## 6 Numerical Results
### Energy Consumption of Measurements Formation in MLMS Construction and Maintainance  
We randomly disperse \(N\) nodes in an 800 \(\times\) 800 two-dimensional plane. Each node has the same transmission distance of \(r\). Only the two nodes, whose Euclidean distance (\(d\)) between each other is less than \(r\), can communicate with each other. In order to guarantee the network connectivity, the communication distance of nodes is fixed at \(800\sqrt{5}/N\)[22], and the collection node is located at (400,800). The detailed parameters are shown in Table 2. We compare our MLMS method with the schemes mentioned in the Section 2, including Plain-CS, Hybrid-CDG, and the PB-CDG [14]. Measurement coefficients for all of these methods obey Bernoulli distribution (entries take value of either \(1\) or \(-1\), each with probability of \(1/4\), and value \(0\), with probability of \(1/2\)). We first test the energy consumption of various measurement formation methods with the increasing number of measurements and the nodes in the network (density). Note that, similar to reference [14], we measure the node's energy consumption by the number of packets it transmitted. The experiment results are shown in Figure 9. We can see that Non-CS and Hybrid-CS incur higher energy cost than that of PB-CDG and the proposed MLMS method, because all measurements in these two methods are collected through a single fixed forwarding tree, which does not consider the sparse structure and derivative routing path of each measurement. Despite the fact that both MLMS-CDG and PB-CDG are derived from the idea of a Steiner tree, MLMS-CDG consumes less energy than PB-CDG (up to 14.52% and 17.76% in Figure 9A,B, respectively). This is largely because MLMS incurs lower \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\Delta\textbf{M}\) & **Errors** & **Upper Bound** & **Average Measurement** & **Standard Deviation** \\ \hline
1 & 128 & 200 & 31.0733 & 6.9430 \\
2 & 2 & 200/3 & 34.8133 & 4.8461 \\
3 & 0 & 200/9 & 36.7350 & 4.8882 \\
4 & 0 & 200/27 & 37.9867 & 4.8428 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Data reconstruction experiment on sparse Bernoulli measurement by using Proposition 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline size of network & 800 \(\times\) 800 \\ number of nodes & 100–1200 \\ max transmission radius & 800\(\sqrt{5}\)/(number of nodes) \\ sink node position & (400,800) \\ packet length & 120 Byte \\ PHY layer & Free space propagation model \\ MAC layer & CSMA/CA \\ E\_elec & 50 nJ/bit \\ E\_start & 250 nJ/packet \\ channel type & erasure channel \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the simulation experiments. overhead of measurement coefficient distribution than PB-CDG (with an average of 11.4% in different network densities). We then experiment on the number of control messages of PB-CDG and MLMS measurement-formation methods with the increasing value of the node density in the network. Because the PB-CDG method did not specify the generation method of measurement coefficients as well as the processing strategy of the node failures and the packet losses, we treat the PB-CDG scheme in the same manner of measurement generation as MLMS. Figure 10A shows the overhead of constructing the measurement-formation paths by using original methods and compressed construction method (MLMS-CC). With the growth of the number of sensor nodes, we observe that PB-CDG increases more sharply than both MLMS and MLMS-CC methods do. This is mainly because PB-CDG builds several line-shape paths for each measurement formation which will take a greater number of control messages in distributing global seeds (overheads of MLMS and MLMS-CC are averagely 53.33% and 68.70% less than that of PB-CDG). Then, we experiment on the influence of the intensity of interested nodes.