## 3 Analysis
### Stationary Property  
For illustration simplicity, we consider a scenario shown in Figure 3. This simple setup consists of a beacon (called \(Beacon_{L}\)) and two sensors (called \(Sensor_{C}\) and \(Sensor_{R}\)). They are placed on the X-axis where \(Beacon_{L}\), \(Sensor_{C}\), and \(Sensor_{R}\) are located at \(-1\), 0 and 1 on the X-axis respectively.  
We use the notation \(C_{n}\) and \(R_{n}\) to represent the computed sensor coordinates after \(n\) rounds of iterations in the algorithm. Assume the algorithm is initialized using the true locations, we have \(C_{0}=0\), \(R_{0}=1\). Beacons are always in fixed location, and in this setup \(L=-1\). The estimated distance \(d\) from a particular node to another is calculated according to the path loss model [18] based on the received signal strength. Due to the log-normal shadowing [18] resulting the log-distance path loss model [12; 22], the estimated distance \(d\) from a node to another is a log-normal random variable of mean \(d\) and a particular fixed variance \(\sigma^{2}\)[22]. Therefore, in the scenario of Figure 3, we have three log-normal random variables for the estimated distances, where \(d_{1}\) is the estimated distance from \(Sensor_{C}\) to \(Sensor_{R}\), \(d_{2}\) is the estimated distance from \(Sensor_{C}\) to \(Beacon_{L}\), and \(d_{3}\) is the estimated distance from \(Sensor_{R}\) to \(Sensor_{C}\). We use the notation \(d_{k,n}\) to represent the \(k\)-th estimated distance updated in the \(n\)-th round of iterations. From the true locations of three nodes, these three random variables have the parameters as \(E[d_{k,n}]=1\), \(Var(d_{k,n})=\sigma^{2}\), where \(k=1,2,3\).  
Figure 3: A scenario for a beacon node and two sensor nodes.  
According to our algorithm, since both \(Beacon_{L}\) and \(Sensor_{R}\) are neighbors to \(Sensor_{C}\), we get  
\[C_{n+1}=(1-2\delta)C_{n}+\delta R_{n}+\delta(d_{2,n}-d_{1,n}+L) \tag{6}\]  
where \(\delta\) is the step size constant specified in the algorithm. Similarly, \(Sensor_{R}\) only has the neighbor \(Sensor_{C}\), and we have  
\[R_{n+1}=(1-\delta)R_{n}+\delta C_{n}+\delta d_{3,n}. \tag{7}\]  
Formulas (6)-(7) can be viewed as a type of first order vector autoregression systems. The main purpose of this analysis is to show that the above system is stable. Precisely, we wish to show the variances of \(C_{n}\) and \(R_{n}\) after a long run remain finite. To solve (6)-(7) for \(Var(C_{n})\) and \(Var(R_{n})\), we first determine their covariance to be  
\[cov(C_{n},R_{n})=\frac{(1-2\delta)Var(C_{n})+(1-\delta)Var(R_{n})}{3-3\delta}.\]  
With the above result, using time series analysis, we further get  
\[Var(C_{n}) = \frac{-\delta(3\delta^{2}-13\delta+10)}{\delta^{3}-9\delta^{2}+2 2\delta-12}\hat{\sigma}^{2}\] \[Var(R_{n}) = \frac{-2\delta(3\delta^{2}-11\delta+7)}{\delta^{3}-9\delta^{2}+ 22\delta-12}\hat{\sigma}^{2}.\]  
From the last result, the variances of \(C_{n}\) and \(R_{n}\) are positive and remain finite when \(\delta<3-\sqrt{5}\approx 0.764\). This concludes that the computed locations are wide sense stationary with a proper setting of the \(\delta\) parameter. The \(\delta\) parameter controls the step size in the algorithm, thus a large step size increases the fluctuation of \(C_{n}\) and \(R_{n}\) which in turns increases their variances.  
From the above simple scenario, we see that while the measured signal strength fluctuates over the time, the estimated location also fluctuates and stays stationary around its true location, even for the sensor where none of its neighbors is a beacon. This stationary property suggests that cooperative approach for the spring-relaxation technique maintains stability in location estimation.