Time-Varying Causal Graphs

• Time-varying causal graphs are dynamic models that capture evolving causal relations among variables with structures that change over time or context.
• They employ methodologies like discrete-time graph dynamics, time-indexed DAGs, and dynamic mixtures to model nonstationarity and regime shifts.
• These frameworks enhance applications in neuroscience, economics, and climate science by offering rigorous estimation, identifiability, and policy analysis capabilities.

A time-varying causal graph is a mathematical and algorithmic construct that encodes dynamic patterns of causal relations among a collection of variables, where the existence, strength, or directionality of edges are allowed to change over time or in response to evolving context. Such models generalize the classical static causal graph (DAG or more general mixed graphs) by allowing the underlying structure to be indexed by time, regime, or other exogenous processes. They are essential for phenomena where the mechanism of interaction itself is nonstationary, including neuroscience, climate, economics, and systems biology. Time-varying causal graphs are formalized and estimated in several mathematically rigorous frameworks, allowing analysis of evolving causal effects, dynamic policies, and transient dependencies—often in the presence of latent confounders or abrupt change points.

1. Formal Definitions and Foundations

Time-varying causal graphs can be specified via several paradigms, depending on the domain and the type of dynamism modeled.

A. Discrete-Time Causal Graph Dynamics:

The Causal Graph Dynamics (CGD) framework models evolution of labeled, bounded-degree graphs under the constraints of shift-invariance and locality (causality, i.e., bounded speed of information propagation). A causal graph dynamic is a function $F: \mathcal{G} \to \mathcal{G}$ (where $\mathcal{G}$ is a space of pointed labeled graphs up to isomorphism) satisfying:

• Shift-invariance: acting "everywhere the same" under relabeling.
• Causality: For any region, the updates to nodes within a finite radius depend only on the input within a time- and space-bounded neighborhood.
• Localizability: Existence of a local update rule $f$ such that $F(G) = \bigcup_{v\in V(G)} f(G^r_v)$, where $G^r_v$ is a disk of radius $r$ around $v$ (Arrighi et al., 2012).
• Reversible Causal Graph Dynamics (RCGD) further require invertibility: both $F$ and $F^{-1}$ are CGDs, admitting a block-local circuit decomposition (Arrighi et al., 2015).

B. Time-Indexed Dynamic DAGs and Ancestral Graphs:

For multivariate time series, time-varying causal graphs are modeled as collections of time-indexed nodes, with edges indexed by time and possibly lag:

• In the "ts-DAG"/"ts-DMAG" formalism, each vertex is a pair $(i,t)$, where $i$ is a variable index and $t$ a time point. Edges $(i,t_1) \to (j,t_2)$ encode specific lagged or contemporaneous effects (Gerhardus, 2021).
• Marginalization over latent series induces mixed graphs (bidirected edges) encoding the presence of latent confounding.
• The class of ts-DMAGs is strictly characterized by constraints such as "repeating orientation" and "repeating separation sets," and their Markov equivalence classes are represented by ts-DPAGs, which retain all causal knowledge that can be learned from conditional independence structure, time ordering, and stationarity.

C. Dynamic Mixtures and Regimes:

In population or regime-switching scenarios, the joint distribution of observed variables at any point in time is modeled by a (possibly latent) mixture over a finite set of DAGs, with the mixture index capturing time or other regime variables. The overall process is represented by a "mixture graph" (acyclic when including mixture variables), and the fused summary graph may be cyclic at the population level (Strobl, 2019).

2. Causality, Locality, and Temporal Structure

A defining property of time-varying causal graphs is that they formalize causal influence as both a function of structure (edges) and time.

• Causality and Locality: The effect on each variable at time $t$ can only depend—directly or indirectly—on covariates within a bounded temporal (and often spatial) neighborhood. This is formalized by a "speed" parameter $r$ such that the outcome at a node can only be influenced by variables within distance $r$ in prior time steps (Arrighi et al., 2012).
• Shift- or Isomorphism-Invariance: In the absence of a global translation (as in irregular graphs), the dynamics are required to be isomorphism-conjugate: relabeling the entire input graph must commute with the dynamics, and the evolution must not depend on arbitrary node labeling (Arrighi et al., 2012, Arrighi et al., 2015).
• Time and Causal Ordering: All rigorous frameworks assert that causal influence must respect the arrow of time: a cause at time $t_1$ can only influence effects at time $t_2 \ge t_1$ (Reisach et al., 31 Jan 2025, Gerhardus, 2021). This notion underlies both acyclicity conditions (time-acyclicity and effect-acyclicity) and the construction of time-ordered graphs.

