## 3 Network Connectivity Analysis
### Link Probability Analysis  
Depending on different radio link model, the link probability \(P(L|s)\) is of different values. For Boolean model, either disk or irregular shape as shown in Figure 1(a) and Figure 1(b), \(P(L|s)\) is a logical function, equal to either 0 or 1 depending on the communication range. Hence,  
\[\xi=\pi r^{2}\rho \tag{12}\]  
where \(r\) is the real communication radius or equivalent radius for Boolean disk model and Boolean shadowing model. For probabilistic model as shown in Figure 1(c), \(P(L|s)\) is not deterministic 0 or 1, but defined as the packet reception rate (PRR) \(\psi\) over a period of time \(\tau\). In practice, \(\psi\) can be estimated over a moving time window. However, we can use it in simulation as a stationary variable to indicate the probability of instant link success or failure. Empirical studies have shown that the lognormal shadowing model provides more accurate multipath channel models than other models like Nakagami and Rayleigh models [13]. According to this model, the received power \(P_{r}\) in dB is given by:  
\[P_{r}(s)=P_{t}-PL(s_{0})-10\eta\log_{10}(\frac{s}{s_{0}})+\mathcal{N}(0,\sigma) \tag{13}\]  
where \(P_{t}\) is the output power, \(\eta\) is the path loss exponent, \(\mathcal{N}(0,\sigma)\) is a Gaussian random variable with mean 0 and variance \(\sigma^{2}\), and \(PL(s_{0})\) is the power decay for the reference distance \(s_{0}\). \(\sigma\) can be high in the case of severe signal fluctuations due to irregularities in the surrounding of the receiver and transmitter's antennas. (13) does not include anisotropic properties of the radio. To incorporate it, we add a coefficient \(K_{i}\) to represent the difference in path loss in different directions. (13) can be modified as :  
\[P_{r}(s)=P_{t}-\left(PL(s_{0})+10\eta\log_{10}(\frac{s}{s_{0}}) \right)\times K_{i}+\mathcal{N}(0,\sigma) \tag{14}\] \[K_{i}= \left\{ \begin{matrix}1&\text{if }i=0\\ K_{i-1}\pm rand\cdot DOI&\text{if }0<i<360\,\text{ and }i\in N\end{matrix} \right.\] \[DOI=0.01821\]  
DOI represents the degree of irregularity. The reference value is suggested by [16] to be 0.01821, which is measured using Mica2 more. We can generate 360 values for 360 different directions. However, to make the simplicity, we just adopt 4 directions in the simulation. For a modulation method \(M\) and encoding method \(E\), the PRR \(\psi\) is defined as a function of bit-error rate \(\beta_{\mathrm{M}}\), while \(\beta_{\mathrm{M}}\) is a function of SNR \(\gamma\). Takeing Manchester encoding for example:  
\[\varphi(\gamma)=(1-\beta_{M}(\gamma))^{8(2/-h)} \tag{15}\]  
where \(f\) is the frame size (preamble, payload and CRC), \(h\) is the preamble (both in bytes). The value of \(\beta_{\mathrm{M}}\) depends on modulation, and more generally on transmitting and receiving techniques (diversity, equalization, etc.). Here we take non-coherent FSK for example, which is the default modulation scheme in Mica2 Mote. Expressions for \(\beta_{\mathrm{M}}\) is:  
\[\beta_{M}=\frac{1}{2}e^{-\frac{\gamma}{2}},\ \gamma=10^{\gamma_{AB}/10} \tag{16}\]  
Plugging (16) into (15), we have:  
\[\Psi(\gamma)=(1-\frac{1}{2}e^{-\frac{\gamma}{2}})^{8(2/-h)} \tag{17}\]  
The SNR \(\gamma\) can be obtained from (14) and is given by:  
\[\gamma_{AB}(s)=P_{r}(s)-P_{n} \tag{18}\]  
where \(P_{n}\) is the noise floor which depends on both the radio and the environment. Considering an ambient temperature of 27\({}^{\circ}\)C and no interference signals, the noise floor is -115 dB [13]. Note that \(\gamma\) in (18) can also be regarded as a Gaussian random variable \(\mathcal{N}(\mu(s),\sigma)\) with mean \(\mu(s)\) and variance \(\sigma^{2}\), where \(\mu(s)\) is given by:  
\[\mu(s)=P_{r}-(PL(s_{0})+10\eta\log_{10}(\frac{s}{s_{0}}))\cdot K_{i}-P_{n} \tag{19}\]  
According to (19) and (17), given the distance and some related parameters, the PRR can be calculated. Figure 2 shows the analytical model of PRR to distance in one direction. It is clear that there is a large transitional region, in which the PRR is unstable.