## 6 Results
### Latency Results
The difference of the MIP latency values obtained for the two nodes is minimal, making it difficult for the GRASP to actually find the optimal value, and, instead, obtaining a slightly worse one with a different node pinpointed as critical. Figure 2(a) depicts the results of the previously explained tests in which node density is increased by adding more nodes to the same network deployment area. Each of the lines drawn A-C shows the average impact when removing the most C critical nodes from the network. There are two main conclusions that can be extracted. Firstly, it can be clearly seen that, as the number of nodes inside the network increases, its deterioration when removing a fixed quantity of critical nodes decreases. As there are more nodes inside the network, the paths to which a node can transmit increase, the connections between the nodes are more distributed and, thus, network flexibility increases, reducing the impact of critical node removal. Specifically, it can be seen that latency deterioration in 500 node networks is half that of 100 node networks. Secondly, when looking at the values of a single network size, it can be seen that the removal of more critical nodes has higher impact on the latency increment. The theory behind such result is the following: let \(c\) be the critical node removed and \(Parents_{c}\) the nodes that send data to it. When \(c\) is removed, latency is reduced by the amount of hops \(c\) needs to send data to base station, but, at the same time, it increases by at least \(|Parents_{c}|\) since the new path for such nodes will be at minimum one hop larger. If it happens to be a faster path, it would have been selected either during the construction or search phase. This explanation is not true when \(c\) shares two paths with the exact same length or the network does not have inner nodes. However, such scenarios are very specific and only possible in networks with very few nodes, which is not the case. The explanation is also extensible when  
Figure 3: Average and maximum latency increase values and execution times for networks with nodes distributed in a 200 m radius area. more than one critical node is removed. It is worth mentioning that these are theoretical calculations and they might slightly differ from the obtained results in some cases due to the inexcattitudes of the GRASP meta-heuristic. Additionally, in order to clearly see how much impact the removal of \(C\) critical nodes can have in a network, the maximum values are shown in Figure 3b. Even though the tendency of the curves remains almost the same as before, the latency increment is much higher, arriving to double the average in some cases. Figure 3c,d show, respectively, the average and maximum execution time needed for calculating the previous values. As it can be seen, the time needed to remove a fixed amount of critical nodes increases exponentially with the number of nodes, mainly due to the amount of neighborhood checks to perform. Moreover, when looking at a fixed network size, it can also be seen that the tendency when removing a higher amount of critical nodes is exponential because of the combinatorial factor introduced for calculating all possible combinations of critical nodes' subsets. Figure 4a shows the average results after scaling the \(N=100\) node network deployed within a 200 m radius area. Performing a scaling to keep the same characteristics as the \(N=100\) and 200 m radius network allows us to conclude that the latency deterioration remains the same, with very small variation, when removing the same amount of critical nodes. However, when looking at the maximum results in Figure 4b, the horizontal tendency disappears. Even though the average is maintained, it can be seen that the worst scenarios can have huge deterioration variations. For instance, the worst network of \(N=500\) nodes increases its latency by 25% when removing more than four critical nodes, and this percentage is drastically reduced to around 10% for the \(N=700\) network.