# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 8 Discussion and Conclusions  
This paper presented an analytical study of detecting the transmitting state of multiple sources using a secondary user network in the presence of noise. We employed tools from statistical mechanics, namely the replica approach, to calculate the error detection rate for the network and showed that its results were in numerical agreement with those of the message-passing approach. Based on these results, we were able to analyze and quantitatively predict the error detection behavior in large-scale networks using only a few parameters related to the statistical characteristics of the connections, such as SUs' and primary sources' density, path loss exponent, and fading, which was the main novelty of this paper. Furthermore, driven by the need for a practical network implementation, we introduced a general framework for mapping our bipartite graph to a network structure. The entire framework we developed is fully distributed and presents an excellent approximation to the inference solution.  
In the future, we intend to expand this approach to fault detection and diagnosis in sensor networks, i.e., the identification of faulty sensors or sensors with abnormal behavior, incorporating the dependencies between sensor signals. In parallel, we expect to modify the proposed framework by targeting the prediction of the powers of the primary sources. In this case, the exchanged messages are real-valued variables and thus differentiate most of the aspects of our analysis.  
Additionally, we could extend our results to wireless networks where the sources emit dynamically, i.e., in several time frames. This scenario introduces the time variable into our problem of multi-source detection, creating a structured graph with many short loops, Therefore, it requires a new methodology. Interesting applications of the proposed framework include sensor localization, where neighboring sensor nodes exchange information based on the received signal strength; time of arrival; or other localization-related measurements. Apart from the sensor-localization application, which constitutes a direct continuation of our work, we plan to introduce the statistical mechanics-inspired solution to the target-localization problem, where belief propagation can be deployed to estimate the trajectory or current location of a moving target in a wireless sensor network.  
Finally, as an interesting approach to detecting the emitting state of sources in a sensor network, we plan to combine the belief propagation algorithm with hidden Markov models to target improved inference in more complex graphical structures. Belief propagation can propagate beliefs through the HMM's graphical structure, allowing for a more accurate estimation of the hidden states and model parameters. Moreover, the belief propagation algorithm can effectively deal with noisy or missing data in the observation sequence and thus compensate for the uncertainties in the observed data.  
Conceptualization, S.E. and A.L.M.; methodology, A.L.M. and S.E.; validation, S.E. and A.L.M.; formal analysis, S.E. and A.L.M.; resources, S.E. and A.L.M.; data curation, S.E.; writing--original draft preparation, S.E. and A.L.M.; writing--review and editing, A.L.M.; visualization, S.E.; supervision, A.L.M.; project administration, S.E.; funding acquisition, S.E. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the European Union's Horizon 2020 research and innovation program under grant agreement number 957406 (TERMINET).  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.