## 4 Simulation Results
### First Step: Empirical Determination of Alpha  
The first step of the simulations aims to define the value of \(\alpha\) to be used by EBF for comparisons with DBF in the second step of the simulations.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Step 1** & **Step 2** \\ \hline \(\alpha\) & \(\{0.01,0.02,0.04,0.06,\) & 0.1 \\  & \(0.08,0.1,0.2,0.4\}\) & \\ \(N\) & 100 & \(\{50,100,150,\) \\  & & \(200,250,300\}\) \\ \(k\) & \(\{4,6,8,10\}\) & 8 \\ Transmission Range & 295 m & 295 m \\ Interference Range & 887 m & 887 m \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
Figure 3 displays the average number of messages (sent and received) per node during the execution of EBF. The reduction in the number of messages is quite significant up to \(\alpha=0.1\). From this point, the average number of messages tends to stabilize. It is also observed that the higher the network average degree \(k\), the greater the reduction in the number of messages. This is because the performance of DBF worsens as the network becomes denser.  
Figure 4 displays the time spent by the algorithm for the tree construction. Again, there is a significant reduction up to \(\alpha=0.1\). Although this reduction is greater as the network average degree increases, from \(\alpha=0.1\), the runtime is almost the same for all network degrees evaluated.  
Figure 5 displays the average distance (in meters) from nodes to the sink. Up to \(\alpha=0.1\), we can observe that the distance increases slightly. However, from \(\alpha=0.1\) the increase in path length becomes more significant. We can also observe that this increase is more pronounced the lower the network degree. This is because the lower the network degree, the lower the number of possible paths. For the same reason, the distance to the sink increases as the network degree decreases. The graph shows that the increase in path length as a function of \(\alpha\) is approximately linear, which suggests that we should use the lowest possible value of \(\alpha\).  
Figure 4: Runtime.  
Figure 3: Average messages per node.  
Figure 6 shows that \(\alpha\) causes little impact on the average number of hops from sensor nodes to the sink. Up to \(\alpha=0.1\), the difference does not exceed one hop and reaches a maximum of two hops when \(\alpha=0.4\). We also identified that the parameter \(\alpha\) does not influence the number of alternative parents, which is only affected by the network average degree. This is due to the operation of the algorithm. Nodes inevitably receive a paternity offer from all of their neighbors. Therefore, the denser the network, the greater the number of alternative parents. Parameter \(\alpha\) influences only the number of subsequent messages and, thus, the distance of alternative parents to the sink.  
Considering the above results, we set \(\alpha=0.1\) for the comparisons between EBF and DBF. This choice is justified by the following observations:  
* The most significant reduction in the number of messages occurs up to \(\alpha=0.1\) and from this point tends to stabilize;
* The most expressive reduction in the runtime also occurs up to \(\alpha=0.1;\)
* The increase in the distance and number of hops from sensors to the sink is accentuated from \(\alpha=0.1\).  
Figure 5: Average distance to the sink.  
Figure 6: Average hops to the sink.