# Optimization Algorithms for Detection of Social Interactions
## 4 Results
### Large Synthetic Networks  
In order to assess the scalability of Hybrid-IA, as last step of this research work, we considered larger synthetic networks with 1000 and 5000 vertices. The clear advantages of using them are given by the knowledge of the communities and consequently by the possibility of evaluating the goodness of the detected communities, as well as the possibility of testing the algorithm on different scenarios and complexities. The validity of this benchmark is given by faithfully reproducing the keys features of real graphs communities. The new benchmark was produced using the \(LFR\) algorithm proposed by [PERSON] and [PERSON] in [PERSON] and [PERSON] (2017). Two different network instances were considered with \(|V|=1000\) and \(|V|=5000\), respectively, considering \(k=20\) as average degree and \(k_{max}=50\) as maximum degree. For both values of \(|V|\), we considered \(\tau_{1}=2\) as exponent of the degrees distribution and \(\tau_{2}=1\) for the one of the communities' sizes. Moreover, about the community dimension, we fixed \(min_{c}=10\) and \(max_{c}=50\), respectively, minimum and maximum sizes. Finally, the mixing parameter \(\mu_{t}\) was varied from 0.1 to 0.7, which identifies the relationships percentage of a node with those belonging to different communities. Therefore, the larger the value of \(\mu_{t}\), the larger are the relationships that a vertex has with other nodes outside its community. More details on the \(LFR\) algorithm, the key parameters, and the generated benchmark can be found in [PERSON] and [PERSON] (2017); [PERSON] and [PERSON] (2017).  
Thanks to the advantages offered by the synthetic networks, we took into account a new evaluation metric in order to confirm the efficiency of the proposed Hybrid-IA in detecting strong communities. In particular, we considered the _Normalized Mutual Information_ (\(NMI\)) [PERSON] and [PERSON] (2017), which is a widely used measure to compare community detection methods since it discloses the similarity between the true community and the detected community structures. Thus, while the modularity allows for assessing how cohesive the detected communities are, the \(NMI\) allows for evaluating how similar they are with respect to the real ones.  
In Table 3, the outcomes of Hybrid-IA on these new synthetic datasets are then reported and compared to the ones obtained by Louvain. Modularity (\(Q\)) and \(NM1\) values are presented and considered for the comparisons. In the first column of the table, the features of each instance tackled are also shown.  
Analyzing the comparison, it is possible to see how Hybrid-IA outperforms Louvain in all networks with 1000 vertices with respect to the \(Q\) modularity metric, whilst the opposite happens for those instances with 5000 vertices, where instead Louvain outperforms Hybrid-IA. This gap is due to the combination between the random search and local search that, together with the diversity produced by the immune operators, requires a longer convergence time than the Louvain one. It is important to point out that very likely, with a larger number of generations, Hybrid-IA would reach comparable results to Louvain in terms of \(Q\) modularity also on these larger networks.  
Different instead is the assessment of the comparison if it is analyzed with respect the \(NM1\) metric [34]: Hybrid-IA outperforms Louvain in all networks, and this proves a better ability of the hybrid immune algorithm proposed in detecting communities closer to the true ones than the greedy optimization algorithm. Note that, although modularity assesses the cohesion of the communities detected, maximizin \(Q\) might not correspond to detect the true communities.