Causal Structure Graphs

• Causal structure graphs are directed acyclic graphs that encode direct causal effects among variables using structural causal models and d-separation criteria.
• They integrate methodologies such as constraint-based, score-based, hybrid, and functional models to infer causality from data with enhanced scalability and robustness.
• Their applications span fields like biology, epidemiology, and economics, improving intervention planning and policy analysis using metrics such as SHD and SID.

Causal structure graphs are directed graphical representations—typically Directed Acyclic Graphs (DAGs)—that encode direct causal relationships among a set of variables, serving as the backbone for formalizing and reasoning about causality in complex systems. In causal inference, these structures allow both the interpretation of observed statistical dependencies and the prediction of counterfactual and interventional outcomes. Over the past decades, research advancing the theory, identification, estimation, and evaluation of causal structure graphs has matured into a multi-disciplinary field encompassing statistics, computer science, biology, and the social sciences.

1. Formal Definition and Causal Semantics

Causal structure graphs are represented mathematically as directed graphs $G = (V, E)$, where $V$ is a set of random variables and $E$ is a set of directed edges, and in most frameworks, $G$ is constrained to be acyclic (DAG). Each edge $i \to j$ reflects a direct causal effect of $X_i$ on $X_j$, often formalized via Structural Causal Models (SCMs) or Structural Equation Models (SEMs) as:

$X_j = f_j(X_{pa(j,G)}, \epsilon_j)$

where $pa(j,G)$ denotes the set of parent nodes of $j$. The acyclicity property ensures well-posed recursive causal assignments.

The encoding of conditional independence (CI) relationships in a causal structure graph is governed by d-separation: for node sets $A$, $B$, and $C$, $A$ and $B$ are d-separated given $C$ in $G$ if and only if every path between a node in $A$ and a node in $B$ is "blocked" by $C$ according to Pearl's graphical rules. This d-separation criterion serves as the basis for mapping observed CI relations to graphical structure (Heinze-Deml et al., 2017, Sadeghi et al., 2021, Squires et al., 2022).

Extensions of standard DAGs are required for incomplete observability or latent confounding; these include Mixed Ancestral Graphs (MAGs) and Partial Ancestral Graphs (PAGs), which incorporate bidirected edges to represent the presence of unmeasured confounders (Heinze-Deml et al., 2017, Chen et al., 2021).

2. Classes of Causal Structure Learning Algorithms

Methodologies to learn causal structure graphs from data fall into several broad classes, each with characteristic assumptions and computational implications (Heinze-Deml et al., 2017):

Class	Principle	Typical Output
Constraint-based	CI testing	(Markov equiv.) CPDAG/PAG
Score-based	Graph scoring (e.g., BIC)	CPDAG/DAG
Hybrid	Score & CI test combination	CPDAG/PAG
Functional model	Exploiting functional form	DAG

Constraint-based methods (e.g., PC, FCI) employ systematic CI tests to infer absent edges and apply orientation rules to identify v-structures, delivering a completed partially directed acyclic graph (CPDAG) or a PAG when latent variables are considered (Sadeghi et al., 2021, Chen et al., 2021). Score-based approaches (e.g., GES, NOTEARS) search for the optimal graph with respect to a penalized likelihood or information criterion, and may incorporate continuous optimization (Heinze-Deml et al., 2017, Fang et al., 2020). Hybrid methods such as MMHC leverage the strengths of both. Functional causal model–based estimators (e.g., LiNGAM, ANMs, CPCM) utilize properties of the generating mechanism—such as nonlinearity, non-Gaussian noise, or conditionally parametric distributions—to enhance identifiability beyond conditional independence (Bodik et al., 2023, Duong et al., 2023).

Bayesian approaches (e.g., variational inference on graph space (Annadani et al., 2021), random graph priors (Gonzalez-Soto et al., 2020)) model uncertainty over causal graphs by maintaining distributions over possible structures, facilitating intervention planning and uncertainty quantification.

3. Identifiability and Equivalence Classes

From observational data alone, causal structure is often only identifiable up to a Markov Equivalence Class (MEC)—the set of DAGs sharing the same skeleton and v-structure configuration (Squires et al., 2022, Kocaoglu, 2023). The essential graph or CPDAG provides a graphical summary of this equivalence class. The incorporation of interventional data (from hard do-interventions) shrinks the equivalence class to the interventional Markov equivalence class (I-MEC) (Zhou et al., 2 May 2025, Squires et al., 2022). The theoretical framework in (Zhou et al., 2 May 2025) defines I-Markov equivalence and provides graphical characterizations involving augmented pair graphs and "twin augmented MAGs." These representations track the full set of causal graphs indistinguishable by the invariances of do-calculus derived from multiple interventions.

An innovation in (Kocaoglu, 2023) introduces $k$-Markov equivalence: restricting the set of considered CIs to those with conditioning sets of size at most $k$, a necessary adaptation for small-sample, high-dimensional regimes, as typical CI tests lose power with large conditioning sets.