3. Estimation and Learning Methodologies

A diversity of methods has been developed to estimate time-varying causal graphs from data, often leveraging parameter expansions, penalized optimization, and nonparametric procedures:

A. Basis Expansions and State Space Models:

• Dynamic causal models (e.g., neuroimaging DCM): time-varying connectivity matrices are expanded in a (small) set of slow temporal basis functions, with the expansion coefficients inferred via variational Bayes under Gaussian-process or Gamma hyperpriors. Posterior distributions over entire connectivity trajectories $A(t)$ are computed, yielding fully probabilistic time-resolved causal graphs (Medrano et al., 2024).
• In dynamic SVAR/SEM, edge weights are parametrized via local basis expansions (e.g., splines or discrete cosines), allowing smooth evolution, with acyclicity and causal effect constraints enforced by differentiable penalties (e.g., NOTEARS-style constraints within a VAE) (Wang et al., 11 Jan 2025).

B. Local and Nonparametric Estimation:

• In high-dimensional locally stationary VAR, parameters are estimated via penalized local linear methods with time-varying LASSO or group LASSO, followed by CLIME precision estimation for contemporaneous partial correlations. Support recovery (edge detection) is consistent under sparsity and smoothness assumptions (Chen et al., 2023).
• Information-theoretic frameworks define the causal graph at each time by evaluating time-local (windowed) conditional mutual information, typically via kernel or k-NN entropy estimation; strong theoretical consistency and practical recovery are observed even in high-dimensional, nonlinear, and quickly changing networks (Etesami et al., 2023).
• Non-stationarity and regime-switching are accommodated by models that combine kernel-based changepoint detection (e.g., HSIC or GP-residuals with PELT) and context clustering, segmenting both temporal regimes and contextual groups before or during the search for optimal causal structure; minimum-description-length based criteria provide consistent global model selection (Mameche et al., 17 Jan 2025).

C. Amortized and Generative Approaches:

• Amortized Causal Discovery trains neural inference networks to recover latent sample-varying graphs from sequence observations, leveraging shared transition dynamics for improved sample efficiency and robustness to noise and confounding (Löwe et al., 2020).
• Dynamic factor models (e.g., time-varying graph as a superposition of K latent graphs with time-varying weights) allow both nonlinear and cyclic interactions and yield tractable learning schemes with proven empirical performance in high-dimensional, state-dependent neural timeseries (Brown et al., 27 May 2025).

4. Theoretical Guarantees and Structural Results

Multiple frameworks provide rigorous theorems and identifiability conditions for time-varying causal graphs:

• Axiomatic Equivalence and Localizability: In the discrete CGD setting, causal dynamics are equivalent to parallel synchronous application of a local rule; they are stable under composition, restriction, and inversion if defined over finite alphabets. Reversible CGDs admit block-local circuit decompositions (Arrighi et al., 2012, Arrighi et al., 2015).
• Characterization of Time-Indexed Graphs: ts-DMAGs are characterized exactly as those mixed graphs whose stationarification under a canonical construction equals the original, with latent confounders represented as additional time-series (Gerhardus, 2021). Markov equivalence, partial orientation, and reducibility to DPAGs follow as corollaries.
• Identifiability via Nonstationarity: Time-varying filter SCMs with stationary noise allow identifiability of directionality: unless the time-varying filters are pathologically constant, independence and stationarity criteria identify true direction and parent structure, both in bivariate and network settings (Du et al., 2020).
• Consistency of Estimation Procedures: Penalized regression approaches (local-linear, CLIME, basis expansion with LASSO) are shown to recover the correct support (adjacency matrix) uniformly over time, with oracle properties and explicit error bounds under high-dimensional scaling (Chen et al., 2023, Wang et al., 11 Jan 2025). Information-theoretic graph recovery is provably consistent under mixing and smoothness assumptions (Etesami et al., 2023).

