## 3 Modified SOSK-Means
### Modification in the SOS Phase  
The modifications in the SOS phase affect three major parts of the standard algorithm, the initialization phase, the mutualism phase, and the commensalism phase. During the initialization phase, the population size is required to be specified as an input parameter. This is common to most metaheuristic algorithms, with no specific rule for determining the best population size for optimum algorithm performance. In the improved SOSK-means, the rule for determining the population size suggested by [39] was adopted. The population size was constructed as \(40+2g\) where \(g>1\), representing the minimum number of possible groupings of the datasets for an optimum search in the metaheuristic algorithm to achieve a well-distributed space for initial population of solutions that scales well with the data size. In this work, the value for \(g\) was given as 2. The idea was coined from generalizing the central limit theorem (CLT) for a mixture distribution that infers sample population parameters. The central limit theorem states [57] that the mean (average) of a random sample follows a normal distribution as the mean, \(\mu\) and variance, \(\sigma\) of the population from which it is selected According to [39], the selection of a data sample having a size that is sufficiently large enough to be well above the total number of clusters has a high possibility of containing data objects that belong to each of the clusters. At each iteration, new population samples are randomly selected as initial populations. The use of the population size of \(40+2g\) ensured that the initial selection of candidate solutions was well spread across the solution search space for the generation of optimum values in the SOS phase for a faster search for the optimum cluster centroids, which, subsequently, served as the K-means algorithm's initial centroids [39]. The different initial population samples at each iteration in the optimization process represented the varied samples of the initial solution space. Selecting such a sufficiently large population size has a high probability of containing the cluster centroids for all the clusters within the dataset. This invariably increased the convergence rate of the SOS algorithm, and the computational time was, consequently, reduced. To upgrade the SOS algorithm's performance, a three-part mutualism phase, with the random weighted reflection coefficient introduced by [28], was incorporated. Three-part mutualism reflects the possibility of three organisms interacting mutually, each deriving benefits that sustain their existence in the ecosystem. A three-part mutual relationship can be observed in the interaction between sloths, algae, and moths. A three-part mutualism is incorporated alongside the use of two interacting organisms in the mutualism phase. In three-part mutualism, three organisms are chosen randomly in the ecosystem to interact for the generation of newer organisms. Each organism is simulated using Equations (20)-(22), respectively. Their mutual benefits are simulated using Equation (23), involving contributions from the three organisms. At the commencement of the mutualism phase for the organism's update, a random probability is used to determine whether an organism will be engaged in three-part mutualism or not. Thus, a choice is required between the normal dual mutualism interaction and the three-part mutualism. The flowchart depicting the three-part mutualism phase is shown in Figure 2.  
\[x_{ineuv} =x_{i}+rand(0,1)\times(x_{best}-x_{mutual}\times MBF_{1}) \tag{20}\] \[x_{ineuv} =x_{j}+rand(0,1)\times(x_{best}-x_{mutual}\times MBF_{2})\] (21) \[x_{hetv} =x_{k}+rand(0,1)\times(x_{best}-x_{mutual}\times MBF_{3})\] (22) \[x_{mutual} =\frac{x_{i}+x_{j}+x_{k}}{3} \tag{23}\]The BF is also modified using the modified benefit factor (_MBF_) in [28] to increase the quality of the solution.