LoCoMapper: Local Structural Uncertainty Analysis of Mapper Graphs under Cover Perturbations

Authors

Xinyuan Yan (Scientific Computing and Imaging Institute), Jixian Li (University of Texas at Austin), Bei Wang (University of Utah)

Keywords

Mapper graphs, topological data analysis, uncertainty visualization, structural uncertainty, parameter sensitivity

Abstract

Mapper graphs provide compact topological summaries of high-dimensional data, but small changes in cover parameters can substantially alter their topology, making it difficult to distinguish persistent structures from those that arise only under particular parameter configurations. Despite growing interest in uncertainty visualization, methods for representing uncertainty in Mapper graphs remain largely unexplored. We present LoCoMapper, a framework that takes a first step toward uncertainty-aware visualization of Mapper graphs by characterizing their local structural sensitivity to perturbations of the cover parameters. Given a reference Mapper graph, LoCoMapper aligns local regions of the reference graph with corresponding regions in an ensemble of perturbed graphs and summarizes their variation using two node-level measures: local structural consistency, which quantifies the preservation of local topology across reliable alignments, and ensemble support, which captures the amount of reliable evidence supporting that estimate. Jointly visualizing these measures on the reference graph enables users to distinguish robust, sensitive, and weakly supported regions. We demonstrate the effectiveness of LoCoMapper on a controlled two-dimensional dataset, a synthetic three-dimensional ant dataset, and word embeddings from a fine-tuned BERT model.