5. Extensions: Latent Confounding, Regime Changes, and Policy Effects

Time-varying causal graphs must contend with unobserved confounders, changing regimes, and complex interventions:

• Latent Time Series and Ancestral Graphs: Projecting time-indexed DAGs with unobserved series induces bidirected edges in the ts-DMAG, capturing ancestral confounding paths at specific lags. The projection and stationarification procedures precisely specify which directed and bidirected links are warranted by the observed data (Gerhardus, 2021).
• Regime Mixtures and Cyclic Summaries: In mixtures of DAGs, longitudinal or regime-switching populations are modeled as a mixture over DAGs, yielding fused population-level graphs that may exhibit cycles even when each underlying regime is acyclic. CIM-algorithm constraint-based learning establishes identifiability under faithfulness with respect to the mixture graph (Strobl, 2019).
• Policy Analysis and Evolving Effects: Structural dynamic SEMs embed mediators, lagged effects, and treatment-outcome structure to estimate the instantaneous and evolving causal impact of interventions. Dynamic coefficient curves track the effectiveness of time-varying policies, with explicit formulas for effect size and interpretable trajectories (Wang et al., 11 Jan 2025).
• Time-Varying Instruments: LSTM+VAE architectures can learn time-varying conditional instrumental variables directly from proxy variables, identifying dynamic causal effects in the presence of time-indexed latent confounders without prior domain knowledge (Cheng et al., 2024).

6. Applications, Case Studies, and Empirical Results

Time-varying causal graphs are foundational for scientific inference in dynamic domains:

• Neuroscience: Time-varying DCM recovers slowly modulating synaptic couplings underlying evoked and spontaneous neural responses, matching observed MMN phenomena. Factor-based dynamic causal graph models uncover state-dependent neural circuits and outperform both static and linear dynamic baselines in f1 accuracy and interpretability (Medrano et al., 2024, Brown et al., 27 May 2025).
• Economics and Finance: Nonparametric TV-DIGs track sectoral and cross-sectoral risk propagation in high-dimensional financial markets, detecting pre-COVID systemic risk from cryptocurrencies, and revealing abrupt changes during crises (Etesami et al., 2023).
• Climate and Environmental Science: SpaceTime algorithm segments river-runoff and biosphere-atmosphere networks into regime periods and spatial contexts, revealing fine-grained dynamical shifts previously unattainable with stationarity-based methods (Mameche et al., 17 Jan 2025).
• Policy and Epidemiology: Dynamic SVAR approaches quantify the declining effectiveness of COVID-19 policy, linking time-resolved intervention coefficients to policy fatigue and shifts in compliance (Wang et al., 11 Jan 2025).
• Causal Inference with Nonstationary Time Series: Estimation of time-varying filters and stationarity-based identifiability is effective even in the presence of smooth high-order or non-smooth modulations, outperforming Granger and static structure methods in both synthetic and real data (Du et al., 2020).

7. Discussion and Future Directions

Time-varying causal graph modeling is an active domain integrating advances from dynamic graphical models, nonparametric statistics, optimization, and machine learning. Open challenges and research directions include:

• Computational scaling to very high-dimensional settings, especially with combinatorial parent selection under nonstationarity.
• Improved identifiability in the presence of rapidly switching or continuous-time changing regimes, and development of piecewise-stationary or continuous-time generalizations.
• Integration of interventions, hybrid context-regime modeling, and principled incorporation of external knowledge or partial observability.
• Theoretical analysis of amortized and generative neural approaches, automated hyperparameter selection, and further development of nonlocal or global regularization metrics for nonlinear graph diversity.

Across application domains, time-varying causal graphs provide a unified and robust framework for tracking, interpreting, and forecasting the evolution of complex, nonstationary systems. Recent theoretical and empirical advances have demonstrated significant gains in accuracy, identifiability, and interpretability compared to static or stationarity-restricted models, with applications reaching from fundamental network science to urgent policy evaluation (Arrighi et al., 2012, Arrighi et al., 2015, Gerhardus, 2021, Medrano et al., 2024, Etesami et al., 2023, Wang et al., 11 Jan 2025, Mameche et al., 17 Jan 2025, Brown et al., 27 May 2025).