## 2 Germinal Center Optimization
### Algorithm Description  
In this section, we explain the GCO algorithm. In Table 1, we present the computational analogy between germinal center and the optimization problem. \begin{table}
\begin{tabular}{c c} \hline \hline
**Germinal Center** & **Optimization Problem** \\ \hline Antigen & Objective function \\ Antibody and B-cell & Candidate solution \\ Affinity & Objective function evaluation \\ T-help cell binding & Incrementation of life-signal \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational analogy between Germinal Center (GC) and Optimization. Figure 1: Germinal Center reaction. The GC reaction has multiple competitive processes, the GCO algorithm does not try to simulate GC reaction per se but to use some of its competitive mechanisms. A key factor in the GC reaction is the distinction between the dark zone and the light zone. The dark zone represents the diversification of the solutions that could be understood like a mutation process, many algorithms, such as Differential evolution and Genetic algorithms [1], already have the idea of mutation, but the dark zone includes not only a mutation process (somatic hyper-mutation), but also the clonal expansion. This clonal expansion is guided by the life signal of the B-cell, denoted by \(\mathbb{L}\in[0,100]\). The B-cells with greater life signal are more likely to clone, increasing the B-cell multiplicity. In the light zone, B-cells with the best affinity are rewarded and the other cells age (lower their life signal). Then, the affinity-based selection in the light zone changes the probability of clone or death of a B-cell. In Figure 2, we present the GCO algorithm flowchart, where \"For each B-cell\" indicates that the process is applied in every candidate solution. As we are dealing with a population-based algorithm [1], there is a particular interest in the cells mutation and which information we use for crossover particles. In the GCO algorithm, we use the distribution of the cells multiplicity, denoted with \(\mathcal{C}\), to select three individuals for crossover. It is important to note that initially, all the cells have a multiplicity of one, then the individuals are uniformly selected, but this distribution changes through iterations modeled by the competitive process. This kind of distribution offers different types of leadership behaviors in the collective intelligence, for example, initially the GCO algorithm behaves like Differential Evolution in the DE/rand/1 strategy [1], and when a particle wins for many iterations, the algorithm behaves like Particle Swarm optimization [1]. However, the leadership in GCO is not only dynamic, but it also includes temporal leadership, this is implemented when a particle mutates to a better solution, this new candidate substitutes the actual particle resetting the cell multiplicity to one. Then, GCO algorithm offers a bio-inspired technique of adaptive leadership in collective intelligence algorithms for multivariate optimization problems. In Algorithm 1, we include an explicit pseudocode, where \(B_{i}\) is the \(i\)-esim B-cell, \(M\) is a mutant B-cell and a new candidate solution, \(F\in[0,2]\) is the mutation factor and \(CR\in[0,1]\) is the cross-ratio, \(\mathcal{C}\) is the distribution of cells multiplicity, and \(\mathbb{L}\) is the life signal. Figure 2: Germinal Center Optimization (GCO) algorithm flowchart.