## 4 Numerical Results
### Probability of Blocking an Authorized Node
We evaluated also the effect of the number of nodes in the area. In scenarios A1-3, we varied the number of nodes in the range \([15,30]\) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **Min (n.Splits)** & **Max (n.Splits)** \\ \hline A1 & 95.28\% (15) & 95.84\% (20) \\ A2 & 95.00\% (15) & 96.04\% (20) \\ A3 & 91.02\% (15) & 96.07\% (25) \\ B1 & 88.19\% (15) & 95.62\% (25) \\ B2 & 86.04\% (15) & 89.89\% (40) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Accuracy variation vs. number of CART splits. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Scenario** & **CART Accuracy** & **Random Forest Accuracy** \\ \hline A1 & 95.43\% & 95.49\% \\ A2 & 95.95\% & 96.14\% \\ A3 & 96.53\% & 96.66\% \\ B1 & 93.91\% & 95.04\% \\ B2 & 87.96\% & 89.74\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average classifiers accuracy for different scenarios. Figure 6: Classification accuracy for scenario A1. and we saw that there is not a performance degradation in terms of accuracy, but obviously the number of splits must be increased. In particular, for up to 25 nodes, the sufficient number of splits is \(N\) increased by the 20%/25%. At 30 nodes instead, with 40 splits, the accuracy decreases down to 79%, and 65 splits are needed to reach the 95% value. In order to further investigate this issue, we considered also different clusters' distributions. We noted that there is not a significant difference if the clusters' position varies, but the number of nodes/clusters is the same. Similarly, leaving unchanged the number of nodes per cluster and increasing the number of clusters up to 10/12 (which corresponds to more than 60,000/70,000 nodes/km\({}^{2}\)), performance is not significantly affected because the number of nodes that can create confusion in the classification process (since AoA and delay attributes are very similar within a cluster) is the same. Obviously, with a higher density of clusters, it is more likely that cluster overlapping occurs (being that the clusters are randomly placed). Thus, a lower number of clusters with a higher number of nodes occurs, and the overall accuracy decreases. We investigated also the case of a higher number of clusters and nodes per cluster as well as the extreme case, where all nodes belong to the same cluster. In the first case, considering 6 clusters with 7 nodes per cluster, there is only a slight reduction in the accuracy, due to the presence of more nodes within the cluster that have similar attributes: the average value of the CART algorithm in scenario B1 is 91.5%, while in the extreme case of a single cluster with 15 nodes, the performance worsens and average accuracy of CART is 88%. In general, up to a certain nodes/clusters density, the reduction in accuracy is limited, but obviously when the density significantly increases, there is a reduction in the accuracy due to the high probability that different nodes have similar attributes. Since the proposed method is based on multiple attributes, it is interesting to evaluate, as these impact the accuracy of the classification. For this reason, the classifiers were used with different sets of attributes, particularly the following:  
* The whole set;
* The whole set without AoA attribute;
* The whole set without delay attribute;
* Only attributes related to the signal intensity (i.e., RSP, peak value and energy) without AoA and delay;
* Only AoA and delay attributes. Table 3 reports the accuracy averaged over the whole dataset for different scenarios. The results show that the classifier using all the attributes outperforms others, using only a subset; in particular, the information provided by AoAs and delays improves drastically the prediction accuracy when compared to a classifier that relies only on \"energy-based\" attributes.