The 1995–2018 global evolution of the network of amicable and hostile relations among nation-states

The paper investigates how networks of amicable and hostile international relations evolve over time and whether structural balance mechanisms govern these dynamics at a global scale. Classical structural balance theory predicts that signed networks evolve toward either all-positive ties or two antagonistic camps. Yet most prior empirical work lacked large-scale longitudinal data and often assumed complete networks without neutral relations. Leveraging a 23+ year dataset (1995–2018) of directed signed interactions among nation-states, the study asks whether reductions in intransitive (unbalanced) relations drive the evolution of international appraisal networks, and whether the system exhibits emergent stability over time.

Structural balance theory originates in sociology and social psychology and was formalized by Cartwright and Harary. It has been applied to diverse domains, including analyses of international crises (e.g., World War I precursors, Middle East 1956), consumer branding, sports, virtual worlds, and animal societies. Classical theory assumes complete signed networks and yields two balanced macro-structures (all-positive or two antagonistic cliques). Empirical tests have often been static or limited in temporal depth; longitudinal studies have typically involved small populations. A line of research relaxes the completeness assumption to allow null/indifferent ties in large-scale networks. Prior works have explored transitivity and clustering-based notions of balance, but comprehensive longitudinal evidence on large, sparse international networks has been lacking. This study builds on and generalizes these models (classical, clustering, and transitivity) to sparse directed networks to evaluate balance dynamics empirically over two decades.

Data: The Integrated Crisis Early Warning System (ICEWS) event dataset (1995-09-01 to 2018-09-30) containing 8,073,921 international events among 250 countries. Each event encodes a directed source→target action with a signed weight in [-10, +10], where negative indicates hostile and positive indicates supportive actions. Aggregation: Time is discretized into periods; primary analyses use quarterly (≈12-week) windows, with robustness checks at monthly, biweekly, and weekly resolutions. For each period t, all events between a country pair are summed and the sign of the sum determines the directed edge sign A_ij(t) ∈ {−1, 0, +1}. Network focus: Each period’s network over 250 nodes yields up to 62,250 directed edges. The network exhibits a core-periphery structure; initially 111 countries in the core with 23 in the periphery, expanding to a strongly connected core of 134 countries within ~4 years. Analyses focus on these 134 countries present across periods. Sparse structural balance: Triads (triplets of countries) are classified allowing null edges, yielding 138 possible directed signed triad types. Three generalized balance models are defined for sparse directed networks: classical (all four Heider axioms), clustering (A1–A3), and transitivity (A1 only). Each triad is labeled balanced or unbalanced per model definitions. Empirical Markov transitions: For each consecutive pair of periods (t, t+1), the triad type transitions are counted to form a 138×138 row-stochastic transition probability matrix P_t via maximum-likelihood estimates (row-normalized counts). Stationary distributions are computed to assess long-run tendencies. Time-varying Markov model: To capture smooth temporal evolution and mitigate sparsity/dependencies among triads sharing edges, the study estimates a sequence of transition matrices {P_t} jointly via a convex optimization with fused and group lasso regularization promoting small, sparse changes over time. Constraints enforce row-stochasticity and probabilities in (0,1]. Estimation uses CVXPY; hyperparameters (λ1=0.01, λ2=0.005) tuned via validation; sliding 2-year windows used for prediction tasks. Stability and exogenous factors: Stability is quantified as the Frobenius norm of differences between consecutive empirical transition matrices. The series is related to exogenous shocks (e.g., wars, 9/11) and to World Bank global trade data (trade as % of GDP). Pearson correlations and Granger causality (lag=1; F and χ2 tests) assess associations between stability and trade. Forecast evaluation: One-step-ahead forecasts of triad proportions compare the proposed time-varying Markov model against baselines (last proportion, average proportion, and time-invariant Markov).

