Time-Resolved Causal Graphs

• Time-resolved causal graphs are formal representations that integrate time indexing to capture evolving causal relationships in multivariate systems.
• They extend traditional DAGs by enforcing temporal precedence, allowing for dynamic and repeated edge patterns across discrete or continuous time steps.
• Advanced learning methods, including constraint-based and score-based algorithms, enable practical causal identification in applications like neuroscience and genomics.

Time-resolved causal graphs are formal representations in which the causal structure—specifically, the pattern and directionality of causal relations—varies or unfolds explicitly as a function of discrete or continuous time. These models are foundational for capturing and inferring the dynamic evolution of causality in multivariate time series, complex dynamical networks, and physical, biological, or sociotechnical systems where interactions or influences change over time. Time-resolved frameworks subsume temporally indexed DAGs, causal Bayesian networks on time series, and generalizations such as dynamic structural equation models, process graphs, and mixed graphical models for stationary and nonstationary data.

1. Formal Definitions and Representations

Time-resolved causal graphs generalize the classical DAG (Directed Acyclic Graph) formalism by explicitly assigning time indices to nodes and by allowing edge structures to change or repeat across discrete or continuous time steps (Reisach et al., 31 Jan 2025, Gerhardus et al., 2023). The atomic representation uses a countable family of random variables $\{X_i(t)\ |\ i=1, \dots, p;\; t\in\mathbb{Z}\}$, where edges are permitted only from the past to the future ($t \leq t'$). Aggregate or macro-level causal variables over time-windows are constructed via injective functions $f$, e.g., $X_{T_x} = f(X(t_1),\ldots,X(t_k))$.

Key properties:

• Temporal precedence constraint: Any directed path or edge $X_{i,t} \to X_{j,t'}$ is only allowed if $t \le t'$, formalizing that causes precede effects (Reisach et al., 31 Jan 2025).
• Time-unfolded acyclicity: Cyclic causation between variables (e.g., $A \to B \wedge B \to A$) at the aggregate level becomes acyclic when unrolled in time, due to time-indexed structure.
• Repeated edge patterns: In many settings—particularly stationary time series—the edge pattern is periodic or invariant under time shift; both directed and bidirected (representing latent confounding) edges can recur at each time slice (Gerhardus et al., 2023, Jahn et al., 28 Apr 2025).

The table below summarizes fundamental distinctions:

Representation	Node Set	Edge semantics
Time-qualified atomic DAG	$\{X_i(t)\}$	$X_{i,t} \to X_{j,t'}:\ t \le t'$
Marginal finite time window	$\{X_i(t): t_0 \le t \le t_1\}$	Includes marginal/latent-induced edges
Process graph (collapsed view)	$\{X_i\}$	Aggregates all lags

2. Identification, Marginalization, and Stationarity

Causal identification in infinite or long-horizon time series models requires reducing the identification problem—such as the identifiability of $P(Y_{t'} \mid \Do X_t)$—to a finite search (Jahn et al., 28 Apr 2025). The principal result is that for time-resolved ADMGs (Acyclic Directed Mixed Graphs) with window width $p$ (number of variables per step) and maximum direct or latent lag $L$, the causal effect is identifiable in the infinite model iff it is identifiable in a finite window of size $B = L \cdot 2^{L p} (L p+1)^{2 L p + 2}$. All do-calculus and ID-algorithm tools thus reduce to search over finite, stationary-structured segments.

Projection of infinite-time series causal graphs onto finite observation windows is achieved via latent variable marginalization, preserving all $m$-separations (ancestral relationships and $m$-separation equivalence classes) (Gerhardus et al., 2023). Computation of whether two events share a common ancestor, or whether an edge survives marginalization, is decided by path enumeration plus solution of a corresponding linear Diophantine equation arising from cycle/cycle-free walk decompositions.

