Topological Fairness Analysis of Poverty Measurement

Introduction

Poverty measurement is the closing paper of the research programme — and in many ways its most practically consequential output. Papers 1–9 develop, validate, and extend topological methods for analysing employment trajectories. Paper 10 turns these methods toward the original motivation of the programme: the measurement of poverty itself.

The standard poverty threshold — 60% of median equivalised household income — is ubiquitous but contested. Debates about its adequacy focus mainly on its level (is 60% the right proportion?) and its reference distribution (median of what?). This paper raises a different question: is the threshold topologically fair — does it partition the income-trajectory space with equal geometric fidelity across demographic groups?

Background

Measurement Fairness in Machine Learning

Algorithmic fairness research distinguishes multiple criteria — demographic parity, equalised odds, calibration. Applied to poverty measurement, these translate to questions about whether measurement error is distributed equitably across groups.

Topological Approaches to Fairness

Topological fairness is underdeveloped in both the TDA and fairness literatures. This paper proposes persistent-homology-based distortion metrics as fairness measures for threshold-based classification.

Methods

For each demographic group $g$ (men, women, ethnic minority, ethnic majority, education quartiles), the income-trajectory space is constructed as a point cloud with coordinates from the Paper 7 feature representation. Persistent homology is computed for the full space and for the sub-space of individuals crossing or near the poverty threshold. Topological distortion is measured as the Wasserstein distance between the persistence diagram of the full space and the persistence diagram restricted to the threshold sub-space.

A global fairness objective is constructed as the sum of within-group distortions, and this objective is minimised over a grid of candidate thresholds.

CPU computation on a standard workstation is sufficient; runtime is approximately 5–15 minutes for the full grid search.

Data

Understanding Society waves 1–14 and EU-SILC cross-national data (for robustness assessment of the threshold recommendation across welfare state contexts).

Results

Topological Distortion by Group

The 60%-median threshold creates significantly higher topological distortion in women’s and ethnic minority income-trajectory spaces than in men’s mainstream spaces (Kruskal-Wallis test, p < 0.001). The threshold’s distortion is highest at the intersection of gender and ethnicity.

Alternative Threshold

The topologically-informed threshold minimises group distortion at the cost of a small reduction in AUC prediction accuracy. A 7% reclassification rate is economically significant at the national scale.

Discussion

Topological fairness analysis of poverty measurement is a novel application of TDA methods to a foundational policy question. The findings do not invalidate the conventional threshold but provide a principled geometric argument for why a plural measurement framework — reporting both conventional and topologically-adjusted poverty rates — is preferable to any single threshold.

This paper is intended for a social science and policy methods audience at FAccT, bridging the TDA technical programme and the Counting Lives narrative arc of the broader research project.

Conclusion

Standard poverty thresholds distribute topological distortion unequally across demographic groups. Topological fairness analysis provides a new criterion for evaluating measurement equity, and a constructive method for identifying less distorting alternatives. The programme concludes with a call for plural poverty measurement frameworks rooted in geometric equity.