# Detection of Transmission State of Multiple Wireless Sources: A Statistical Mechanics Approach
## 1 Introduction
One of the simplest and most robust methods is the use of non-coherent energy detection [7], which, compared to, e.g., time-of-arrival techniques, has low complexity and does not heavily depend on the presence of multiple reflections inside the propagation medium. Figure 1: A CR-enabled IoT network deployment in a smart home environment. A number of previous works have studied the detection problem. More specifically, [8] focused on fixed architectures with feedback from a fusion center over limited bandwidth channels, which undergo multiple scattering and noise, and [9] focused on finite-sized networks with a specific graph structure. In both cases, a global fusion center is assumed, to which each SU transmits a function of its observation over a noisy channel. The existence of a fusion center becomes problematic when the network becomes large, especially when it varies with time. In this case, SUs need to dynamically update their information in a distributed fashion, indicating the necessity of distributed detection algorithms. There are several algorithms that belong to the class of Bayesian inference algorithms and are known as message-passing algorithms [10], which, along with their variations [11, 12, 13], have been proven to be near-optimal in many optimization and inference problems. These algorithms constitute an appealing distributed computational scheme, which takes advantage of their local structure, by conveying iterative messages along the edges of a factor graph, with each message being updated by the messages coming in from the adjacent nodes. This message update process allows for parallelization and local updating [14, 15] since it decomposes the optimization problem into smaller parts that can then be solved independently. However, the performance of the message-passing algorithms is network-specific since they require knowledge of the graph's connectivity structure. Convergence speed is essentially sublinear with respect to the number of sources. Only two conditions are necessary for successful convergence, i.e., the sparsity [16] of the connectivity matrix and the presence of a local tree-like structure that prevents loops, where the algorithm typically stalls. It has mostly seen applications using binary states, in this case, corresponding to whether the sources are active emitters. For such systems, the convergence to the optimal threshold is almost always guaranteed for both random fully connected [17, 18] and sparse (Erdos-Renyi-like) random graphical models. Such algorithms have also been applied to more general alphabets [19] in the case of non-sparse, fully connected graphs in the context of compressed sensing, where they have been shown to converge in the case of random graphs represented by full matrices [20, 21] with only a few further assumptions. In this paper, we apply the idea of message passing to the detection of the operational state of a network of primary sources in the presence of secondary users. In contrast to other approaches that focus on situations with a few primary and secondary sources, we are particularly interested in cases where these become large. Hence, our approach is not limited by the size of the network. We assume that the locations of the primary sources are known through radio neighborhood maps. Due to the multiple interferences that any given SU receives from multiple primary sources in its vicinity, it is not at all obvious how to robustly detect which primary sources present are operational. As demonstrated in this paper in Section 7, myopic approaches using a single SU to detect the state of a source in the presence of others fail spectacularly. Our main contribution in this paper is to analyze and quantitatively predict, for a given environment, the appropriate system's parameter space, such as the node density and signal-to-noise ratio (SNR), where reliable detection of emitting sources can be achieved.