## 4 Simulation Environment and Methodology  
This section presents the simulation environment and methodology used to evaluate the network convergence process of RPL in IEEE 802.15.4 multihop networks. In order to carry out our simulations, we used OMNeT++[29], a well-known C++-based discrete event simulator, jointly with MiXiM, an OMNeT++ framework created for various types of wireless networks [30]. We implemented RPL for this simulation environment. As a side contribution, we have made the simulation code publicly available [31]. The simulated network nodes were static and located following a uniformly random spatial distribution in two-dimensional square areas. We configured the nodes to use the 2.4 GHz band IEEE 802.15.4 physical layer, and the beaconless mode functionality. We set the MAC queue length of the nodes to 1 in order to replicate the value of this parameter in CC2420, a widely used IEEE 802.15.4 radio chip [32]. We used the log-normal shadowing propagation model available in MiXiM. In order to achieve accurate and realistic link behavior in our simulation framework, we carried out experiments and tuned the parameter settings of the simulated propagation model based on the experimental results. The experiments comprised communication in a link between two TelosB nodes [33] running TinyOS [34]. Table 1 shows the main physical and link layer parameter settings used in the simulations. In order to evaluate the influence of the network size in our study, we considered 3 different scenarios, denoted small, medium and large network scenarios. The area of the large network scenario is \(100\times 100\) m\({}^{2}\), which is 5 times greater than the medium network area. There is the same relationship between the medium and small network scenario areas. Figure 2 depicts the nodes' location and existing links in example instances of small, medium and large network scenarios, for a node degree of 10. We also considered the network density in our study, by evaluating node degree values of 5, 10 and 15. Note that this range of network densities covers from relatively sparse networks to highly connected and dense networks. As a result, we simulated a variety of network sizes and densities, ranging from \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Communication range (m) & 9.96 \\ Carrier frequency (GHz) & 2.4 \\ Carrier sense sensitivity (dBm) & \(-\)95 \\ Transmit power (dBm) & \(-\)25 \\ MAC queue length & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main physical and link layer simulation parameters. 8-node to 483-node networks, which cover a wide range of use cases. The number of nodes and the average rank for each network size and density scenario are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Network Size** & **Node Degree** & **Number of Nodes** & **Average Rank** \\ \hline \multirow{3}{*}{Small} & 5 & 8 & 3.09 \\  & 10 & 14 & 3.30 \\  & 15 & 21 & 3.30 \\ \hline \multirow{3}{*}{Medium} & 5 & 34 & 6.52 \\  & 10 & 66 & 6.34 \\  & 15 & 99 & 5.78 \\ \hline \multirow{3}{*}{Large} & 5 & 162 & 16.76 \\  & 10 & 322 & 12.43 \\ \cline{1-1}  & 15 & 483 & 10.74 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of nodes and average rank for each network size scenario and density. Figure 2: Illustration of network nodes and existing links in example instances of (**a**) small, (**b**) medium, and (**c**) large scenarios, for an average node degree of 10. The depicted scenarios have been plotted to scale. For each individual combination of network size scenario, network density and protocol configuration, we simulated 1500 different randomly generated topologies, and evaluated 20 instances of the network convergence process for each topology, yielding a total of 30,000 network convergence instances.