# Brain-Inspired Data Transmission in Dense Wireless Network
## 4 Network Reliability Assessment through the Percolation Theory
### Application of the Percolation Theory
Analogously, the total sum-interference from all \(j=1,2,...,J\) simultaneously transmitting nodes can be computed as \(I=\sum_{j=1}^{J}\frac{P_{j}\lambda_{j}\prod_{k}X_{j,k}}{R_{j}^{2b}}\), where \(X_{j,k}\) represents propagation effects in link to device \(j\). In our case we assume that the network is not synchronized, thus, in order to describe the transmission profile (i.e., the duty cycle) we introduce the random variable \(\Delta\in[0,1]\). Thus, the total aggregated interference \(I\) can be defined utilizing the skewed stable distribution [42]:  
\[I\sim S\bigg{(}\alpha=\frac{1}{b},\beta=1,\gamma=\pi\lambda C_{1/b}^{-1}P_{I}^ {1/b}\mathbb{E}\Big{\{}\Delta_{j}^{1/b}\Big{\}}\prod_{k}\mathbb{E}\Big{\{}X_{ j,k}^{1/b}\Big{\}}\bigg{)}. \tag{1}\]  
where \(\Delta_{j}\) is the duty-cycle factor associated with interferer \(j\). Here, in this generic formula, the term \(C_{\alpha}\) is defined as  
\[C_{\alpha}= \begin{cases}\frac{1-\alpha}{\Gamma(2-\alpha)\cos\left(\pi\alpha/ 2\right)},&\alpha\
eq 1\\ 2/\pi,&\alpha=1\end{cases} , \tag{2}\]  
where \(\Gamma()\) denotes the gamma function. Next, the \(P_{I}\) term stands for the transmit power of the interfering nodes. In our case, however, we assume that the maximum power is the same for the desired node and the interferers. The introduced skewed distribution \(S(\alpha,\beta,\gamma)\), valid for \(\alpha\in(0,2]\), \(\beta\in[-1,1]\) and \(\gamma\geq 0\), can be characterized by its skewness \(\beta\), and it possesses the corresponding characteristic function  
\[\omega(w)= \begin{cases}\exp\big{(}-\gamma|w|^{\alpha}\big{[}1-j\beta\text{ sign}(w)\tan(\frac{\pi\alpha}{2})\big{]}\big{)},&\alpha\
eq 1\\ \exp\big{(}-\gamma|w|\big{[}1+j\frac{2}{\pi}\beta\text{sign}(w)\ln(w)\big{]} \big{)},&\alpha=1\end{cases}  \tag{3}\]  
It is well known that the probability density function of our searched random variable \(I\) can be calculated using inverse Fourier transformation  
\[f(x)=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-isx}\omega(s)ds. \tag{4}\]  
In our case, we assume that the nodes are densely deployed over a certain area, thus the dominant channel effect will be path loss at the line-of-sight link, i.e., \(X_{k}=1\). Again, following the authors of [42], the probability of SINR above-assumed threshold \(\theta^{*}\) can be derived as  
\[P\{SINR\geq\theta^{*}\}=F_{I}\Bigg{(}\frac{P_{0}}{\sigma_{0}^{2b}\theta^{*}}- \sigma_{n}^{2}\Bigg{)}, \tag{5}\]  
where \(\sigma_{n}^{2}\) is a noise level, \(r_{0}\) is a distance between nodes, \(b\) is a propagation environment coefficient, \(P_{0}\) is a transmit power of the desired signal, and \(F_{I}\) is a cumulative distribution function of random variable \(I\). Consequently, Formula (1) can be simplified as  
\[I\sim S\bigg{(}\alpha=\frac{1}{b},\beta=1,\gamma=\pi\lambda C_{1/b}^{-1}P_{I}^ {1/b}\mathbb{E}\Big{\{}\Delta_{i}^{1/b}\Big{\}}\bigg{)} \tag{6}\]  
where \(\alpha\), \(\beta\), and \(\gamma\) are characteristic function coefficients. Consequently, the corresponding characteristic function can be derived as  
\[\omega(w)= \begin{cases}\exp\big{(}-\gamma|w|^{\frac{1}{b}}\big{[}1-j\text{ sign}(w)\tan(\frac{\pi}{2b})\big{]}\big{)},&b\
eq 1(\text{non line of sight})\\ \exp\big{(}-\gamma|w|\big{[}1+j\frac{2}{\pi}\text{sign}(w)\ln(w)\big{]}\big{)},&b=1(\text{line of sight}).\end{cases}  \tag{7}\]In our highly dense wireless network, we assume the presence of line-of-sight, and the above formula for \(b=1\) is selected. Moreover, the value of \(\gamma\) is defined as:  
\[\gamma=\pi\lambda C_{1}^{-1}P_{l}\mathbb{E}\{\Lambda_{i}\}=\frac{\pi^{2}}{2} \lambda P_{l}\mathbb{E}\{\Lambda_{i}\} \tag{8}\]  
as \(C_{1}^{-1}=\frac{2}{\pi}\).