From graphs to point clouds: the tropical abel–jacobi transform and persistent homology for metric graphs
File(s)
Author(s)
Cao, Yueqi
Type
Thesis
Abstract
Metric graphs are important models for representing network-structured data from various domains. The problem of extracting geometric and topological features from metric graphs presents fundamental yet challenging questions that classical graph-theoretic tools often fail to resolve. This thesis develops new computational and statistical methods for the analysis of metric graphs by combining ideas from tropical geometry, topological data analysis and nonparametric statistics. Motivated by the one-to-one correspondence between metric graphs and abstract tropical curves, we study the tropical Abel–Jacobi transform of metric graphs and develop efficient algorithms for its computation. We analyze the resulting vector embeddings and compute pairwise extrinsic distances between points in the tropical Jacobian under different tropical metrics, which are crucial for subsequent persistent homology computations. To address the high computational complexity, we propose practical algorithms for both exact and approximate distance matrix computation. These methods enable the computation of persistent homology for point clouds sampled from the tropical Jacobian. However, for large datasets, such computation remains infeasible, motivating the development of methods for approximating persistent homology representatives. We adapt the classical statistical method of bootstrapping and show that the mean of the persistence diagrams of subsamples is a valid approximation of the true persistent homology of the larger dataset. We give the rate of convergence of the mean persistence diagram to the true persistence diagram in terms of the number of subsamples and size of each subsample. In general, our work advances the interplay among geometry, topology, and statistics in data analysis, offering theoretical insight and practical tools for analyzing data with complex geometric structures.
Version
Open Access
Date Issued
2025-09-03
Date Awarded
2026-03-01
Copyright Statement
Attribution 4.0 International Licence (CC BY)
License URL
Advisor
Monod, Anthea
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)