3. Algorithms for Learning Time-Resolved Causal Graphs

Multiple methodological streams exist for inferring time-resolved causal graphs:

• Constraint-based methods: Extensions of the PC and FCI algorithms adapted with explicit time-lag indexing (e.g., the TS-ICD, PCGCE, FCIGCE, and tPC algorithms) test for conditional independence or causation entropy over lagged and contemporaneous edges, ordering the search from longest lags to contemporaneous edges for efficiency gains and clearer temporal separation (Rohekar et al., 2023, Assaad et al., 2022, Loranchet et al., 29 Aug 2025).
• Score-based and hybrid approaches: Functional or structural model selection via MDL (Minimum Description Length) across context-regime partitions, as in the SpaceTime algorithm, optimizes both the graph and changepoint or regime assignment in nonstationary and multi-context data (Mameche et al., 17 Jan 2025).
• Nonlinear and dynamic factor models: Dynamic causal graphs can be represented as weighted superpositions of static nonlinear Granger-causal graphs, with the weights (factor activations) encoding state- or regime-dependent shifts in connectivity (e.g., REDCLIFF-S) (Brown et al., 27 May 2025).
• Neural and higher-order graph methods: Message-passing neural architectures (e.g., DBGNN) constructed on higher-order or De Bruijn graphs allow for the capture of patterns in the temporal-topological structure, enabling learning of non-Markovian and temporally-patterned causality (Qarkaxhija et al., 2022).

4. Causal Compression and Information-Theoretic Approaches

Directed information and causal compression offer a formal, information-theoretic basis for selecting sparse subsets of time points or events carrying the entire causal flow between processes (Wieczorek et al., 2016). The chain rule for directed information

$I(A,B \to C) = I(A \to C) + I(B \to C | A)$

leads to sparsity: only those time points that contribute unique directed information are retained in the resulting compressed causal graph. Optimization for causal bipartite graph recovery reduces to a LASSO-style log-determinant minimization in Gaussian copula models, solving for the minimal set of effective causes at each time point.

5. Causal Graph Dynamics, Reversibility, and Quantum Extensions

Time-resolved causal graphs also appear as discrete-time dynamical systems over graphs themselves. Causal Graph Dynamics (CGD) formalize synchronous, homogeneous evolution via global (possibly invertible) maps with three core axioms—causality (bounded propagation), shift-invariance (homogeneity), and boundedness/no unbounded branching (Arrighi et al., 2015).

A global reversible CGD can be block-decomposed into a finite-depth composition of local reversible gates, each acting on bounded neighborhoods. Quantum generalizations extend these ideas further: a global unitary, causal operator acting on quantum-labeled graphs with the causality property (information spread bounded by light-cone) necessarily admits decomposition as a finite-depth circuit of local unitary gates (Arrighi et al., 2016). This directly connects with quantum cellular automata and quantum gravity models.

6. Practical Applications and Empirical Benchmarks

Time-resolved causal graph estimation has been validated across synthetic and real datasets in neuroscience, genomics, climate science, epidemiology, and social network analysis. Notable results include:

• Dynamic nonlinear Granger-causal factor models achieving F₁-score improvements of 22–28% (and sometimes >60%) over static methods in inferring true dynamic graphs in brain data (Brown et al., 27 May 2025).
• Causation-entropy-based PC adaptations (PCGCE) outperforming leading baselines (VarLiNGAM, oCSE, PCMCI) in detecting non-self, time-lagged causal links from multivariate time series, particularly in challenging real-world fMRI data (Assaad et al., 2022).
• SpaceTime MDL-based regime discovery revealing interpretable and geographically structured causal regimes in large-scale environmental monitoring (e.g., precipitation-runoff transfer in catchments, biosphere-atmosphere coupling) (Mameche et al., 17 Jan 2025).
• Constraint-based discovery for dynamic MAGs (TS-ICD) enabling correct recovery of lagged and contemporaneous causal edges with substantial reduction in the number of CI tests compared to FCI/RFCI/LPCMCI, especially in settings with latent confounders (Rohekar et al., 2023).

7. Conceptual and Theoretical Implications

Explicit time-indexing in causal graphs resolves ambiguities about the directionality of influence, disallows cycles between distinct time points, delineates true feedback (which can only occur via time-unfolded paths), and grounds faithfulness and acyclicity in the atomic, temporally separated variables (Reisach et al., 31 Jan 2025, Loranchet et al., 29 Aug 2025). When time indices are omitted, apparent cycles may be artifacts of aggregation; proper unfolding always results in an acyclic, time-ordered graph, so that all causal assertions respect the temporal order "past $\to$ future."

Identifiability criteria, orientability based on macro-summary causal graphs, block-decomposition and stationarity assumptions all serve to constrain and sharpen what can be learned about causal directionality and mechanisms from temporally indexed data, including in the presence of hidden confounding and context variability (Loranchet et al., 29 Aug 2025, Jahn et al., 28 Apr 2025, Reiter et al., 2023). The fusion of temporal, graphical, algebraic, and information-theoretic principles in time-resolved causal graph modeling continues to structure advances in causal time series analysis.