The political-topology framework identifies discrete regime clusters using a Gaussian Mixture Model estimated via Expectation-Mmaximisation (EM). The GMM assumes the observed distribution of liberty scores arises from a finite mixture of Gaussian components, each representing a latent regime type. For an observation xi, the likelihood is:
where πk is the mixing weight, μk the mean, and σk2 the variance of component k. The EM algorithm iterates between an E-step (computing posterior responsibilities) and an M-step (updating parameters) until convergence. We select the number of components using the Bayesian Information Criterion (BIC). A three-component solution (K=3) yields a BIC improvement of ΔBIC = 193.9 over K=2, providing strong evidence for three distinct basins: autocratic (μ ≈ 18), hybrid (μ ≈ 48), and democratic (μ ≈ 82). The K=4 solution offers negligible improvement (ΔBIC < 5), confirming the three-basin topology.
The three components correspond to the attractor basins central to the book's argument. Countries cluster naturally around these modes, with the hybrid basin exhibiting the highest variance (σ ≈ 12), consistent with its ccharacterisation as a zone of instability.
To fformalise the attractor-basin landscape, we model liberty dynamics as overdamped Langevin motion in a potential field:
where L is the liberty score, V(L) is the potential function, σ is the noise intensity, and dW is a Wiener process. The drift function −V'(L) is estimated non-parametrically from year-over-year changes in liberty scores, using kernel smoothing (Gaussian kernel, bandwidth selected by Silverman's rule). The potential is then recovered by numerical integration:
where f(x) is the estimated drift. The reconstructed potential reveals three wells (minima) corresponding to the GMM-identified basins, with barrier heights of approximately 0.8 units (autocratic-to-hybrid) and 0.5 units (hybrid-to-democratic). The asymmetry of barrier heights implies that escaping the autocratic basin requires larger shocks than escaping the hybrid zone, consistent with observed transition frequencies.
We test whether liberty scores exhibit strong path dependence using a first-order autoregressive model:
Pooled OLS across 193 countries and 53 years yields β = 0.96 (SE = 0.002) and R2 = 0.87. The near-unit-root coefficient confirms extreme persistence: a country at liberty score 40 has an expected score of 39.4 one year later, barely different from its current position. The AIC of the AR(1) model is more than 300 units lower than a discrete-stage model (with stage dummies), demonstrating that the continuous persistence model provides a far superior fit to the data. Newey-West standard errors with 3-year lag correction account for serial correlation. Hausman tests confirm that fixed-effects and random-effects specifications yield substantively identical estimates for β.
All confidence intervals and standard errors are computed via a country-cluster bootstrap with 1,000 iterations. In each iteration, we resample countries (not country-years) with replacement to preserve the within-country correlation structure. For the GMM analysis, each bootstrap iteration re-estimates all three component parameters. For the AR(1) model, each iteration re-estimates β on the resampled panel. The 95% confidence intervals are computed as the 2.5th and 97.5th percentiles of the bootstrap distribution. This approach is preferred over analytic standard errors because it makes no distributional assumptions and naturally accounts for the clustered, heteroskedastic structure of cross-national panel data.
We aanalyse the duration of regime spells using Kaplan-Meier survival curves and log-rank tests. A regime spell is defined as a continuous period during which a country remains within one of the three GMM-defined basins. Spell termination (the "event") occurs when a country's liberty score crosses a basin boundary and remains in the new basin for at least two consecutive years (to filter noise). Kaplan-Meier estimates reveal that the median survival time for the autocratic basin is 38 years, for the hybrid basin 12 years, and for the democratic basin exceeds the observation window (right-censored). Log-rank tests confirm statistically significant differences in survival functions across all three basin pairs (p < 0.001). The short median survival of hybrid regimes provides quantitative support for the "hybrid trap" concept developed in the text.
