When Are Neural Interaction Discoveries Real? Identifiability, Recoverability, and a Pre-Fit Diagnostic

When a neural time-series model reports that one variable modulates another's effect on a target, is the discovered interaction a property of the data or an artifact of model flexibility? We argue that this is fundamentally a question of identifiability, governed by the geometry of the observed input support rather than by the specific neural architecture. We study the problem in a multiplicative-gating extension of neural additive vector autoregression (GNAVAR), in which source contributions are modulated by other lagged variables. We show that representational capacity is not identifiability: dependent inputs induce leakage between edge-specific interaction terms, and low-dimensional support permits distinct interaction decompositions that agree on the observed data while differing elsewhere. We then prove a population identifiability theorem for normalized minimal GNAVAR decompositions under explicit support conditions, including settings with shared modulators. The theory yields a simple practitioner-facing diagnostic: the effective rank of the joint lag-block covariance predicts, before fitting, whether interaction recovery is feasible for a given candidate set. When the candidate set is unknown, a two-seed stability check provides a practical operational test. The same support condition organizes empirical outcomes into the three states predicted by the theory. Our results show that interaction recoverability depends on support geometry, that effective rank provides a practical pre-fit diagnostic, and that instability across independent fits is a characteristic signature of non-identifiable interaction discovery. The identifiability phenomenon, the support condition, and the instability signature are model-agnostic; GNAVAR is the vehicle that makes them provable.

Trajectory-Aware Node Contributions and the Limits of Static Controllability

A recurring data mining task in complex networks is to determine how individual nodes contribute to system behavior. Existing approaches rely on either static-graph centralities or control-theoretic quantities such as controllability Gramians, which assume linear, time-invariant dynamics. Estimated systems, however, are typically nonlinear and time-varying. We define "emergent contribution (EC)," a finite-horizon measure of a node's dynamical leverage: the metric-weighted energy of its impulse response accumulated along the system trajectory. Computed from the Jacobians of any differentiable model, EC is estimator-agnostic and reduces exactly to average controllability in the linear, time-invariant limit. Our contribution is a characterization of when the two measures agree and diverge. Using a controlled synthetic family with known ground-truth contribution, we construct a phase diagram spanning nonlinearity, regime structure, persistence, and perturbation amplitude. EC and average controllability agree under static or smoothly drifting dynamics and both track ground truth. Divergence emerges under persistent regime switching, is strongest under persistent sign reversal, and disappears when the sign reversal is removed. At extreme perturbation amplitudes, both measures degrade, identifying the limits of local linearization. We place five estimated real systems from several domains within this phase space. Their placement serves as a diagnostic of when EC provides information beyond static controllability and therefore justifies its additional computational cost. On one panel examined in depth, a twenty-seed retraining ensemble reveals a robust variance--leverage dissociation: nodes whose perturbations propagate widely despite low within-system variance, which is not recovered by static centralities nor variance-based summaries.

Function-Valued Causal Influence in Nonlinear Time Series

Causal discovery in time series is increasingly performed using nonlinear machine-learning models, yet the resulting causal relationships are almost always summarized by scalar edge scores. We argue that this practice obscures the true object learned by nonlinear autoregressive models: a state-dependent function whose effect varies across regimes, magnitudes, and contexts. We formalize function-valued causal influence for additive, contribution-decomposable architectures and show that scalar causal scores constitute a severe information bottleneck, conflating between-state variation with within-state residual noise. Using Neural Additive Vector Autoregression as a representative architecture, we introduce a practical framework based on Individual Conditional Expectation for estimating causal response functions directly from trained models. Through controlled synthetic experiments, we demonstrate that edges with indistinguishable scalar scores can exhibit qualitatively different functional behaviors, including monotonic, thresholded, saturating, and sign-changing effects. An applied case study on democratic development further shows that function-valued analysis reveals regime-specific and asymmetric causal structure systematically missed by score-centric approaches.

Trajectory-Aware Reliability Modeling of Democratic Systems

Failures in complex systems often emerge through gradual degradation and the propagation of stress across interacting components rather than through isolated shocks. Democratic systems exhibit similar dynamics, where weakening institutions can trigger cascading deterioration in related institutional structures. Traditional reliability and survival models typically estimate failure risk based on the current system state but do not explicitly capture how degradation propagates through institutional networks over time. This paper introduces a trajectory-aware reliability modeling framework based on Dynamic Causal Neural Autoregression (DCNAR). The framework first estimates a causal interaction structure among institutional indicators and then models their joint temporal evolution to generate forward trajectories of system states. Failure risk is defined as the probability that predicted trajectories cross predefined degradation thresholds within a fixed horizon. Using longitudinal institutional indicators, we compare DCNAR-based trajectory risk models with discrete-time hazard and Cox proportional hazards models. Results show that trajectory-aware modeling consistently outperforms Cox models and improves risk prediction for several propagation-driven institutional failures. These findings highlight the importance of modeling dynamic system interactions for reliability analysis and early detection of systemic degradation.

Beyond Coefficients: Forecast-Necessity Testing for Interpretable Causal Discovery in Nonlinear Time-Series Models

Nonlinear machine-learning models are increasingly used to discover causal relationships in time-series data, yet the interpretation of their outputs remains poorly understood. In particular, causal scores produced by regularized neural autoregressive models are often treated as analogues of regression coefficients, leading to misleading claims of statistical significance. In this paper, we argue that causal relevance in nonlinear time-series models should be evaluated through forecast necessity rather than coefficient magnitude, and we present a practical evaluation procedure for doing so. We present an interpretable evaluation framework based on systematic edge ablation and forecast comparison, which tests whether a candidate causal relationship is required for accurate prediction. Using Neural Additive Vector Autoregression as a case study model, we apply this framework to a real-world case study of democratic development, modeled as a multivariate time series of panel data - democracy indicators across 139 countries. We show that relationships with similar causal scores can differ dramatically in their predictive necessity due to redundancy, temporal persistence, and regime-specific effects. Our results demonstrate how forecast-necessity testing supports more reliable causal reasoning in applied AI systems and provides practical guidance for interpreting nonlinear time-series models in high-stakes domains.

From Causal Discovery to Dynamic Causal Inference in Neural Time Series

Time-varying causal models provide a powerful framework for studying dynamic scientific systems, yet most existing approaches assume that the underlying causal network is known a priori - an assumption rarely satisfied in real-world domains where causal structure is uncertain, evolving, or only indirectly observable. This limits the applicability of dynamic causal inference in many scientific settings. We propose Dynamic Causal Network Autoregression (DCNAR), a two-stage neural causal modeling framework that integrates data-driven causal discovery with time-varying causal inference. In the first stage, a neural autoregressive causal discovery model learns a sparse directed causal network from multivariate time series. In the second stage, this learned structure is used as a structural prior for a time-varying neural network autoregression, enabling dynamic estimation of causal influence without requiring pre-specified network structure. We evaluate the scientific validity of DCNAR using behavioral diagnostics that assess causal necessity, temporal stability, and sensitivity to structural change, rather than predictive accuracy alone. Experiments on multi-country panel time-series data demonstrate that learned causal networks yield more stable and behaviorally meaningful dynamic causal inferences than coefficient-based or structure-free alternatives, even when forecasting performance is comparable. These results position DCNAR as a general framework for using AI as a scientific instrument for dynamic causal reasoning under structural uncertainty.