## 2 Materials and Methods
### Physical-Layer Security Analysis: Non-Colluding Eavesdroppers
#### 2.2.1 Probability of Non-Zero Secrecy Capacity  
To explore the physical-layer security for the three-node wiretap system model under the threat of non-colluding eavesdroppers, the probability of non-zero secrecy capacity is utilized. From (20), the probability of non-zero secrecy capacity is expressed as  
\[P_{\text{nsc}}=1-\int_{0}^{\infty}F_{\gamma_{B}}(y)f_{Y}(y)dy, \tag{24}\]  
where \(Y=\frac{\gamma_{E}}{d\mu}\), in which \(d\) is the distance between the transmitter ([PERSON]) and the \(i^{th}\) closest eavesdropper and \(\alpha\) is the path loss exponent. The PDF of the path loss \(d^{\alpha}\) is distributed as [37]  
\[f_{d^{\alpha}}(x) = \exp\Bigl{(}-A_{e}x^{\delta}\Bigr{)}\frac{\delta A_{e}^{i}x^{ \delta i-1}}{\Gamma(i)}, \tag{25}\]  
where \(A_{e}=\pi\lambda_{e}\), \(\delta=\frac{\mu}{\alpha}\), and \(\Gamma(\cdot)\) is the gamma function. First, one needs to obtain the PDF of \(Y\) as  
\[f_{Y}(y) = \int_{0}^{\infty}y_{b}f_{\gamma_{E}}(yy_{b})f_{d^{\alpha}}(y_{b} )dy_{b}. \tag{26}\]  
Substituting (9) and (25) and with the help of ([34] Equation (2.24.3.1)) yields  
\[f_{Y}(y)=\sum_{r_{1}=0}^{\infty}\sum_{r_{2}=0}^{\infty}\cdots \sum_{r_{n_{e}}=0}^{\infty}C_{1}y^{-1-\delta i}G_{\delta n_{e}}^{1}\begin{array} []{c}\delta n_{e}\\ 1\end{array}\begin{array}{c}F\\ F^{\prime}\end{array}\Bigg{|}\frac{A_{e}\delta^{\delta n_{e}}(\gamma_{E}K)^{ \delta}}{y^{\delta}\Bigl{(}\prod_{j=1}^{n_{e}}E\left[X_{j}^{2}\right]\mu_{ej} K_{j}\left(1+\kappa_{ej}\right)\Bigr{)}^{\delta}}\Bigg{)}, \tag{27}\]  
where  
\[C_{1}=\left(\frac{\prod_{j=1}^{n_{e}}E\left[X_{j}^{2}\right]}{\gamma_{E}K} \right)^{\mu_{1}K+r_{1}}\frac{c_{xe}A_{e}^{i}\delta^{n_{e}(\delta i+\mu_{ \alpha}K+r_{1})+\rho^{*}}}{2\Gamma(i)(2\pi)^{\frac{(\delta-1)n_{e}}{2}}}\left( \frac{\prod_{j=1}^{n_{e}}E\left[X_{j}^{2}\right]\mu_{ej}K_{j}\left(1+\kappa_{ej }\right)}{\gamma_{E}K}\right)^{-\delta i-\mu_{\alpha}K-r_{1}},\]  
\[F=\frac{1-\delta i-\mu_{\alpha}K-r_{2}}{\delta},\cdots,\frac{1-\delta i-\mu_{ \alpha}K-r_{n_{e}}}{\delta},\frac{1-\delta i-\mu_{\alpha}K-r_{1}}{\delta},F^{ \prime}=0,\text{ and }\rho^{*}=\sum_{j=1}^{n_{e}}1-\beta^{\prime}_{e}+1-\frac{n_{e}}{2}.\]  
Using (8), (27), and ([34] Equation (2.24.3.1)), the probability of non-zero secrecy capacity given in (24) is solved as  
\[P_{\text{nsc}}=1-\sum_{r_{1}=0}^{\infty}\sum_{r_{2}=0}^{\infty}\cdots\sum_{r_ {n_{e}}=0}^{\infty}\sum_{r_{1}=0}^{\infty}\sum_{r_{2}=0}^{\infty}\cdots\sum_{ v_{N}=0}^{\infty}C_{2}G_{\delta N+1+\delta}\delta^{N+1}\begin{pmatrix}\zeta\\ \zeta^{\prime}\end{bmatrix}\left\lvert\phi\right\rangle, \tag{28}\]  
where  
\[C_{2}=\frac{C_{1}a_{1}}{2}\Biggl{(}\frac{\prod_{i=1}^{N}E\left[X_{i}^{2} \right]}{\gamma_{B}}\Biggr{)}^{\mu_{1}+p_{1}}\frac{\delta^{N(\mu_{1}+p_{1}- \delta i)-1+q}}{(2\pi)^{\frac{(\delta-1)N}{2}}}\left(\frac{\prod_{i=1}^{N}E \left[X_{i}^{2}\right]\mu_{i}(1+\kappa_{i})}{\gamma_{B}}\right)^{\delta i-\mu_{ 1}-p_{1}},\]  
\[q=\sum_{j=1}^{N+1}\rho+\mu_{1}+v_{1}-\frac{N}{2},\zeta=1,\frac{1+\delta i-\mu_{ 2}-\gamma_{B}}{\delta},\cdots,\frac{1+\delta i-\mu_{N}-\gamma_{N}}{\delta}, \frac{1+\delta i-\mu_{1}-\gamma_{1}}{\delta},\frac{1+\delta i}{\delta},\zeta^{ \prime}=1-\]  
\(F,i,\) and \(\phi=\frac{\left(\prod_{j=1}^{n_{e}}E\left[X_{j}^{2}\right]\mu_{ej}K_{j}\left(1+ \kappa_{ej}\right)\right)^{\delta}}{\left(\frac{\prod_{j=1}^{N}E\left[X_{j}^{2} \right]\mu_{ej}K_{j}\left(1+\kappa_{ej}\right)}{\gamma_{B}}\right)^{\delta}}\).