## 3 Network Connectivity Analysis
### Link Probability Analysis
No matter what kind of radio link model adopted, either Boolean disk model or more realistic model with fading and shadowing, the phenomenon of phase transition exists, which gives us a tool for analyzing and determining resource efficient regime of operation for wireless sensor networks. For example, following the settings in Figure 5(a), it tells us that for the nodes with identical transmission power, distributed in an area of 20000 m\({}^{2}\) according to homogeneous Poisson process, the node density must be higher than 0.007 to form giant component and higher than 0.0125 to reach one-connectivity. The density threshold is an energy-efficient point of operation, in that to the left of this threshold the network is disconnected with high probability, and to the right of this threshold, additional energy expenditure results in a negligible increase in the high probability of connectivity. \[\varphi(\gamma)=(1-\beta_{M}(\gamma))^{8(2/-h)} \tag{15}\]1 Cont. * In all simulation cases, the non-isolation probability serves as the upper bound for probability of one-connectivity. Generally, the difference between the two probabilities is non-negligible. However as node density increases, the two probabilities converge to 1. This result agrees with inequality (3). With respect to critical node density, this means that \[\varphi(\gamma)=(1-\beta_{M}(\gamma))^{8(2/-h)} \tag{15}\]2 (24) For \(p\)=90%, the critical node densities for each setting are listed in \[\varphi(\gamma)=(1-\beta_{M}(\gamma))^{8(2/-h)} \tag{15}\]3 It is observed that when the shadowing effect \(\sigma\) is of a large value (\(\sigma\)=10), the difference \(\delta\) becomes very small over the whole node density range. This can be explained by the fact that as \(\sigma\) increases, the increase in long links and the decrease in short links have reduced the correlation between links. As a result, the geometric random graph behavior approaches generic random graph behavior and the probability of one-connectivity approaches the probability of non-isolation. * It should be noted that, the analytical performance of non-isolation probability closely matches the real non-isolation probability in different settings except when \(\sigma\) is large (\(\sigma\)=10), as shown in Figure 5(c). This is also reflected in Table 1 with minor difference between the simulating and analytical critical density for \(P(\bar{I})\) except when \(\sigma\)=10. In some related papers [10-11], it is argued that a large value of \(\sigma\) helps the network to be connected because the number of added long links is larger than the number of removed short links. However, it should be note that the network also suffers severe link asymmetry as the shadowing variance increases. The analytical approach has underestimated the asymmetry problem, which leads to an overestimate of network connectivity. In practical, the analytical values have to be calibrated with certain asymmetry coefficient to match the simulated data. Besides, in practice, some realistic issue like antenna diversity, battery power difference etc. will also cause link asymmetry and is more complicated to simulate. We will leave it as future work. * Comparing Figure 5(b) with Figure 5(a), we can see that the path loss decreases the connectivity of the network. The reason is obvious since the higher \(\eta\), the faster the decay of the signal strength, resulting in a shortened transmission range. The simulated result agrees with the analytical one. Similar is Figure 5(e) and 5(g) compared with Figure 5(a).