Phase transitions of civil unrest across countries and time

The paper investigates whether collective civil unrest dynamics can be described as repeated latent phase shifts, analogous to phase transitions in complex systems. Motivated by a gap between rich theoretical models of social phase transitions and limited empirical validation, the study aims to detect and measure latent phases of civil unrest that recur across countries and time. It asks: (1) Can civil unrest be plausibly characterized as recurrent latent phases with identifiable and measurable features? (2) Do universal mechanisms underlie aspects of civil unrest despite country-specific idiosyncrasies? (3) How similar or different are countries in long-term unrest intensity per unit time? (4) How does geographic embeddedness relate to a nation’s long-term unrest magnitude? The work situates civil unrest within a broader complex-systems context, referencing self-organized criticality and contagion through social and spatial networks, and emphasizes rigorous statistical detection of latent phases rather than ad hoc descriptions of waves or spikes.

The paper reviews theoretical and empirical research on phase transitions in social systems, including applications in traffic flow, econophysics, evolutionary game theory, opinion and cultural dynamics, and social collapse. It highlights a gap between modeling and empirical verification, noting empirical studies on human dynamics, financial markets, opinion dynamics, elections, and conflict phase forecasting with Markov models. Prior findings include protest recruitment bursts on Twitter, oscillatory political instability, and HMM-based conflict phase forecasting. The study draws on mechanistic perspectives of self-organized criticality and forest-fire-like models for civil unrest, where slow accumulation of societal stress leads to avalanches of unrest via contagion across spatial networks. It emphasizes that observing phased patterns does not prove self-organized criticality but provides a plausible phenomenological framework consistent with observed burstiness, spatial clustering, and time-scale separation between susceptibility buildup, triggering, and spread.

Data: The study uses the Cross-National Time-Series (CNTS) dataset for 170 countries, 1946–2017, derived primarily from New York Times reporting. Eight domestic conflict event types are aggregated: assassinations, general strikes, guerrilla warfare, government crises, purges, riots, revolutions, and anti-government demonstrations.

Weighted Conflict Metric and Magnitude: For each country-year, the weighted conflict (WC) metric is computed as WC = (25e1 + 20e2 + 100e3 + 20e4 + 20e5 + 25e6 + 150e7 + 10e8) / 8 * 100, where ei is the frequency of event type i. Civil unrest magnitude is defined as the logarithm of WC.

Macro-level Phase Model (HMM): The model comprises an unobserved m-state Markov chain Q_t governing latent phases and an observed process X_t (civil unrest magnitude). Emissions P(X_t | Q_t=i) are normal with mean μ_i and standard deviation σ_i, implying WC is lognormal by phase. The number of phases m is country-specific.

Model Parameters and Estimation: The model has m(m+2) parameters (initial state probabilities p_i, transition probabilities a_ij, and emission parameters μ_i, σ_i for each phase). Parameters are estimated via maximum likelihood using the Baum-Welch algorithm (forward–backward recursion), iterating until convergence. Model selection across candidate m uses Bayesian Information Criterion (BIC).

Marginal Distributions and Goodness-of-fit: Marginal distributions of X_t are obtained analytically from the HMM or via Monte Carlo simulation. Goodness-of-fit is assessed using Monte Carlo Kolmogorov–Smirnov (KS) two-sample tests: estimate the model on real data, compute empirical KS between empirical CDF and model marginal CDF; simulate synthetic series from the fitted model, re-estimate the model on each, compute KS statistics, and derive p-values as the proportion of simulated KS exceeding empirical KS. Bootstrap methods (1,000 replications) generate 95% confidence bands for empirical CDFs; over 5,000 kernel density estimates are averaged for model marginal density.

Treatment of Zeros: Zero event counts (likely due to reporting limits) are treated as missing values when taking logs. In Baum-Welch, diagonal emission matrices P(x_t) corresponding to missing observations are replaced by the identity matrix.

Derived Metrics: The civil unrest magnitude scale is the long-run mean magnitude per unit time Σ_i π_i μ_i, where π is the stationary distribution of A. Average durations in highest and lowest intensity phases are computed from transition rates under the estimated Markov chain. The Viterbi algorithm provides the most probable latent phase sequence for each country.

Spatial and Comparative Analyses: Spatial autocorrelation is tested using Moran’s I with (a) a 100 km distance-band weights matrix and (b) contiguity (shared borders). Hotspots/coldspots are identified using the Getis–Ord Local G statistic with random permutation tests. Cross-continental universality is examined via kernel density estimates of model features (unrest intensity ratio UIR = μ_H/μ_L; unrest fluctuation ratio UFR = σ_H/σ_L; durations in high/low phases), Kruskal–Wallis tests, and Bonferroni-adjusted pairwise KS tests (alpha 0.0083).