Chapter 12 introduces a parsimonious model linking sovereign bond yields to regime characteristics. The specification is:
| Variable | Coefficient | Std. Error | t-stat | p-value |
|---|---|---|---|---|
| Intercept (α) | 8.42 | 0.91 | 9.25 | <0.001 |
| Liberty Score (β1) | −0.35 | 0.06 | −5.83 | <0.001 |
| Debt/GDP (β2) | 0.02 | 0.005 | 4.00 | <0.001 |
| Reserve Currency (β3) | −2.08 | 0.42 | −4.95 | <0.001 |
| Regime Velocity (β4) | 1.15 | 0.31 | 3.71 | <0.001 |
N = 87 countries. R2 = 0.79. Standard errors are heteroskedasticity-robust (HC1). Reserve Currency is a binary indicator for currencies composing >1% of global reserves. Regime Velocity is the 5-year rolling standard deviation of liberty-score changes.
The liberty coefficient (β1 = −0.35) implies that a 10-point increase in liberty score is associated with a 350-basis-point reduction in sovereign yields, holding other factors constant. The reserve-currency premium of approximately 2,080 basis points captures the "exorbitant privilege" of reserve-currency issuers, whose bonds serve as global safe assets regardless of fiscal fundamentals. Regime velocity captures the market penalty for political instability: countries with rapidly changing liberty scores pay higher borrowing costs even if their current score is relatively high.
The Human Capabilities Index aggregates 15 indicators across 7 domains to measure the breadth of substantive freedoms available to a country's population. The domains and their constituent indicators are:
1. Health: life expectancy at birth, infant mortality rate. 2. Education: mean years of schooling, expected years of schooling, literacy rate. 3. Economic Access: GNI per capita (log), labour force participation rate. 4. Political Voice: Freedom House political rights score, V-Dem participatory component. 5. Legal Protection: rule of law index (WGI), property rights score. 6. Social Inclusion: gender inequality index (inverted), Gini coefficient (inverted). 7. Information Access: internet penetration rate, press freedom score.
Each indicator is nnormalised to a 0-1 scale using min-max nnormalisation across the full country-year panel. Domain scores are computed as the arithmetic mean of their constituent indicators. The overall HCI is the arithmetic mean of the seven domain scores. This simple aggregation is chosen for transparency and replicability, following the methodology of the UNDP's Human Development Index.
The HCI correlates with the HDI at r = 0.92 (p < 0.001, N = 186), validating its coverage of human development. However, it diverges meaningfully from the HDI for capable autocracies (e.g., Singapore, UAE) where economic and health indicators are high but political voice and press freedom scores are low. This divergence is precisely the analytical leverage the HCI provides: it reveals the capability deficits that the HDI, with its narrower focus, does not capture.
The eight-stage regime classification used throughout the book codes each country-year into one of the following stages based on liberty score thresholds, supplemented by qualitative assessment for borderline cases:
| Stage | Label | Liberty Range | Description |
|---|---|---|---|
| 1 | Closed Autocracy | 0–12 | No political competition; single party or personal rule |
| 2 | Hardening Autocracy | 13–25 | Consolidating authoritarian control; eliminating opposition |
| 3 | Competitive Autocracy | 26–37 | Elections held but systematically unfair; opposition tolerated |
| 4 | Hybrid: Autocratic Leaning | 38–50 | Mixed institutions; elections meaningful but constrained |
| 5 | Hybrid: Democratic Leaning | 51–62 | Genuine competition with significant institutional weaknesses |
| 6 | Electoral Democracy | 63–75 | Free elections; emerging rule of law; civil liberties gaps |
| 7 | Liberal Democracy | 76–90 | Strong institutions; protected rights; independent judiciary |
| 8 | Full Liberal Democracy | 91–100 | Comprehensive rights protection; robust checks and balances |
Two trained coders independently classified a random sample of 200 country-years. Inter-coder reliability was assessed using Cohen's kappa (κ = 0.78), indicating substantial agreement. Disagreements were resolved through discussion and reference to the original Freedom House and V-Dem source data. The 0.78 kappa exceeds the conventional threshold of 0.70 for acceptable inter-coder reliability in political science classification tasks.
