# A Topology Optimization Method for Reducing Communication Overhead in the Kalman Consensus Filter
## 4 Performance Evaluation and Discussion  
In this section, the performance of the proposed network topology optimization method is evaluated. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Communication Overhead** & **Computational Complexity** \\ \hline The proposed method & \(\sum\limits_{b=1}^{K}\left(|\text{del}^{\prime}_{N_{i}}|+|\text{add}^{\prime}_{N_ {p}}|\right)\) & \(O(KI^{2})\) \\ The WWKJ method [29] & \(I^{2}\sum\limits_{b=1}^{K}\left(|\text{del}^{\prime}_{i}|+|\text{add}^{\prime} _{N_{p}}|\right)\) & \(O(T_{m}I^{2})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of the communication overhead and computational complexity. This section considers a WSN \(\mathcal{G}=(\mathcal{V},\mathcal{E})\) covering a region of \(100\)m \(\times\)\(100\)m, \(I=15\), \(K=40\), and \(D=60\) m, as shown in Figure 1, where the algebraic connectivity \(\lambda_{2}(\mathcal{G})=0.78\). Figure 2 shows the corresponding adjustments of the original network \(\mathcal{G}\), where black lines and blue lines represent the deleted links and the added links, respectively. The normalized topology structural parameters as the corresponding adjustments of \(\mathcal{G}\) are illustrated in Figure 3. From Figure 3, we observe that the degree distribution of nodes (\(d\)) becomes more uniform, and the average path length decreases (\(l\)), while clustering coefficient (\(c\)) decreases. Figure 4 shows the optimized topology \(\mathcal{G}^{\prime}\), where \(\lambda_{2}(\mathcal{G}^{\prime})=1.96\). Figure 1: Original network topology. Figure 2: The corresponding adjustments of the original topology. From Figures 1-3, the proposed network topology optimization method is effective in that the algebraic connectivity of network topology is improved. Furthermore, the performance improved by \(d\) and \(l\) is larger than the performance decreased by \(c\) because of the locality of clustering coefficient (\(c\)). In the tracking problem of a dynamic interested object, the state equation of the object is  
\[\mathbf{x}(k+1)=\mathbf{A}\mathbf{x}(k)+\mathbf{B}\mathbf{w}(k)\]  
with parameters  
\[\mathbf{A}=\mathbf{I}_{3}+\gamma\mathbf{A}_{0}+\frac{\gamma^{2}}{2}\mathbf{A}_{0}{}^{2}+\frac{ \gamma^{3}}{6}\mathbf{A}_{0}{}^{3}\]  
\[\text{where}\ \mathbf{A}_{0}=\left[ \begin{array}{ccc}0&0&-1\\ 0&-1&0\\ 1&0&0\end{array} \right],\gamma=10^{-3},\text{ and }\mathbf{B}=0.005\text{I}_{3}.\]  
Figure 4: Optimized network topology. Figure 3: Normalized topology parameters. The initial position of the object is arbitrary. For example, \(x(0)=\left(5,15,10\right)^{\text{T}}\). The initial covariance of the prediction error \(P(0)=10\text{I}_{3}\) is adopted according to the parameters in [5]. The measurement equation of the object is  
\[\mathbf{z}_{i}(k)=\mathbf{H}_{i}\mathbf{x}(k)+\mathbf{v}_{i}(k)\]  
where \(\mathbf{H}_{i}\) has three forms, namely \(\mathbf{H}_{i}=\left[ \begin{array}{cccc}1&0&0&0\end{array} \right],\mathbf{H}_{i}= \left[ \begin{array}{cccc}0&1&0\end{array} \right]\text{or}\,\mathbf{H}_{i}=\left[ \begin{array}{cccc}0&0&1\end{array} \right]\), \(R_{i}=400\sqrt{i}\) at node \(i\), and \(i\)\(\in\)\(\left[1,2,\,\ldots,\,I\right]\).