## 5 Results and Discussion
### Real-World Benchmark Network Datasets  
The results obtained for the real-world benchmark datasets compared the time difference between the FCM focal-node identification in normal mode and in parallel mode, as the proportions of the network grew. The results depicted that while there was no  
Figure 5: One of the directed acyclic graphs of the parallel FCM processing.  
considerable difference in the time taken to detect the focal nodes in the case of smaller networks, the time taken by the simple FCM increased exponentially as the size of the network increased, whereas the parallel FCM took just fractions of a second to process even the largest network being considered. The results are depicted in Table 1.  
The focal nodes identified by the fuzzy cognitive map were given as input to the genetic algorithm. The proposed GA identified the community structure in the network by maximizing the modularity (Equation (4)) values. To assess the performance of the proposed model in detecting the community structure of the network, a comparison of the execution times of the distributed GA with a parallel FCM and other genetic algorithms in the literature for community detection is depicted in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Input Network** & **No. of Nodes** & **No. of Edges** & **No. of Focal Nodes** & **FCM** & **Parallel FCM** \\ \hline Karate club network [62] & 34 & 78 & 13 & 300 ms & 350 ms \\ \hline Dolphin social network [63] & 62 & 159 & 32 & 500 ms & 600 ms \\ \hline Books about U.S. politics [64] & 105 & 441 & 41 & 574 ms & 651 ms \\ \hline U.S. college football [28] & 115 & 613 & 50 & 692 ms & 664 ms \\ \hline Les Miserables [34] & 77 & 254 & 37 & 887 ms & 809 ms \\ \hline C. Elegans metabolic Network [65] & 453 & 2025 & 244 & 0 m 1 s & 890 ms \\ \hline Human protein (Figeys) [66] & 2239 & 6452 & 1260 & 0 m 2 s & 786 ms \\ \hline U.S. power grid [67] & 4941 & 6594 & 2427 & 1 m 20 s & 793 ms \\ \hline Pretty good privacy [65] & 10,680 & 24,314 & 5040 & 3 m 5 s & 853 ms \\ \hline Cora citation [68] & 23,166 & 91,500 & 12,306 & 10 m 15 s & 903 ms \\ \hline Online social network
\begin{tabular}{c} opinions [69] \\ \end{tabular} & 75,879 & 508,837 & 25,498 & 1 h 48 m 25 s & 941 ms \\ \hline \hline \end{tabular}
\end{table}
Table 1: Focal node identification in a normal FCM vs. the parallel FCMs.  
Figure 6: Comparison of the execution times of a distributed GA with a parallel FCM and other genetic algorithms for community detection.  
The results show that the use of a distributed genetic algorithm with parallel FCM significantly reduced the execution time. It is visible that while there was not much improvement in the execution times when smaller networks were considered, as the network size increased, there was a massive difference between the execution times. In the case of community detection, when the size of the network was small, the use of the distributed algorithm did not yield a considerable difference in the execution time because, in smaller networks, the overhead of distributing the network across the nodes outweighed the processing time. In larger networks, the time necessary to disseminate the data was trivial in comparison to the time required for processing. It can be observed that while the time taken by other algorithms increased exponentially as the scale of the network expanded, the time taken by the distributed GA with a parallel FCM increased linearly with the size of the network. Additionally, the comparison study demonstrated that when compared to alternative techniques for calculating chromosomal fitness in a GA, such as the community score, NED index, and so on, using modularity as the fitness function produced the best outcome. Furthermore, in all the cases, we observed that the adoption of a parallel fuzzy cognitive map to determine the initial community composition, along with a distributed genetic algorithm, considerably reduced the program's overall computational time and helped it to converge faster.