4. Algorithmic Developments and Structural Constraints

Recent advances have tackled challenges such as computational scalability, robustness to high degrees (hubs), and appropriate regularization for dense or large networks:

• Low-rank adaptations exploit the observation that many practical DAGs (particularly scale-free networks) have low-rank adjacency matrices, even if not strictly sparse (Fang et al., 2020). By parameterizing the weighted adjacency as $W = UV^\top$ or penalizing nuclear norm, algorithms retain interpretability and scalability in dense graphs.
• Local constraints, such as the $(\eta, \gamma)$–local path property, replace global sparsity assumptions (Chen et al., 2021), enabling consistent structure learning in graphs with hubs.
• In federated settings where central data aggregation is prohibited, secure federated CI tests (FedC²SL, FedPC/FedFCI) have been developed, using secure aggregation and random projection to learn global causal structures without exposing raw data (Wang et al., 2023).

5. Extensions for Functional, Dynamic, and Interventional Settings

Functional extensions enhance structure learning via leveraging non-additive noise, heteroscedasticity, or conditionally parametric mechanisms. Models such as CPCM admit broad classes of conditional effect distributions (e.g., Gaussian, Gamma, Pareto), going beyond mean shifts and enabling identifiability where classical Additive Noise Models (ANMs) fail (Bodik et al., 2023, Duong et al., 2023).

Dynamic causal graphs accommodate time-varying structure. Basis approximation techniques expand each time-varying parameter as a linear combination of smooth basis functions (e.g., B-splines), ensuring parameter parsimony and tractable inference (Wang et al., 11 Jan 2025). This paradigm is vital for understanding causal change in settings such as epidemiology (e.g., COVID-19 policy impacts).

Interventional structure learning leverages the extra constraints provided by experimental manipulations. Algorithms explicitly integrate the invariances from hard interventions, generalize do-calculus, and use orientation rules conspicuously designed for interventional data, as in (Zhou et al., 2 May 2025). These methods characterize interventional equivalence via graphical representations (e.g., I-augmented MAGs) and offer soundness guarantees given h-faithfulness.

6. Evaluation Metrics, Robustness, and Empirical Findings

The performance of causal structure learning algorithms is evaluated using structural Hamming distance (SHD), F-score for orientations and skeleton recovery, and for causal evaluation, the structural intervention distance (SID) or its improved variants (e.g., adjustment identification distance, AID) (Henckel et al., 13 Feb 2024). Adjustment-based distances directly measure discrepancies in the identifiability of causal effects (e.g., via parent, ancestor, or optimal adjustment sets) between graphs, and recent implementations (as in the gadjid package) scale to thousands of nodes in polynomial time, a substantial improvement over classic SID.

Empirical studies reveal nuanced findings:

• Including domain knowledge (e.g., separating primary and secondary variables in conditional DAGs) improves recovery and causal orientation, as demonstrated in molecular biology applications (TCGA proteomics) (Oates et al., 2014).
• Interventional data (even if limited) yields significant gains in identifiability.
• Variability between graphs estimated from similar subsamples or via different algorithms can be high, undermining overinterpreted conclusions from a single graph; robust reporting should average findings across multiple estimations (Hulse et al., 18 May 2025).
• Bayesian and deep-learning methods (e.g., variational causal networks, transformer-based supervised learners) produce well-calibrated uncertainty estimates and can generalize to new graph structures under substantial data heterogeneity (Annadani et al., 2021, Ke et al., 2022).

7. Applications and Broader Implications

Causal structure graphs are used in molecular biology (gene regulatory networks, proteomics), epidemiology (infectious disease dynamics, policy analysis), economics (instrumental variables, shift interventions), and clinical studies (treatment effect identification). In modern computational biology, the ability to identify causal orientations in signaling networks underpins the design of targeted therapies. In social sciences, dynamic structure graphs provide insights into evolving policy effects and system adaptation.

The rapid progress in scalable algorithms (local FCI, low-rank, federated inference), practical evaluation metrics (adjustment identification distance), and functionally rich model classes (CPCM, heteroscedastic models, dynamic basis expansion) is broadening the class of systems amenable to causal analysis and increasing the trustworthiness of causal conclusions from complex, high-dimensional, and time-evolving data.

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Causal structure graphs thus represent a foundational pillar of modern causal inference. Ongoing developments continue to refine the precision and scalability of structure learning, improve the correspondence between graphical features and interventional/identification properties, and address practical limitations encountered in empirical research and real-world application scenarios (Oates et al., 2014, Heinze-Deml et al., 2017, Squires et al., 2022, Bodik et al., 2023, Duong et al., 2023, Henckel et al., 13 Feb 2024, Wang et al., 11 Jan 2025, Zhou et al., 2 May 2025).