Structured Hypothesis Testing for Persistent Homology of Longitudinal Social Data

Dorman, Stephen

Abstract

Topological summaries such as persistence diagrams are increasingly applied to social-science data, but their use has often outpaced the inferential machinery needed to decide when an observed topological feature is real. This paper develops a framework for structured hypothesis testing of persistent homology computed from longitudinal panel data. It formalises a ladder of permutation null models — beginning with memoryless and higher-order Markov nulls — and a survey-design-aware null based on pool-draw and spanning-individual constructions, together with Wasserstein-distance and persistence-landscape test statistics under calibrated resampling. The framework is the methodological (Series B) companion to an applied treatment of UK career inequality; it is assembled from, and supersedes, the earlier core persistent-homology and zigzag papers in this programme.

Plain-Language Summary

When researchers find an interesting "shape" in data using topology, how do they know it is not just noise? This paper builds the statistical tools to answer that question for the kind of long-running social survey data used elsewhere in the programme. It sets out how to compare an observed topological pattern against carefully constructed random benchmarks that respect how the survey was actually collected, so that claims about structure in people's life trajectories can be tested rather than merely asserted.

Programme note. This paper is the methodological (Series B) half of a companion pair, assembled in April 2026 from the earlier core persistent-homology paper and the zigzag paper, both of which it supersedes (see Papers 1–3). Its applied companion is The Geometry of UK Career Inequality (Paper 11). Zigzag persistence appears here as motivating context rather than as the paper's subject.

Overview

The paper addresses a gap between practice and inference: persistent homology is now routinely computed from social data, but the question of when a topological feature is statistically significant has frequently been answered informally. The contribution is a set of explicit null models and test statistics that make the question answerable with calibrated error control, adapted to the sampling structure of longitudinal panels.

Status

This is an active draft (in-progress, Stage 0), in lockstep revision with its Series A companion. Methodological details are still being finalised and reproducibility-locked; specific numerical results are deliberately not reported here until the manuscript stabilises toward submission.

Key Findings

Methods

Computational Requirements

Hardware: CPU
⏱ Runtime: Hours

Position in Research Programme

■ This paper

Downloads & Citation

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