# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 1 Introduction
To achieve the above goal, we apply ideas from the physics of random media, obtaining self-consistent equations, which we then solve using the so-called population dynamics methodology. This allows us to calculate the probability of detection of the operational state of the primary sources in the presence of Rayleigh fading channels using a set of equations that corresponds to the so-called replica symmetric ansatz [22]. While this approach is known to work only in the case of random graphs without loops, we show that our results, obtained in two-dimensional, loopy graphs, are in good agreement with those for random Erdos-Renyi graphs [23]. This application of statistical mechanics in the context of wireless networks represents an innovative approach to analyzing and understanding the system's behavior. The extensive simulations performed (described in Section 6) allow for a comprehensive assessment of our method's performance and provide valuable insights into its effectiveness, verifying its accuracy, scalability, and computational complexity. The above-mentioned replica method from physics allows us to obtain results for large-scale systems and estimate the theoretical limits for successful multi-source detection. As a result, our proposed method does not depend on the specific network structure, and one can easily obtain the error detection rate of an arbitrary network given the distributions of the density connections between primary sources and secondary users. For completeness, in Appendix A, we provide a concrete example of how this algorithm can be used in a distributed fashion. In Table 1, a brief comparison of the proposed framework with various state-of-the-art methods is presented, accompanied by the relevant advantages and drawbacks of each method. In conclusion, our method seems superior since it combines high levels of accuracy, computational complexity, robustness to noise, and scalability. The utilization of BP allows for efficient message passing and inference, enabling more reliable and precise primary source detection compared to alternative techniques. Our method also offers additional advantages such as adaptability to various network topologies and flexibility in incorporating prior knowledge. These features set our proposed approach apart and position it as a reliable and accurate solution for the detection of emitting sources in wireless sensor networks. \begin{table} \begin{tabular}{p{227.6 pt}} \hline \hline
**Thresholding-based methods** compare the received signal strength or other relevant parameters with a predefined threshold to infer the emitting state of the primary sources [24, 25, 26]. In summary, thresholding-based methods offer simplicity and speed but they may lack adaptability and robustness to complex data distributions. Belief propagation, on the other hand, provides adaptability, accuracy, and flexibility but has higher computational complexity and potential convergence challenges. \\ \hline Advantages Compared to the Belief Propagation Algorithm \\ \hline \hline \end{tabular}   
* Lower complexity, since they are straightforward and easy to implement, thus they are more computationally efficient. * Rapid decision making, since they do not require extensive computations or iterations. * Robustness to noise and fluctuations that affect the emitting state detection by setting appropriate thresholds. \\ \hline Drawbacks Compared to the Belief Propagation Algorithm \\ \hline \hline
* Lack of adaptability, since they rely on fixed thresholds, which may not capture the full complexity of the data distribution and thus can lead to suboptimal results. * Sensitivity to threshold selection, since they heavily rely on choosing appropriate thresholds. Selecting thresholds that are too low or too high can lead to false positives or false negatives, respectively, impacting the accuracy of the emitting state detection. * Lack of incorporation of prior knowledge, which in many cases, is crucial for the improvement of the accuracy of the emitting state detection.