Cross-National Welfare State Topology

Introduction

Welfare state typologies are a foundational organising concept in comparative sociology. Since Esping-Andersen’s (1990) identification of three welfare state worlds — liberal, social-democratic, and conservative-corporatist — the framework has been elaborated, critiqued, and extended. New contenders (Southern European, East Asian, post-socialist) have been proposed and debated. These typologies are typically constructed from aggregate institutional indicators: replacement rates, decommodification scores, benefit generosity.

This paper takes a different approach: rather than characterising welfare states by their institutional rules, it characterises them by the topological shape of the individual employment histories they produce. If institutions shape careers, the topology of career space should vary systematically across welfare state regimes.

Background

Comparative Welfare State Research

The debate about welfare state typologies has been methodologically limited by its reliance on aggregate indicators. Individual-level comparative data — harmonised employment sequences from EU-SILC — now make it possible to compare career structure directly across countries.

Topological Comparison Across Groups

To compare topology across countries, this paper follows the framework developed in Paper 1: for each country, a Rips complex is constructed from the employment trajectory microdata, and persistent homology is computed. Country-level persistence diagrams are then compared using the Wasserstein distance and summary statistics (entropy, sum of barcode lengths).

Methods

Country-level employment trajectory datasets will be constructed from EU-SILC longitudinal microdata for 16 European countries. Harmonised monthly activity sequences will be coded into a five-state schema (employed full-time, employed part-time, unemployed, inactive, and in education/training) following the EU-SILC activity variable definitions. Trajectories will be restricted to individuals aged 25–55 observed for a minimum of 10 annual waves and weighted using the EU-SILC cross-sectional sampling weights to account for complex survey design.

Trajectory distance metric. Pairwise distances between individual trajectories will be computed using optimal matching (OM) distance with a substitution cost matrix derived from observed transition rates in each country, and an indel cost of 1.5. To ensure comparability across countries with different activity-state distributions, substitution costs will be re-estimated separately per country and trajectories will be length-normalised before distance computation. Sensitivity analyses will be conducted using Dynamic Time Warping (DTW) and Hamming distances as alternative metrics.

Rips complex and persistent homology. For each country, a Vietoris–Rips complex will be built over the pairwise OM distance matrix of individual trajectories. Persistent homology will be computed for H₀ (connected components, capturing employment state clustering) and H₁ (one-cycles, capturing cyclical employment patterns) using the Ripser library. The filtration range will be determined by the 95th percentile of pairwise distances to avoid noise from outlying trajectories.

Persistence diagram comparison. Country-level persistence diagrams will be compared using the 2-Wasserstein distance (W₂), which balances sensitivity to persistent features against robustness to topological noise. W₂ distances will also be used to construct the country-level distance matrix entering the multidimensional scaling. Summary statistics (H₁ persistence entropy and total barcode length) will serve as scalar representations for regression analyses.

Regression model. The reduced-form regression will use the first principal component of the country-level W₂ distance matrix as the dependent variable. The key independent variable is the four-category welfare state regime classification (social-democratic, conservative-corporatist, liberal, Southern European) entered as dummy variables. Controls will include: country-level GDP per capita (log), aggregate unemployment rate, female labour force participation share, mean educational attainment (ISCED), age distribution (share aged 25–34 and 45–55), and survey year. Given the small number of countries (N ≈ 16), heteroskedasticity-robust standard errors will be reported alongside a non-parametric permutation test of regime-group differences as the primary inference procedure.

Data

EU-SILC longitudinal microdata (2004–2020) provides harmonised employment, income, and household data for EU member states. Country samples are restricted to individuals observed for a minimum of 10 annual waves.

Results

Topological Welfare State Clusters

Multidimensional scaling of country-level persistence diagram distances will be used to test whether an extended four-category welfare state typology is recoverable from topology alone. The analysis will examine whether Nordic, conservative-corporatist (Germany, France, Netherlands), Southern European (Spain, Italy, Greece, Portugal), and liberal (UK, Ireland) countries occupy distinct regions of topology space, and whether H₁ entropy tracks expected regime differences.

Regression Analysis

A regression of the first principal component of country topology on regime type dummy variables for all four categories (social-democratic, conservative-corporatist, liberal, and Southern European) will quantify how much cross-national topological variance is explained by regime classification.

Discussion

The topological approach to welfare state comparison is complementary to existing aggregate-indicator methods. It operates directly on individual employment histories, avoiding the aggregation choices that complicate composite index construction.

Conclusion

This paper will test whether employment trajectory topology varies systematically across welfare state regimes. A positive result would validate topological methods for comparative welfare state research and motivate the intergenerational extension in Paper 6.