# Clustered Distributed Learning Exploiting Node Centrality and Residual Energy (CINE) in WSNs
## 4 Performance Evaluation
### Numerical Results
In the same table, we report the standard deviation of the residual energy in each cluster and, for the sake of comparison, we report the standard deviation of the residual energy that would be obtained without considering the multi-criteria definition of the penalty \(\mathcal{G}\). We observe that by applying the proposed approach, significantly lower standard deviation values are obtained, which means that there is an improvement in fairness. Note that Clusters 1 and 2 comprise nodes with lower centrality values. On the contrary, Cluster 3 in the center of the network comprising nodes with higher centrality metrics also has the lowest residual energy when compared to the other clusters. In Figure 7, we report the average residual energy versus the number of iterations for different values of threshold \(\sigma_{TH}\). Obviously, the residual energy increases when the threshold value increases. This is because by increasing the threshold, the number of chunk transmissions decreases. This is confirmed by the values in Table 2.  
Figure 5: Average loss vs. number of iterations for different values of \(\alpha\). Figure 6: Average loss vs. number of iterations for different thresholds. Therefore, from the obtained numerical results, we derive that the residual energy and centrality metrics play an important role in enhancing the distributed learning in a network. In Figure 8, we show the average residual energy in each cluster. We observe that cluster '3' has consumed higher energy than the other clusters. This can also be seen in Figure 9, where we show the residual energy in each node. The residual energy is lower in cluster '3' as the nodes in cluster are in the center of the network with higher centrality values. Thus, they have the opportunity to gossip about the model parameters a few times more than the nodes in other clusters. Similarly, the nodes in cluster '1' exhibit higher energy consumption as they are not well connected like the other nodes, and they have lower centrality values. This shows how the centrality of the nodes can affect the energy consumption of the nodes. In Figure 10, we compare CINE with other cases in which we exploit neither the centrality information nor the residual energy information. On comparing the three cases, we observe that our CINE protocol that exploits both the centrality and the residual energy information outperforms the cases in which either the the centrality information or the residual energy information is not utilized. Though the average loss obtained by CINE is comparatively higher in the initial iterations, the average loss reduces significantly after 10 iterations and converges quickly when compared to the other two cases without the centrality information or the residual energy information. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Cluster** & **1** & **2** & **3** & **4** & **5** \\ \hline avg. \(rE_{CH}\) & 84.18 & 72.36 & 46.13 & 66.53 & 68.06 \\ std. \(rE_{CH}\) & 3.31 & 3.62 & 16.52 & 10.09 & 5.18 \\ std. \(rE_{CH}\) (benchmark) & 4.82 & 7.29 & 20.08 & 16.65 & 8.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Residual energy, \(rE_{CH}\), after 80 iterations. \begin{table}
\begin{tabular}{c c} \hline \hline
**Threshold** & **Average Data Chunk Transmissions** \\ \hline
0.5 & 4.68 \\
1 & 2.72 \\
1.5 & 1.92 \\
2 & 0.32 \\
2.5 & 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average data chunk transmissions. Figure 7: Average residual energy vs. number of iterations as a function of the threshold \(\sigma_{TH}\). Furthermore, we compare CINE with an ordinary gossiping approach in Figure 11. In case of ordinary gossiping, each node is selected as a cluster head to periodically broadcast the model parameters (we call this approach _standard gossiping_). We show the average loops with respect to the number of iterations for both CINE and the _standard gossiping_. It is evident that the performance in CINE is significantly better than the _standard gossiping_.