• Latent phases: Of 170 countries, 81 exhibit two latent phases (low and high), 5 exhibit three phases (low, intermediate, high), and 64 are best described by a single phase (lognormal magnitude without phase switching).
• Country example (Spain): High-intensity phase X ~ N(μ=8.37, σ=0.35); Low-intensity phase X ~ N(μ=6.84, σ=0.96). Transition probabilities: P(high→high)=0.94; P(low→low)=0.87. An observed large spike during a low phase can be a random fluctuation within that phase’s distribution.
• Goodness-of-fit: For representative countries (United States, France, China, Ethiopia), Monte Carlo KS tests yield p-values > 0.05, and model-generated marginal distributions closely match empirical distributions (means, medians, quartiles, SDs), supporting model plausibility across diverse cases.
• Universality across regions: Kernel density distributions of UIR, UFR, and phase durations show no significant differences across Africa, Americas, Asia, and Europe. Kruskal–Wallis tests are non-significant for all evaluated features (e.g., UIR p=0.97). Pairwise KS tests with Bonferroni adjustment (alpha=0.0083) also show no significant differences. This suggests universal distributions for phase intensity ratios, fluctuation ratios, and durations across continents.
• Geographic clustering: The civil unrest magnitude scale varies across regions and exhibits spatial clustering. Moran’s I indicates significant spatial autocorrelation: I=0.15 (p=0.04) for 100 km distance-band neighbors; I=0.185 (p=0.01) for contiguity neighbors. Hotspots (high magnitude scale) are concentrated in North-Central Africa (Algeria, Tunisia, Libya, Chad, Sudan, Egypt), Middle East (Israel, Lebanon, Syria, Jordan, Saudi Arabia), Eastern Europe (Belarus, Moldova), South Asia (India, Pakistan, Sri Lanka), Southeast Asia (Myanmar), and Central Asia (Tajikistan). Coldspots include Southern Europe (e.g., Albania, Bosnia and Herzegovina, Croatia, Montenegro, North Macedonia, Serbia, Kosovo), Eastern Europe (Czech Republic), Western/Northern Europe (Austria, Belgium, Germany, Netherlands, Denmark, Sweden), East Asia (South Korea), South America (Suriname), Middle Africa (Angola), Southern Africa (South Africa).
• Continental differences in long-run intensity: Despite universality in phase features, the long-run magnitude scale differs across continents (Kruskal–Wallis χ²(3)=14.48, p=0.002), indicating geographic embedding influences overall unrest intensity per unit time.

The findings support the central hypothesis that civil unrest dynamics can be represented by recurrent, measurable latent phases with Markovian transitions between phases. The macro-level HMM reproduces observed distributions of unrest magnitudes and their variability across countries, distinguishing latent phases from short-term spikes. Cross-continental invariance in intensity and fluctuation ratios and in phase durations suggests universal mechanisms operate despite country-specific contexts. Spatial analyses reveal significant geographic clustering of long-run unrest magnitude scales, consistent with historical patterns (e.g., the Arab Spring) and with mechanisms of contagion and shared regional conditions. While spatial clustering does not establish causal contagion, it aligns the model with known regional dynamics. The approach offers a framework for forecasting both magnitude distributions and phase transitions, enabling early-warning applications. It also complements early-warning signal research (e.g., rising autocorrelation or variance near transitions) by providing a latent-phase perspective that is less sensitive to noise and can be integrated with high-resolution data to probe resilience indicators around detected phase boundaries.

This study introduces and empirically validates a macro-level phase model of civil unrest that captures recurrent latent phases across 170 countries (1946–2017). The model fits marginal distributions well, quantifies phase-dependent magnitude and variability, estimates transition probabilities and long-run phase occupancy, and yields a magnitude scale enabling cross-country comparisons. Evidence for universality in phase features across continents and for geographic clustering of long-run unrest magnitudes suggests a combination of universal mechanisms and geographically embedded drivers. Applications include forecasting of magnitudes and phases, continuous monitoring, and early warning of heightened instability, informing policy and risk management by governments and international organizations. Future research directions include: distinguishing apparent versus true contagion via alternative (socio-cultural/economic) contiguity matrices; analyzing finer temporal resolutions (e.g., daily) to examine resilience indicators (autocorrelation, variance) near detected phase transitions; integrating covariates via classification models (e.g., multinomial logistic regression) to explain and predict phase transitions; and extending the methodology to other collective human phenomena.

• Data source and reporting bias: The CNTS dataset is based primarily on New York Times reporting, which can introduce geographic and temporal biases and undercount events; zero counts are treated as missing values.
• Model assumptions: Emissions are modeled as Gaussian in log-space (lognormal on original scale), which may not capture all distributional features; the Markov assumption and stationarity may be simplifications.
• Causality: Spatial clustering analyses (Moran’s I, Getis–Ord G) identify correlation, not causation; they cannot distinguish apparent versus true contagion without additional data.
• Temporal resolution: Annual aggregation may mask short-lived transitions and early-warning signatures; finer temporal data could reveal dynamics not captured here.
• External validity: The presence of phased patterns does not prove self-organized criticality; the model is phenomenological and consistent with, but not definitive of, SOC mechanisms.
• Multiple comparisons: While goodness-of-fit tests are performed per country, broader cross-country inference without correction could inflate Type I error if aggregated naively.