## 1 Introduction
[PERSON], [PERSON], and [PERSON] [30] combined SOS with simple quadratic interpolation (SQI) for handling large-scale and real-world problems. Although their proposed hybrid SOS algorithm increased the algorithms' complexity, it provided an efficient and effective quality solution, with an improved convergence rate. [PERSON] and [PERSON] [31] improved and extended the classical SOS using three mutation-based local search operators to improve population reconstruction, exploration, and exploitation capability and to accelerate the convergence speed. Other improvements reported in the literature include [32; 33; 34; 35; 36; 37; 38]. According to [PERSON], [PERSON], and [PERSON] [9], most of the proposed improvements could not guarantee finding the best optimal solution. As such, they proposed using a non-linear benefit factor where the mutual vector was calculated, based on the weights of two organisms for a broader and more thorough search. They also altered the parasitism phase to reduce the computational burden of the algorithm. This paper proposes an improved version of the hybrid SOSK-means algorithm [24], called ISOSK-means, to address the common limitations of the individual classical algorithms and their hybridization variants for a more robust, efficient, and stable hybrid algorithm. The algorithm adopted a population size of \(40+2g\) for a well distributed initial population sample, based on the assumption that selecting the right sample size produces a sample mean that approximates the true centroid on Gaussian distribution [39]. This ensured a substantially large population size at each iteration for early convergence in the SOS phase of the hybrid algorithm. The modified algorithm also employed a global threshold of point-to-centroid distance distribution in the K-means algorithm phase to detect outliers for subsequent exclusion in calculating the new centroids [3]. The detection and exclusion of outliers in the K-means phase assisted in improving the cluster result. The performance of the SOS algorithm was enhanced by introducing self-adaptive benefit factors, with a three-part mutualism phase, as suggested in [28]. These techniques enhanced the search for the optimum centroids in the solution space and improved the convergence rate of the hybrid algorithm. The performance of the proposed algorithm was evaluated on 42 datasets with varying sizes and dimensions. Davies Bouldin (DB) index [40] and cluster separation (CS) index [41] were used as cluster validity indices for evaluating the performance of the algorithm. The choice of the two clustering validity indices was based on the duo's cluster validity approach of minimizing the intra-cluster similarity and maximizing the inter-cluster similarity, which is the same as the primary objective of data clustering. The CS index was only applied to 12 datasets for comparison with the classical version of the improved hybrid algorithm. The two indices, based on their validity approach, are credited with the ability to produce clusters that are well separated, with the minimum intra-cluster distance and maximum inter-cluster distance [42; 43; 44]. The DB index can guide the clustering algorithm using a partitioning approach without the need to specify the number of clusters, and the CS index is known for its excellent performance in the identification of clusters with varying densities and sizes [40; 41; 43]. The enhanced outcomes were compared with the existing hybrid SOSK-means algorithm and other comparative non-hybrid and hybrid metaheuristic algorithms on large dimensional datasets. The performance of the proposed improved hybrid algorithm was further validated using the nonparametric Friedman mean rank tests and Wilcoxon signed-rank statistical tests. The results of the extensive computational experiments showed that the proposed improved hybrid algorithm (ISOSK-means) outperformed the existing SOSK-means algorithm, and demonstrated superior performance over some of the competing hybrid and non-hybrid metaheuristic algorithms.