## 6 Experimental Results
### Performance Evaluation of CLAC-KHM
\[\theta=\frac{d}{r} \tag{15}\]  
The haversine distance is calculated via the haversine function which equals \(\text{hav}(\theta)=\text{sin2}(\theta)\), in which we are given a direct calculation of the longitude and latitude of the two points, as elaborated below:  
\[\textit{hav}(\Phi)=\textit{hav}(\Phi_{2}-\Phi_{1})+\textit{cos}(\Phi_{1}) \textit{cos}(\Phi_{2})\textit{hav}(\lambda_{2}-\lambda_{1}) \tag{16}\]  
To solve for the distance d, we apply the archaversine (inverse haversine) to h = hav(\(\theta\)) or use the arcsine (inverse sine) function:  
\[d=r*archav(h)=2*r*\textit{aresin}\left(\sqrt{h}\right)= \tag{17}\]  
\[2*r*arcsin\sqrt{\textit{hav}(\varnothing_{2}-\varnothing_{1})+\text{cos}( \varnothing_{2})\text{cos}\left(\varnothing_{1}\right)\textit{hav}(\lambda_{ 2}-\lambda_{1})}=\]  
\[2*r*\text{arcsin}(\sqrt{\textit{sin}^{2}(\frac{\phi_{2}-\phi_{1}}{2})+\text {cos}(\phi_{1})\text{cos}(\phi_{2})\textit{sin}^{2}(\frac{\lambda_{2}-\lambda _{1}}{2})})\]  
The \(\Phi_{1}\) and \(\Phi_{2}\) are the latitudes and \(\lambda_{1}\) and \(\lambda_{2}\) are the longitudes of data_point\({}_{1}\) and data_point\({}_{2}\) in our function, respectively. Since LAC is based on k-means, the CLAC-KHM algorithm is evaluated with standard k-means in addition to LAC. Spectral clustering is a method with roots in graph theory, whereas k-means, k-medoids, and k-median are methods based on the distance between the centers and data points. Therefore, they are different from spectral. That is why we also compared CLAC-KHM with the spectral algorithm on four topologies (Dialtecom, Intellifiber, Iris, and Aarnet) with four different sizes. Table 4 shows the topologies' details studied throughout our evaluations [42]. Table 5 presents the results. As mentioned previously, the Aarnet topology comprises 19 nodes, and for accommodating this topology, the number of controllers was examined within the range of 3 to 10. Consequently, in the table, you will observe a shaded area representing the number of controllers exceeding 10 for this particular topology. The given values demonstrate that the delay between switches and controllers in k-means is less compared to that of spectral. However, the values of OF are not higher than spectral in all topologies with any number of controllers. These results determine that the CLAC-KHM achieves better results than the other algorithms as well. For the sake of a better explanation, the OF values are depicted for all algorithms in Figures 1-3 for the four topologies. In this study, it was observed that Dial_telecom is the largest topology, and the spectral algorithm achieved better results compared to k-means. However, as shown in Figure 2, LAC performed as well as the spectral algorithm. Nevertheless, it is noteworthy that CLAC-KHM achieved the best performance in this topology. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number of Nodes** & **Number of Switches** & **Number of Edges** & **Geographical Area** & **Geographical Location** \\ \hline Dial telecom & 193 & 151 & country & Czech Republic \\ Aarnet & 19 & 24 & country & Australia \\ Intellifiber & 73 & 97 & country & USA \\ Iris & 51 & 64 & region & Tennessee, USA \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Features of the topologies. Figure 2: Objective functions for all algorithms on Dial_Telecom. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Dial Telecom**} & \multicolumn{4}{c}{**Intelligiber**} & \multicolumn{4}{c}{**Iris**} & \multicolumn{4}{c}{**Aarnet**} \\ \hline  & **NOC** & **S-C** & **C-C** & **OF** & **S-C  
Figure 3 indicates the fact that the CLAC-KHM is the most efficient rather than the other representative algorithms. Meanwhile, it depicts that when there are a small number of controllers, the spectral clustering outperforms k-means in terms of OF, and contrarily, with increasing controller numbers, k-means has better results. It is worth bringing up the fact that they have no predictable behaviors in Intellifber.