## 6 Results  
This section presents the results obtained for both the latency and the lifetime models by initially comparing the MIP results obtained for small networks with the GRASP meta-heuristic ones. After that, results for bigger networks are presented. All the calculations have been performed using a MacBook Pro early 2015 (Apple, Cupertino, CA, USA) with a 2.7 GHz Intel Core i5 processor (Santa Clara, CA, USA) and 8 GB 1867 MHz DDR3 memory.  
The initial MIP and GRASP comparison results are obtained by studying 10 networks of \(N=100\) nodes evenly distributed in a 200 m radius area. For each of these networks, we show the latency and lifetime values for both methods as well as the nodes pinpointed as critical. Such metrics allow us to conclude whether the effectiveness of the GRASP is good enough.  
The study of larger networks has been initially performed with the same 200 m radius area for even node distribution. For \(N=\{100,200\}\), results are shown for up to five simultaneous critical node removals. In the case of \(N=300\), results are reduced to four critical nodes. Finally, for \(N=500\), the three most influential nodes are calculated.  
Networks with more node density have not been studied because of the already small values obtained during the aforementioned tests, and, since GRASP is not an exact tool, it is difficult to extract correct conclusions with such little margin error. Moreover, the limitation in the area to deploy the sensors highly increases the complexity of the problem due to the amount of neighbors each node can send data to, increasing the explorable neighborhood during construction and search phases.  
For all that, it has been decided to scale the \(N=100\) and 200 m radius network in order to study larger ones with the same node density. Moreover, the _hops to base station_ threshold feature is tuned to also consider this scaling. Table 6 shows the radius and hops to base station thresholds used for networks with \(N=\{100,300,500,700,1000\}\) nodes. The rest of the feature thresholds remain the same. Since these new networks share the same characteristics in terms of node density and feature thresholds, but, scaled up, we want to see whether the impact of critical nodes also remains the same. The decision for not studying more critical node impact relies in the trade-off between the time needed for finding the solution and the network degradation variation.  
Table 7 shows a comparison between the MIP and GRASP execution times needed for selecting the most critical node in a network of \(N=100\) nodes. As it can be seen, the difference in time is substantial for such small networks. When increasing the network size, not only the complexity of the problem increases, but also the number of iterations to perform. That is, for a network of \(N=100\) sensors, it is needed to execute the MIP model \(N\) times in order to find the most critical node. For this, and, due to the inner high exponential complexity that MIP presents versus the one that GRASP offers, it has been decided to develop the meta-heuristic for obtaining results for bigger networks and a wider set of critical nodes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Nodes** & **Radius** & **Hops to Base Station** \\ \hline
100 & 200 & 1 \\
300 & 350 & 2 \\
500 & 450 & 2.5 \\
700 & 530 & 3 \\
1000 & 630 & 3.5 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Larger network parameters scaled from base \(N=100\) node network.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **MIP** & **GRASP** \\ \hline
100 Nodes & 2 min & 5.4 s \\ \hline \hline \end{tabular}
\end{table}
Table 7: Comparison of the execution times for the MIP model and the GRASP meta-heuristic for \(N=100\) nodes and one critical node removal.