# Optimization Algorithms for Detection of Social Interactions
## 4 Results
It is important to note that, for the MA-Net outcomes, the values reported, and taken from [30], have been rounded to third decimal places unlike the others that are instead based on four decimals. Obviously, all 10 algorithms optimize the same fitness function reported in Equation (1) and rewritten in a simpler way in Equation (2). From Table 2, it is possible to note that all algorithms reach the optimal solution in the first two networks _Zebra_ and _Karate Club_, except Louvain, which fails on the second; on all the other network instances, both Opt-IA and Hybrid-IA outperform all other compared algorithms, matching their best values only with HDSA and MA-Net ones. It is important to point out, which proves even more the efficiency of the two proposed immunological algorithms, how the _mean_ values obtained by  
\begin{table}
\begin{tabular}{l l c c c c c c c c c c} \hline \hline \multirow{3}{*}{_Networks_} & \multicolumn{10}{c}{_Algorithms_} \\ \cline{2-13}  & & Louvain & **Opt-IA** & Hybrid-IA** & **HDSA** & **BADE** & **SSGA** & **BB-BC** & **BA** & **GSA** & **MA-Net** \\ \hline \multirow{6}{*}{Zebra} & _Best_ & **0.2768** & **0.2768** & **0.2768** & **0.2768** & **0.2768** & **0.2768** & **0.2768** & **0.2768** & **0.2768** & - \\  & _Mann_ & - & 0.2768 & 0.2768 & 0.2768 & 0.2768 & 0.2768 & 0.2766 & 0.2768 & 0.2768 & - \\  & _Worst_ & - & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & - \\  & \(k\) & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & - \\ \hline \multirow{6}{*}{Karate Club} & _Best_ & 0.4188 & **0.4198** & **0.4198** & **0.4198** & **0.4198** & **0.4198** & **0.4198** & **0.4198** & **0.4198** & **0.4200** \\  & _Mann_ & - & 0.4198 & 0.4198 & 0.4198 & 0.4198 & 0.4198 & 0.4196 & 0.4133 & 0.4170 & 0.419 \\  & _Worst_ & - & 0.4198 & 0.4198 & 0.4198 & 0.4156 & 0.4198 & 0.4188 & 0.3946 & 0.4107 & - \\  & _σ_ & - & 0.0000 & 0.0000 & 0.0000 & 0.0018 & 0.0000 & 0.0004 & 0.0105 & 0.0037 & 0.002 \\  & \(k\) & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 & 4 \\ \hline \multirow{6}{*}{ Dolphins} & _Best_ & 0.5185 & **0.5285** & **0.5285** & **0.5285** & 0.5268 & 0.5257 & 0.5220 & 0.5157 & 0.4891 & 0.529 \\  & _Mann_ & - & **0.5285** & 0.5273 & 0.5282 & 0.5129 & 0.5200 & 0.5141 & 0.4919 & 0.4677 & 0.523 \\  & _Worst_ & - & 0.5268 & 0.5220 & 0.5276 & 0.4940 & 0.5156 & 0.5049 & 0.4427 & 0.4517 & - \\  & _σ_ & - & 0.0003 & 0.0009 & 0.0005 & 0.0120 & 0.0040 & 0.0068 & 0.0289 & 0.0155 & 0.004 \\  & \(k\) & 5 & 5 & 5 & 5 & 4 & 5 & 5 & 4 & 6 & 5 \\ \hline \multirow{6}{*}{ Polittical Books} & _Best_ & 0.5205 & **0.5272** & **0.5272** & **0.5272** & 0.5239 & 0.5221 & 0.4992 & 0.5211 & 0.4775 & 0.527 \\  & _Mann_ & - & 0.5267 & 0.5270 & **0.5272** & 0.5178 & 0.5203 & 0.4914 & 0.5200 & 0.4661 & 0.526 \\  & _Worst_ & - & 0.5063 & 0.5246 & 0.5272 & 0.5137 & 0.5167 & 0.4799 & 0.4815 & 0.4558 & - \\  & _σ_ & - & 0.0028 & 0.0005 & 0.0000 & 0.0042 & 0.0024 & 0.0084 & 0.0149 & 0.0079 & 0.002 \\  & \(k\) & 4 & 5 & 5 & 5 & 4 & 5 & 9 & 3 & 5 & 5 \\ \hline \multirow{6}{*}{ Football} & _Best_ & **0.6046** & **0.6016** & **0.6046** & **0.6046** & 0.5646 & 0.5330 & 0.5171 & 0.5523 & 0.4175 & 0.605 \\  & _Mann_ & - & 0.5989 & **0.6039** & 0.6033 & 0.5513 & 0.5277 & 0.5061 & 0.5272 & 0.4032 & 0.601 \\  & _West_ & - & 0.5736 & 0.6031 & 0.6019 & 0.5430 & 0.5189 & 0.4986 & 0.4742 & 0.3905 &Opt-IA and Hybrid-IA, on all tested networks, are better than the best modularity found by the other algorithms, such as: BADE, SSGA, BB-BC, BA, and GSA; even on the _Dolphins_ and _Football_ networks, the worst modularity value obtained by Opt-IA is equal to or greater than the best one obtained by the same algorithms. On the _Dolphins_ network, Opt-IA reaches a better mean value than Hybrid-IA, HDSA, and MA-Net, since, because of its random/blind exploration of the search space, it jumps out from local optima more easily than the other three. The opposite behavior of Opt-IA occurs when the size and complexity of the networks increase.