A Comprehensive Review of the Mapper Algorithm, a Topological Data Analysis Technique, and Its Applications Across Various Fields (2007-2025) (2504.09042v1)

Abstract: The Mapper algorithm, a technique within topological data analysis (TDA), constructs a simplified graphical representation of high-dimensional data to uncover its underlying shape and structural patterns. The algorithm has attracted significant attention from researchers and has been applied across various disciplines. However, to the best of the authors' knowledge, no comprehensive review currently exists on the Mapper algorithm and its variants as applied across different fields of study between 2007 and 2025. This review addresses this gap and serves as a valuable resource for researchers and practitioners aiming to apply or advance the algorithm. The reviewed literature comprises peer-reviewed articles retrieved from major academic databases, including Google Scholar, Web of Science, Scopus, JSTOR, PubMed, and IEEE Xplore, using the keywords 'topological data analysis,' 'mapper algorithm,' and 'topological graph.' The study further provides an overview and a comparative analysis of the suitability of the most commonly used filter functions and clustering algorithms within the Mapper framework. Additionally, it examines current trends, identifies limitations, and proposes future research directions for the Mapper algorithm and its variants, emphasizing the need for developing effective methodologies to streamline the analysis of high-dimensional data in the age of big data proliferation.