The primary liberty metric. Freedom House rates countries on Political Rights (1–7) and Civil Liberties (1–7), where 1 = most free. We invert and rescale the aggregate score to a 0–100 liberty scale: Liberty = 100 − ((PR + CL − 2) / 12) × 100. This transformation maps the most repressive possible score (7, 7) to 0 and the most free (1, 1) to 100. Coverage: 195 countries and territories, updated annually. Methodological note: Freedom House revised its methodology in 2003 (adding sub-questions) and in 2017 (shifting to a 100-point aggregate). We use the sub-question-based aggregate where available and the rescaled PR/CL scores for pre-2003 years, ensuring continuity.
The Liberal Democracy Index (v2x_libdem) provides a complementary measure coded by country experts using a Bayesian item response theory model. V-Dem's continuous 0–1 scale is rescaled to 0–100 for comparability. We use V-Dem primarily for the deep historical series (pre-1972) and as a robustness check for the Freedom House-based results. Correlation between the two measures in the overlap period (1972–2025) is r = 0.93. V-Dem version 14 is used throughout.
Published by the Fund for Peace, the FSI aggregates 12 conflict, and governance indicators across social, economic, and political dimensions. Scores range from 0 (most stable) to 120 (most fragile). We use the FSI primarily in Chapter 9 (state fragility analysis) and Chapter 15 (early warning systems). The FSI's short time span (20 years) limits its utility for historical analysis but provides granular, annually updated measures of institutional weakness.
The Polity5 dataset (Marshall and Gurr 2020) provides the polity2 composite score (−10 to +10) for the longest continuous time series of regime characteristics. We convert to our liberty scale using: Liberty = (polity2 + 10) × 5. The Polity project ceased updates in 2018, so we use it only for historical analysis, and robustness checks. Periods of transition (−77), interruption (−66), and interregnum (−88) are treated as missing data.
The World Bank's WGI provides six governance dimensions: Voice and Accountability, Political Stability, Government Effectiveness, Regulatory Quality, Rule of Law, and Control of Corruption. Each is measured on a −2.5 to +2.5 scale. We use individual dimensions as covariates in the sovereign credit model and as inputs to the HCI. Coverage: 215 economies, biennial before 2002, annual thereafter.
Economic variables (GDP per capita, debt-to-GDP ratio, trade openness) are drawn from the World Bank's World Development Indicators and the IMF's World Economic Outlook. Human development indicators (life expectancy, schooling, GNI per capita) come from the UNDP's Human Development Reports. Reserve currency status is coded from IMF COFER (Currency Composition of Official Foreign Exchange Reserves) data.
Where country codes differ across datasets, we hharmonise using ISO 3166-1 alpha-3 codes. Temporal gaps are not interpolated; missing years are treated as missing data. The crosswalk between Freedom House aggregate scores and the eight-stage classification achieves a 67% exact-match rate. The remaining 33% of cases fall within one stage of the crosswalk prediction, with discrepancies concentrated at stage boundaries (stages 3–4 and 6–7). All crosswalk mappings and the master country-year panel are provided in the replication data files.
All analyses in this book can be reproduced using 26 Python scripts oorganised in five phases. The complete codebase, data files, and documentation are available in the open-source repository.
Scripts: phase1_*.py, fix-series_*.py
These scripts ingest raw data from Freedom House, V-Dem, and Polity, construct the unified liberty-score panel, and perform the crosswalk validation. Output: the master country-year panel used by all subsequent phases.
Scripts: b1_gmm.py, b2_potential.py, b4_survival.py, b8_stage.py
b1_gmm.py estimates the Gaussian Mixture Model for K = 2 through 6 and computes BIC for model selection. b2_potential.py estimates the Langevin drift function and reconstructs the potential landscape. b4_survival.py computes Kaplan-Meier survival curves and log-rank tests for regime spells. b8_stage.py implements the eight-stage classification and computes inter-coder reliability statistics.
Scripts: phase3_*.py, fix01.py through fix07.py
These scripts apply the topology framework to the United States as a detailed case study, generating the time series, stage classifications, and institutional analysis presented in Chapter 14.
