Zigzag Persistence for Business Cycle Topology

Introduction

Employment trajectory topology is not static. As macroeconomic conditions shift, the structural patterns of career paths change: new instability patterns emerge, stable employment clusters contract, and cyclical unemployment loops proliferate. Standard persistent homology cannot capture this temporal variation, because it analyses a fixed point cloud. Zigzag persistence — introduced by Carlsson and de Silva (2010) — generalises persistent homology to sequences of spaces connected by inclusion maps, enabling the tracking of topological features through time.

This paper applies zigzag persistence to time-indexed slices of employment trajectory data, measuring how the topology of career space evolves across the macroeconomic cycle.

Background

Zigzag Persistence

In standard persistent homology, a sequence of nested spaces $\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_n$ is analysed via inclusion maps. Zigzag persistence relaxes this to arbitrary sequences of spaces connected by maps in either direction: $X_0 \leftrightarrow X_1 \leftrightarrow \cdots \leftrightarrow X_n$. Features (connected components, loops, voids) are tracked across this sequence, with birth and death times in the corresponding barcode.

Business Cycle Effects on Employment

The sociological literature documents business cycle effects on employment outcomes, but characterises them via changes in rates (unemployment, inactivity) rather than changes in structural patterns. Zigzag persistence offers a complementary structural characterisation.

Methods

The BHPS/Understanding Society panel is partitioned into five-year rolling windows advancing by one year. For each window, the Rips complex of the trajectory sub-population is computed and added to the zigzag sequence. Zigzag persistence is computed using the Gudhi library. A topological complexity index is constructed from the sum of H₁ barcode lengths within each window.

Computing zigzag persistence across the full panel is computationally demanding; runtime is estimated at 8–24 hours on a modern 32-core CPU.

Data

The full BHPS/Understanding Society linked panel is used, following the construction protocol from Paper 1. Macroeconomic comparison series (OECD UK unemployment rates) are obtained from OECD.Stat.

Results

Survey Design Artefact: The Post-2008 Discontinuity

The dominant topological signal in the 32-year zigzag sequence is a sharp increase in H₀ fragmentation at the 2008–2009 transition. This signal initially appeared consistent with the Global Financial Crisis as a topological regime change. Diagnostic analysis — era-specific sub-series computation and systematic sub-sampling within each survey era — demonstrates that the discontinuity reflects the change in state-coding schema between BHPS and Understanding Society, not the macroeconomic shock. The BHPS uses a six-category state scheme; USoc’s successor scheme modifies category boundaries and includes a distinct part-time state. The change in categorical distances creates a structural break in the embedding that manifests as inflated H₀ fragmentation after 2009.

Genuine Secular Flexibilisation (BHPS Era, 1991–2008)

Isolating the BHPS-era sub-series reveals a genuine secular trend: H₀ barcode count increases monotonically from 1991 to approximately 2003, then stabilises. This trend is robust to the 11 × 4 sensitivity grid (ε × landmark count) and sub-sampling within the era. The chronology is consistent with labour market deregulation: the proliferation of fixed-term contracts from the early 1990s produces increasing trajectory fragmentation. The 1993 recession produces a temporary spike above the secular trend but does not interrupt it.

Macroeconomic Correlations and H₁ Absence

H₁ (loop) features remain non-significant across both eras, consistent with the Paper 1 negative result. Within-era correlation finds the topological complexity index tracks GDP growth (r = −0.41, BHPS era, p < 0.05) but not unemployment rates — suggesting trajectory topology captures income-band cycling rather than employment-status cycling as the primary instability signal.

Discussion

The central finding — that the most visually dramatic signal is a measurement artefact — is a substantive methodological contribution. Survey panel transitions create structural breaks that topological methods amplify, because they capture the full geometric structure of trajectory space. The paper develops a diagnostic protocol for distinguishing genuine temporal variation from survey-design artefact applicable to any multi-wave zigzag analysis. The genuine secular flexibilisation result, stripped of the artefact, is consistent with the labour market de-standardisation literature.

Conclusion

Zigzag persistence applied to UK employment panel data reveals a genuine secular flexibilisation trend within the 1991–2008 BHPS era and confirms the Paper 1 negative result for H₁; the dominant post-2008 discontinuity is a survey-design artefact. The diagnostic protocol developed here provides a template for future multi-wave zigzag analyses.