## 4 Algorithmic Approaches
The density is calculated by dividing the total number of edges in the network by the total amount of edges in the graph as if it were a fully connected graph. Finally, the fourth row indicates the number of users to be connected to the SDN network. The terms \(N_{1}\), \(N_{2}\), \(N_{3}\), \(N_{4}\), \(N_{5}\), \(N_{6}\), \(N_{7}\), \(N_{8}\), \(N_{9}\), \(N_{10}\), \(N_{11}\), \(N_{12}\), and \(N_{13}\) represent the following networks: Arpanet196912, Abiline, Aarnet, Ans, HurricaneElectric, Atmnet, Bbnplanet, Bics, CrlNetworkServices, Internet2 OS3E, Geant2009, NetworkUsa, and Vtl-Wavenet2008, respectively. In Table 2, numerical results are reported for models \(M_{1}\) and \(M_{2}\) across the three proposed scenarios using real benchmark networks. The first column of the table lists the names of the studied networks. From the second to the seventh column, the following information is presented: the objective value of model \(M_{1}\), the best bound, the MirGap, the processing time in seconds, the number of controllers (NC), and finally, the number of nodes from the branch and bound (\(B\&B\)) method, respectively. Similarly, from the eighth \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline
**Network Name** & \(N_{1}\) & \(N_{2}\) & \(N_{3}\) & \(N_{4}\) & \(N_{5}\) & \(N_{6}\) & \(N_{7}\) & \(N_{8}\) & \(N_{9}\) & \(N_{10}\) & \(N_{11}\) & \(N_{12}\) & \(N_{13}\) \\ \hline Nodes & 4 & 11 & 13 & 18 & 20 & 21 & 27 & 33 & 33 & 34 & 34 & 35 & 87 \\ Edges & 4 & 14 & 24 & 50 & 30 & 22 & 28 & 48 & 38 & 41 & 52 & 39 & 89 \\ Density (\%) & 66.7 & 25.5 & 30.8 & 32.7 & 15.8 & 10.5 & 8 & 9.1 & 7.2 & 7.3 & 9.3 & 6.6 & 2.4 \\ Users & 24 & 66 & 78 & 108 & 120 & 126 & 162 & 198 & 198 & 204 & 204 & 210 & 522 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Real benchmark networks ([[https://www.topology-zoo.org/](https://www.topology-zoo.org/)]([https://www.topology-zoo.org/](https://www.topology-zoo.org/)) accessed on 4 June 2024). to the thirteenth column, the same information is listed as in columns two to seven, but for the \(M_{2}\) model. First, we observe that the solutions obtained using models \(M_{2}\) and \(M_{3}\) decrease when the network node increases. It should also be noted that the data in the solution, best bound, MipGap, and CPU time fields are generally approximated to two decimal places, except for the scenario with a controller density of 40%, where, for the solution, best bound, and MipGap columns, it is necessary to have approximations to three decimal places. It can be noted that in almost all the studied networks, the optimal solution was found. Only in the case of the VtlWavenet2008 network was a close-to-optimal solution found. For this network, the solution time used exceeds the given maximum time, which is much higher than that used for solving the other networks under study. It should also be mentioned that the number of \(B\&B\) nodes used to solve model \(M_{1}\) does not follow the trend of the network under study, and it is related to the CPU used. For example, in the scenario with a controller density of 20%, the networks CrlNetworkServices and Bics have the same number of nodes. However, Bics has a larger number of edges than CrlNetworkServices, and due to this difference, the resolution time and \(B\&B\) nodes are significantly larger than those of the Bics network. From the information presented in Table 2, we can also mention that Model \(M_{2}\) is more efficient in most of the studied networks, except for the