Scripts: c3_*.py through c10_*.py
This phase generates analyses that were identified as gaps during peer review: regional diffusion effects (c3), economic correlates (c4), military-institutional interactions (c5), capable autocracy profiles (c6), democratic recession patterns (c7), safe-asset analysis (c8), hybrid-trap duration models (c9), and backsliding early-warning indicators (c10).
Scripts: phase5_*.py, rp_*.py
Final robustness checks, sensitivity analyses, and recalibration of models after incorporating reviewer feedback. Includes bootstrap resampling, alternative clustering methods, and out-of-sample prediction tests.
political-topology-data.xlsx — Master dataset with all country-year observations, liberty scores, regime classifications, and covariates. political-topology-flat.csv — Flat CSV export of the master dataset for use with any statistical software. human_capabilities_index.xlsx — HCI scores, domain scores, and constituent indicators for all available country-years.
All scripts require Python 3.x with the standard scientific stack (NumPy, SciPy, pandas, matplotlib, scikit-learn, statsmodels, lifelines). No proprietary software is needed. A requirements.txt file is provided for environment setup via pip install -r requirements.txt.
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Acemoglu, Daron, ch. 2, ch. 4, ch. 7
AR(1) model, ch. 3, ch. 5, Appendix A
attractor basins, ch. 1, ch. 3, ch. 5, ch. 16
autocratic consolidation, ch. 6, ch. 8
autocratic equilibrium, ch. 3, ch. 5, ch. 6
backsliding, democratic, ch. 9, ch. 10, ch. 14, ch. 15
barrier heights, ch. 3, ch. 5, Appendix A
Bayesian Information Criterion (BIC), ch. 3, Appendix A
Bermeo, Nancy, ch. 9, ch. 10
bootstrap resampling, ch. 3, ch. 5, Appendix A
Brazil, ch. 4, ch. 10, ch. 11
Cambridge Governance Labs, Preface, ch. 16
capable autocracies, ch. 6, ch. 8, ch. 12
Chenoweth, Erica, ch. 7, ch. 9
China, ch. 6, ch. 8, ch. 12, ch. 13
civil resistance, ch. 7, ch. 9
Cohen's kappa, Appendix D
competitive authoritarianism, ch. 4, ch. 6
consolidation, democratic, ch. 7, ch. 10, ch. 14
corruption, ch. 4, ch. 8, ch. 11
credit model, sovereign, ch. 12, Appendix B
crosswalk, data, ch. 3, Appendix
decoupling, capability-liberty, ch. 8, ch. 13
democratic recession, ch. 9, ch. 10, ch. 15
Diamond, Larry, ch. 2, ch. 9, ch. 10
diffusion effects, regional, ch. 7, ch. 11
drift function, ch. 3, ch. 5, Appendix A
Egypt, ch. 4, ch. 7, ch. 9
eight-stage classification, ch. 3, ch. 5, Appendix D
electoral authoritarianism, ch. 4, ch. 6
EM algorithm, ch. 3, Appendix A
Ethiopia, ch. 6, ch. 11
event horizon, ch. 5, ch. 9, ch. 15
exorbitant privilege, ch. 12, ch. 13
Fragile States Index, ch. 9, ch. 11, ch. 15
Freedom House, ch. 1, ch. 3, ch. 5, Appendix
Fukuyama, Francis, ch. 2, ch. 9
Gandhi, Jennifer, ch. 6, ch. 8
Gaussian Mixture Model (GMM), ch. 3, ch. 5, Appendix A
Geddes, Barbara, ch. 6, ch. 7
