# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 6 Simulation Results
In our case, there was no substantial difference in the value of the error detection rate, either for the dense case--where many sources were connected to a few SUs--or for the sparse case. Finally, having justified the accuracy of the replica-symmetric approximation, we obtained the values of the error detection rate using the population dynamics algorithm for various values of \(T\) and \(R\), as shown in Figure 7. On the left-hand side, we plot the probability of error, given that the source is connected to some SU, i.e., \(\mathbb{P}_{\text{err}}\). Here, we find the approximate symmetry around the line \(T=R\); above that, the error is higher, whereas below that, the error is much lower. The figure on the right-hand side shows the total value of error \(\mathbb{P}_{\text{tot}}\). We can see that for this set of parameter values, the detection rate is dominated by the probability of non-connection. As an SU's density \(n_{s}\) increases, the probability of non-connectivity is expected to decrease since more SUs in the same area yield better connectivity conditions between SUs and sources. Thus, both of the diagrams will have similar behavior, and we can constrain the error detection rate analysis to the case of \(\mathbb{P}_{\text{err}}\). The below Figure 7 depicts the results of our work, as it provides insights into how dense a wireless network should be for adequate source detection. Given some parameters such as the SU and source density, the communication range of the SUs, and the desired SNR, one can simply estimate the error detection probability. For example, if we have a density of \(R=6\) SUs/area and a fixed \(\texttt{SNR}=0\) dB, we can increase the detection reliability by almost 2 dB by taking \(R\) to be 7 instead of 6. Additionally, from a different perspective, \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Path Loss Exponent & \(\gamma=4\) \\ \hline Communication Range & \(r_{c}\) \\ \hline Average Density of Sources & \(n_{t}\) \\ \hline Average Density of SUs & \(n_{s}\) \\ \hline Source Activity Probability & \(\mathbb{P}_{\text{\tiny{$\alpha$}}}=0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings for the obtained simulations. Figure 6: Performance comparison of the population dynamics algorithm and belief propagation algorithm, applied in 200 different random networks, all in an area \(\mathcal{A}\) of \(5\,m^{2}\). The network consists of (**a**) 5 sources and 5 sensors, and (**b**) 10 sources and 10 sensors, all randomly connected to each other, and both path loss and Rayleigh fading have been considered. if we have a density of \(T=5\) sources/area, increasing \(R\) by a factor of 1 will result in a substantial decrease in the error detection rate. Concluding our discussion of the simulation results, we point out the use of diagrams or error detection maps, such as the one in Figure 7, to determine how dense the connections between SUs and sources should be for a given SNR. In practice, they can be used to determine how many SUs should be scattered in a field of interest so that all emitting sources in the field can be adequately detected. Surprisingly, the fact that we modeled our system to behave well when the network forms a loop-less Cayley tree seemed not to be exceedingly important. Despite the fact that in our randomly instantiated two-dimensional networks, the neighborhood matrices contained many short loops, this did not substantially degrade the overall network performance. Additionally, we did not notice any phase transitions that could pose algorithmic barriers to the multi-source detection problem.