 5(f) shows the connectivity performance as the packet size is doubled. It can be observed that the performance merges to 1 at a slightly higher node density. The impact of packet size on the global connectivity is not obvious compared to other parameters. * The impact of area size \(\|A\|\) on the connectivity is straightforward. As the sub-area size gets larger, to reach the global connectivity requires higher node density. The quantity can be estimated via equation (10). However, to form a giant component, the probability has not been affected by the area size \(\|A\|\). In summary, a higher transmission power, larger shadowing variance and more complicated encoding method can improve the network connectivity. However, it is energy inefficient to sacrifice the transmission power for better QoC unconditionally; higher shadowing variance \(\sigma^{2}\) can increase the chance of link asymmetry and typically comes with a higher path loss exponent \(\eta\); complicated encoding method consumes more system resource (energy, memory space etc.) and takes more time. With reference to the improvement space of each parameter, one should better balance the connectivity performance and the sacrifices of resource and energy. Other techniques such as collaborative radio [31] and multiple antennas [10], although effective in theory to improve global connectivity, need more detailed infrastructure design and proof of their effectiveness in practice. In all simulation cases, we find that the giant component probability converges to 1 much faster than one-connectivity. In some applications, a few isolated nodes or small clusters outside the giant component do not change the overall performance. In that case, a lower node density is sufficient for operation. Besides, the largest component size is also an alternative of connectivity measurement. It not only provides information about the network is one-connected or not, but also the fraction of the connected part. Thus a random size of largest component can be designated satisfying different requirement of QoC.