Ghana, ch. 7, ch. 10, ch. 11
Gini coefficient, ch. 8, Appendix C
governance indicators, ch. 4, ch. 11, Appendix
Guriev, Sergei, ch. 6, ch. 8
Haggard, Stephan, ch. 9, ch. 10
Human Capabilities Index (HCI), ch. 8, ch. 13, Appendix C
Human Development Index (HDI), ch. 8, Appendix C
Hungary, ch. 9, ch. 10, ch. 14
Huntington, Samuel, ch. 2, ch. 7
hybrid regimes, ch. 4, ch. 5, ch. 9
hybrid trap, ch. 4, ch. 5, ch. 9, ch. 11
India, ch. 4, ch. 10, ch. 11, ch. 14
Indonesia, ch. 7, ch. 10, ch. 11
institutional persistence, ch. 3, ch. 5, ch. 10
inter-coder reliability, Appendix D
Japan, ch. 7, ch. 12, ch. 13
Jones, Benjamin, ch. 7
Kaplan-Meier survival, ch. 5, Appendix A
Kenya, ch. 4, ch. 11
kernel smoothing, Appendix A
Langevin equation, ch. 3, ch. 5, Appendix A
Levitsky, Steven, ch. 4, ch. 9, ch. 14
liberty score, ch. 1, ch. 3, ch. 5
Linz, Juan, ch. 2, ch. 7
Lipset, Seymour Martin, ch. 2, ch. 4
log-rank test, ch. 5, Appendix A
Mahoney, James, ch. 5, ch. 10
Mexico, ch. 4, ch. 7, ch. 11
military, role of, ch. 7, ch. 8, ch. 9
min-max nnormalisation, Appendix C
mmodernisation theory, ch. 2, ch. 4
Myanmar, ch. 6, ch. 9, ch. 11
Nigeria, ch. 4, ch. 7, ch. 11
North, Douglass, ch. 2, ch. 4, ch. 7
Nussbaum, Martha, ch. 8, ch. 13
oil-wealth effect, ch. 6, ch. 8, ch. 12
path dependence, ch. 3, ch. 5, ch. 10
Philippines, ch. 7, ch. 9, ch. 11
Pierson, Paul, ch. 5, ch. 10
Poland, ch. 7, ch. 9, ch. 10
Polity Project, ch. 3, Appendix
potential function, ch. 3, ch. 5, Appendix A
Przeworski, Adam, ch. 2, ch. 4, ch. 6
regime spells, ch. 5, Appendix A
regime velocity, ch. 12, Appendix B
reserve currency, ch. 12, ch. 13, Appendix B
Robinson, James, ch. 2, ch. 4, ch. 7
rule of law, ch. 4, ch. 7, ch. 10, ch. 14
Russia, ch. 6, ch. 8, ch. 9, ch. 13
Rwanda, ch. 6, ch. 8
safe assets, ch. 12, ch. 13
Saudi Arabia, ch. 6, ch. 8, ch. 12
Schedler, Andreas, ch. 4, ch. 6
selectorate theory, ch. 6, ch. 8
Sen, Amartya, ch. 2, ch. 8, ch. 13
Singapore, ch. 6, ch. 8, ch. 12, ch. 13
South Africa, ch. 7, ch. 10, ch. 11
South Korea, ch. 7, ch. 10, ch. 12
sovereign yields, ch. 12, Appendix B
spin dictators, ch. 6, ch. 8
stochastic differential equation, ch. 3, ch. 5, Appendix A
survival analysis, ch. 5, Appendix A
Svolik, Milan, ch. 6, ch. 9, ch. 10
Taiwan, ch. 7, ch. 10, ch. 12
third wave, ch. 2, ch. 7
topology, political, ch. 1, ch. 3, ch. 5, ch. 16
transition pathways, ch. 5, ch. 7, ch. 11
Treisman, Daniel, ch. 6, ch. 7
Tunisia, ch. 7, ch. 9, ch. 11
Turkey, ch. 4, ch. 9, ch. 10, ch. 14
UAE, ch. 6, ch. 8, ch. 12, ch. 13
Ukraine, ch. 4, ch. 9, ch. 11
United States, ch. 12, ch. 13, ch. 14
V-Dem (Varieties of Democracy), ch. 1, ch. 3, Appendix
Venezuela, ch. 6, ch. 9, ch. 11
Waldner, David, ch. 9, ch. 10
Wiener process, Appendix A
Wintrobe, Ronald, ch. 6, ch. 8
World Bank, ch. 4, ch. 11, Appendix
World Governance Indicators, ch. 4, ch. 11, Appendix
Ziblatt, Daniel, ch. 9, ch. 14
Zimbabwe, ch. 4, ch. 6, ch. 9