## 3 Contextual Received Signal Strength Approach
### CRSS Approach
#### 3.4.3 Probabilistic Approach in Interpreting RSS  
In our examination of the network's connectivity, we consider \(\bar{R}\) to represent the average effective communication radius, assuming a homogeneous propagation environment. This radius is crucial for estimating the area \(\bar{A}\) within which a typical sensor node can communicate, calculated as the area of a circle, \(\bar{A}=\pi\bar{R}^{2}\). This average range is pivotal in our probabilistic approach for interpreting the RSS values. The expected average number of neighbors falling within a radius R around a given node is described as  
\[E(\xi)=\lambda\pi R^{2}. \tag{14}\]  
Acknowledgments \(\bar{R}\) as an average range allows us to probabilistically interpret the RSS data from sensors, providing a basis for inferring the likelihood of node connectivity within this range. This average range, and the area that it defines, become key in localizing nodes and optimizing the performance of the CRSSA method in a WSN setting to find the number of adjacent or one-hop neighbors for each node, which is pivotal for deducing its communication range. This calculation represents the expected count of adjacent neighbors for any given node \(i\), providing an assessment of the network's connectivity. On the other hand, the number of connected neighbors, \(n_{i}\), to node \(i=1,\cdots,N\), can be obtained from the connectivity context data as  
\[n_{i}=\sum_{i=1}^{N}C_{ij}. \tag{15}\]Further, the average number of adjacent neighbors per node is derived by the connectivity matrix considering the individual communication ranges, which may vary due to environmental dynamics but are averaged out across the network:  
\[\bar{n}=\frac{1}{N}\ \sum_{i=1}^{N}n_{i}=\frac{1}{N}\ \sum_{i=1}^{N}\ \sum_{j=1}^{N}C_{ij}. \tag{16}\]  
By leveraging the connectivity matrix, we can accurately determine the average number of connected neighbors for each node, thereby enabling a detailed analysis of network density and communication coverage. This level of specificity is crucial for the effective implementation of the CRSSA method as it directly impacts the localization accuracy within the WSN. * Range Estimation, \(\hat{R}\) The values in Equations (14) and (16) are equivalent; hence, based on these equations, the average range \(\hat{R}\) is estimated as \[\hat{R}=\sum_{i=1}^{N}\frac{\hat{R}_{i}}{N}=\sqrt{\frac{\sum_{i=1}^{N}\sum_{j= 1}^{N}C_{ij}}{\lambda\pi N}}.\] (17)
* PLE Estimation Theoretical and measured-based propagation models indicate that average received signal power decreases logarithmically with distance [29]. Hence, based on the log-distance path loss model, and assuming the estimated average communication range obtained in Equation (17) as the maximal distance reached by a transmitted signal when the received power measured at this node is equal to the power detection threshold \(P_{threshold}\), the PLE, \(\hat{\gamma}\), can be estimated as \[\hat{\gamma}=\frac{-P_{threshold}+PL(d_{0})}{10 log_{10}(\hat{R})}.\] (18)
* Inter-node distance estimation Our proposed approach is an anchor-free solution; thus, it estimates the relative positions with respect to a coordinate system established by a reference group of nodes. Relative positioning information is sufficient for some applications, such as location-aided routing or direction-based routing algorithm [36]. Moreover, the recovery of the network's key propagation characteristics and its geometry can be addressed by relative positioning mainly from the connectivity information. In relative localization, nodes are localized with respect to each other; thus, we estimate the distance separating them. In this work, we will focus on estimating the distance separating two neighbors nodes at one hop. Since the objective is to prove the efficiency of the proposed CRSSA, distances between nodes at more than one hop are beyond the scope of this work and will be addressed in future publications.