- Dataset composition: 8,073,921 international events among 250 countries (1995–2018): 74% positive (5,974,283), 17% negative (1,333,646), and 9% neutral (765,992). Each period’s network can include positive, negative, and null edges. - Core-periphery structure: Initial period shows a core of 111 countries and a periphery of 23 (Afghanistan, Angola, Guinea, Haiti, Sierra Leone, Zimbabwe, Bolivia, Paraguay, Rwanda, Armenia, Azerbaijan, Congo, Grenada, Guatemala, Guyana, Kuwait, Malawi, Mozambique, Nicaragua, Nigeria, Panama, Sudan, Timor-Leste). Within ~4 years all periphery members join the core; the core stabilizes at 134 countries. In the core, positive tie proportion rises from ~4% to ~18% and negative from ~1% to ~4% over time. - Triad concentration: Across 23+ years, 91% of 392,084 observed triads fall into only 10 of the 138 possible types. Among these, types 6, 8, and 9 are intransitive (unbalanced), while others are transitive/balanced. - Temporal balance trajectory: The proportion of balanced triads initially decreases until about 2006 due to conversion of null ties to positive ties (inflating intransitive triad 9), then increases steadily thereafter. Only ~8% of triads (aggregate of types 6, 8, 9) are unbalanced over more than two decades. - Markov dynamics: Transition matrices are robust across period lengths (seasonal, monthly, biweekly, weekly; flattened-matrix Pearson correlations 0.99, 0.98, 0.86 vs. seasonal, all p<0.05). Triads 1, 4, 9, 10 exhibit high self-transition probabilities. Stationary distributions assign >0.85 probability to balanced triads, indicating a strong long-run tendency toward balance. - Time-varying Markov estimation: Balanced triads show high probability of remaining balanced with low variance; unbalanced triads show high probability of transitioning to balanced triads with low variance, consistently across balance definitions (transitivity, clustering, classical). - Stability: The Frobenius norm of differences between consecutive transition matrices declines over time, indicating increasing stability of appraisal dynamics, with noticeable perturbations at major exogenous events (e.g., 9/11). - Trade relation: The Frobenius norm difference series is strongly and negatively correlated with global trade volume (% of GDP) over 23+ years (|r|≈0.88; p<1e−07). Granger causality tests (lag=1) show that higher global trade Granger-causes increased stability (p<1e−03), with feedback from stability to trade (p<1e−02). - Generality: Applying the time-varying Markov analysis to two Bitcoin trust networks yields similar movement toward balance, suggesting ubiquity across settings and actor types. - Prediction: The proposed time-varying Markov model yields lower RMSE in forecasting next-period triad proportions than baselines (last proportion, average proportion, time-invariant Markov) in most years.

The findings demonstrate that the global network of international appraisals evolves primarily through reductions of intransitive (unbalanced) configurations rather than converging to the all-positive or two-camp structures predicted by classical balance theory on complete networks. Allowing null ties, the network supports complex topologies with more than two antagonistic groups and hierarchical positive relations. Empirically, unbalanced triads are short-lived: they transition to balanced configurations with high probability, while balanced triads persist—evidence consistent with balance mechanisms at the micro-dynamic level. The system exhibits increasing dynamic stability over time, as shown by declining differences between consecutive transition matrices, punctuated by perturbations associated with major global shocks. The strong negative association and Granger causality between global trade and the magnitude of relational changes suggest that economic interdependence is linked to stabilization of international relations. The robustness of transition structures across different temporal aggregations and their generalization to other domains (Bitcoin trust networks) support the broader relevance of these dynamics.

Using the largest longitudinal dataset to date on international signed interactions, the study generalizes structural balance theory to sparse, directed networks and provides empirical evidence that international appraisal networks tend toward balance and dynamic stability. A novel convex, time-varying Markov modeling framework with convergence guarantees quantifies micro-dynamics of triad transitions, revealing persistence of balanced states and rapid resolution of unbalanced ones. The system’s stability increases over time and is intertwined with global economic activity, with shocks temporarily disrupting trajectories. Future research could enhance predictability by relaxing the Markov assumption to incorporate longer memory (e.g., recursive neural models) and by integrating additional exogenous covariates to explain temporal variability in transition structures.

- Dependence among triads: Triads share edges, potentially introducing dependencies that can bias simple transition counts; the study mitigates this with smoothness regularization in the time-varying Markov model, but residual dependence may remain. - Event-based measurement: Reliance on ICEWS news-coded events may reflect reporting biases and uneven coverage across countries and time. - Temporal aggregation: Period selection (weekly to quarterly) involves trade-offs between data sufficiency and granularity; although robustness was shown, finer dynamics may be obscured. - Node coverage and focus: Not all 250 countries appear in every period; analyses focus on a core of 134 countries present across periods, which may limit generalizability to countries with sparse data. - Model assumptions: The Markov property assumes transitions depend only on current triad states; non-Markovian influences are not modeled.