# Optimization Algorithms for Detection of Social Interactions
## 4 Results
In this plot, the curves represent the evolution of the best and average fitness of the population; the standard deviation of the fitness values of the population is superimposed onto the average fitness and gives an idea about how heterogeneous the elements in the population are. From Figure 1, one can note how Opt-IA converges more slowly towards the best solution, as expected, always keeping a certain variability within the population. This allows the algorithm to better explore the search space. When the population is composed of very different elements, i.e., when the standard deviation is high, the algorithm discovers new solutions, significantly improving the current best solution. However, after about 250 generations, Opt-IA reaches the optimal solution and the curves (best and average fitness) tend to overlap. Moreover, the achievement of the optimal solution helps the creation of better clones, reducing the variability of the population. Unlike Opt-IA, Hybrid-IA converges easily thanks to the local search applied to the elements after the selection phase. As can be noted from the inset plot in Figure 1, Hybrid-IA reaches the optimal solution after a few generations. Even in this case, once the best solution is reached, the population follows the same trend of the curve of the best fitness, and both curves continue (almost) as a single line. If on one hand the local search helps to quickly discover good solutions, on the other, it reduces the diversity inside the population, reducing then the exploration of the search space. In particular, as demonstrated by the worst value found in the _Jazz Musicians_ network, reported in Table 2, Hybrid-IA prematurely converges towards local optima, from which it will hardly be able to get out. At the end of the analysis of Figure 1, it is possible to conclude that the stochastic operators designed in Opt-IA guarantee an excellent and large exploration of the search space, but with the disadvantage of requiring a longer evolution time; however, the local search developed in Hybrid-IA, and relative sorting criteria, allow for quickly discovering good solutions to exploit during the evolutionary process. In order to evaluate the performances and reliability of both immunological algorithms with respect the state-of-the-art, a wide comparison has been performed with several metaheuristics, and a well-known deterministic algorithm, on the all dataset reported in Table 1. In particular, Opt-IA and Hybrid-IA have been compared with: Louvain[21], a greedy optimization method; HDSA, a Hyper-Heuristics Differential Search Algorithm based on the migration of artificial  
Figure 1: Convergence behavior of Opt-IA and Hybrid-IA on the _Books about US Politics_ network. superorganisms [22]; BADE, an improved Bat Algorithm based on Differential Evolution algorithm [23; 24]; SSGA, a Scatter Search [25; 26] based on Genetic Algorithm; BB-BC, a modified Big Bang-Big Crunch algorithm [27]; BA, an adapted Bat Algorithm for community detection [28]; GSA, a Gravitational Search Algorithm re-designed for solving the community detection problem [29]; and MA-Net, a Memetic Algorithm [30]. For all algorithms, we considered \(MaxGen=100\), except for Opt-IA, as described above, and MA-Net whose stopping criterion has been set to 30 generations without improvement (see [30] for details). All results of the above methods, as well as the experimental protocol, have been taken from [PERSON] et al. [31], except for Louvain and MA-Net. The comparison performed on all dataset is reported in Table 2, where we show for each algorithm (where possible) the _best_, _mean_, and _worst_ values of the \(Q\) modularity; standard deviation (\(\sigma\)), and, finally, the created communities number (\(k\)). Furthermore, whilst the experiments for Opt-IA and Hybrid-IA were performed on 100 independent runs, for all other compared algorithms, only 30 independent runs have been considered, excepts for MA-Net with 50 runs.