## 2 Materials and Methods
### Physical-Layer Security Analysis: Non-Colluding Eavesdroppers
#### 2.2.2 Asymptotic Probability of Non-Zero Secrecy Capacity  
In this section, the security is evaluated as the wiretap channel's conditions are extremely strong. Particularly, we evaluate the probability of non-zero secrecy capacity as \(\gamma_{E}^{*}\rightarrow\infty\). Hence, (28) is rewritten as  
Figure 4 depicts the effect of varying the number of antennas at \(E\) (\(K\)) over the security. It can be seen that increasing the number of antennas improves the eavesdropper's reception capabilities and aids in effectively decoding the tapped messages owing to the use of the MRC method. Hence, the shared information's privacy is compromised. Additionally, the figure illustrates that improving the wiretap channel conditions in terms of the average received SNR at \(E\) (\(\gamma_{E}\)) will eventually result in an extremely low probability of non-zero secrecy capacity. This indicates the \(P_{nsc}\) asymptotic case derived in (22). However, the privacy of shared information may be enhanced by raising the value of \(\gamma_{E}\), which can be achieved by having fewer scatters (\(N\)) obstructing the main channel path. To take the path loss effect over the secrecy into considerations, Figure 5 shows a two-dimensional (2D) graph where the transmitter Alice (A) is the reference location. That  
Figure 4: The probability of non-zero secrecy capacity \((P_{nsc})\) versus the average received SNR at Bob \((\gamma_{B})\). For the main channel, \(\kappa=1,\mu=2\), and for the wiretap channel, \(\kappa_{e}=1,\mu_{e}=2\). \(\gamma_{E}=10\) dB. is, Alice is located at \((0,0)\) and the other receivers \((B\) and \(E)\) have different distances from Alice, and \(B\) stands for the legitimate receiver Bob. Assume \(d_{XY}^{-\beta}=\frac{1}{2\lambda_{I}}\), where \(\beta\) is the path loss exponent, \(X\in\{A,p_{1},p_{2},\cdots\}\), \(Y\in\{B,E,p_{1},p_{2},\cdots\}\), and \(J\in\{B,E\}\). \(\lambda_{I}=\frac{1}{2\sigma_{I}^{2}}\) is the Rayleigh fading parameter, and \(\sigma_{I}\) is the scale parameter of the distribution. \(d_{XY}\) represents the distance from node \(X\) to node \(Y\) in meters \((m)\). \(p_{i}\) (for \(i=1,2,\cdots,N-1\)) are the locations of the obstacles in the main channel. This is to note the effect of the cascade level between A and B. Figure 6 corresponds to the 2D graph and it indicates that regardless of the number of antennas at \(E\) and the effectiveness of the MRC technique, as the eavesdropper moves further away from the transmitter Alice, i.e., as \(d_{AE}\) becomes larger, the privacy of the transferred information improves. This can be interpreted by the fact that as \(d_{AE}\) rises, the wiretap channel's conditions worsen and the received SNR at \(E\) deteriorates accordingly. This graph demonstrates the importance of considering the impact of distances between nodes on privacy. Figure 7 illustrates the intercept probability \((P_{int})\) versus the density of eavesdroppers \((\lambda_{e})\) for two different values of \(i\), in which \(i\) represents the selection of the closest eavesdropper. For example, \(i=1\) denotes choosing the first nearest eavesdropper to the transmitter, and thus the wiretap channel will be the one between Alice and this selected eavesdropper. Figure 5: The 2D graph. Figure 6: The lower bound of the secrecy outage probability \((OP_{sec}^{L})\) versus the distance between Alice and \(E\) for different number of antennas \(K\). For the main channel, \(\kappa=0,\mu=1\), and for the wiretap channel, \(\kappa_{e}=0,\mu_{e}=1\). \(\gamma_{E}=1\) dB, \(\gamma_{B}=10\) dB, \(C_{th}=1\), \(N=2\), \(n_{e}=1\), \(\beta=3\), \(d_{Ap_{1}}=5\) m, and \(d_{p_{10}}=5\) m.  
We assume a 2D area \((U=2)\), and we generate \(10^{5}\) realizations of the positions of the eavesdroppers in a square area of 20 m\({}^{2}\). The figure shows that the probability of the information interception grows as the density of eavesdroppers increases. This is owing to the fact that the probability of a more harmful eavesdropper is rising as the \(\lambda_{e}\) grows.