A distribution-guided Mapper algorithm.

Background: The Mapper algorithm is an essential tool for exploring the data shape in topological data analysis. With a dataset as an input, the Mapper algorithm outputs a graph representing the topological features of the whole dataset. This graph is often regarded as an approximation of a Reeb graph of a dataset. The classic Mapper algorithm uses fixed interval lengths and overlapping ratios, which might fail to reveal subtle features of a dataset, especially when the underlying structure is complex.

Results: In this work, we introduce a distribution-guided Mapper algorithm named D-Mapper, which utilizes the property of the probability model and data intrinsic characteristics to generate density-guided covers and provide enhanced topological features. Moreover, we introduce a metric accounting for both the quality of overlap clustering and extended persistent homology to measure the performance of Mapper-type algorithms. Our numerical experiments indicate that the D-Mapper outperforms the classic Mapper algorithm in various scenarios. We also apply the D-Mapper to a SARS-COV-2 coronavirus RNA sequence dataset to explore the topological structure of different virus variants. The results indicate that the D-Mapper algorithm can reveal both the vertical and horizontal evolutionary processes of the viruses. Our code is available at https://github.com/ShufeiGe/D-Mapper .

Conclusion: The D-Mapper algorithm can generate covers from data based on a probability model. This work demonstrates the power of fusing probabilistic models with Mapper algorithms.