## 4 Experimental Results
### Comparative Study between the Spectral Clustering Algorithms  
In our experiments, we compared the efficient evolutionary spectral clustering frameworks with the normalized efficient spectral clustering algorithm, which we named IND [36]. The IND method applies partitioning on the snakes at current time period \(t\) and ignores all historic data before the current time period. The number of clusters for the two efficient evolutionary spectral clustering frameworks (PCQ and PCM) and the IND method is automatically detected using the density peak method. First, we compute the average link speeds and the standard deviation for time period \(t=6\) with the objective of validating the effectiveness of the efficient evolutionary spectral clustering frameworks in separating homogeneous connected clusters. The results of the evaluation metrics are presented in detail in Table 3.  
By comparing the mean values of link speeds and the standard deviation in different clusters, it is clear that the mean link speeds between the partitions are dissimilar for PCQ and PCM because they give a good separation between partitions. Moreover, the small value of the standard deviation for each cluster defines good intra-cluster homogeneity. Next, we consider all time periods from \(t=1\) to \(t=48\), and we evaluate the total cost values for all three methods. For all costs, a lower value means better results. Figure 17 shows the results of the total cost for the IND method and for both PCQ and PCM frameworks. As can be seen, the PCQ and PCM are better at minimizing the total cost measure over the IND method. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Cluster** & **Red** & **Green** & **Blue** & **Magenta** \\ \hline IND & 6.03/0.79 & 7.17/0.74 & 9.98/2.86 & 10.03/3.63 \\ PCQ & 7.25/1.38 & 9.52/4.05 & 10.03/2.86 & 13.72/3.06 \\ PCM & 7.05/1.32 & 9.44/2.82 & 10.02/3.81 & 13.72/3.06 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Average speed values (m/s) over standard deviation (average speed/standard deviation) for time period \(t=6\) in the case of the IND, PCQ and PCM frameworks. Figure 17: Variation of the total cost for each time period in the case of time period = 10 min. We compare the average runtime in seconds for one period using the PCQ and the PCM frameworks. The time is computed for the 48 periods, and we take the average time using MATLAB on a PC with two Intel CPUs at 2.33 GHZ and eight gigabytes of RAM. It is clear that the two frameworks of the new methodology induce a better performance in computational time than does the evolutionary spectral algorithm (Figure 18). In order to validate the performance and the smoothness of the evolutionary spectral clustering frameworks in finding clusters that fit the current data while simultaneously not deviating from previous history, in this experiment, we plot the partitioning results of the Amsterdam network for the PCQ, PCM and IND methods. In order to match the clusters between consecutive time-steps, we used the Kuhn-Munkres Hungarian algorithm, which performs one-to-one cluster matching between consecutive partitioning based on the number of common links between clusters [37; 38]. This algorithm, which is the most popular polynomial time algorithm for solving classical assignment problems, allows us to follow the evolution of the clusters over time. Figure 19 illustrates the partitioning results with the number of clusters detected automatically by the density peak method. The analyzed time periods are from \(t=2\) to \(t=5\), which corresponds to day hours from 7:10-7:40 a.m, one of the peak times of a weekday. The bold color represents the links specifying the deviation from one cluster to another. For time periods \(t=2\) to \(t=5\), we find that, for the IND method, the number of links in a cluster changes rapidly. However, the PCM and PCQ frameworks are more stable when compared to the historic data. It is shown that the PCQ and PCM frameworks assure the best performance and give more stable and consistent cluster results compared with the IND method.