Title: Single-cell trajectory analysis with V-Mapper

Abstract: During cell differentiation, cells undergo intricate changes in their gene expression levels, forming complex topologies in high-dimensional gene expression space. To infer the trajectory of cell differentiation, Mapper, a topological data analysis method for high-dimensional data, has been applied to single-cell gene expression data. While Mapper can extract the topology of cell differentiation, such as cyclicity and plasticity, it cannot infer the flow on it. We propose V-Mapper (velocity Mapper), an extension of Mapper, designed for high-dimensional data with velocity information. V-Mapper simultaneously describes both the topological structure and flow as a weighted directed graph (V-Mapper graph) by embedding velocity into the edges of the Mapper graph. We apply V-Mapper to single-cell gene expression data to infer the trajectory of cell differentiation. Additionally, by incorporating combinatorial Hodge decomposition, we enhance the interpretation of the flow within the V-Mapper graph. Furthermore, we address the challenges posed by the curses of dimensionality in topological data analysis and present our resolutions to these challenges.