Cyclic Causal Graphs

• Cyclic causal graphs are directed graphs that allow cycles, enabling the modeling of feedback loops and mutual influences in systems like biology and economics.
• Advanced discovery algorithms, such as the CCD method and matrix diagonalization techniques, provide rigorous means to uncover cyclic structures from observational and interventional data.
• Novel identifiability conditions and tailored experimental designs help overcome the inherent challenges of inferring causal relationships in systems characterized by equilibrium and recursive interactions.

A cyclic causal graph is a directed graph in which edges may form cycles, allowing for the modeling of feedback loops and mutual influences. Unlike directed acyclic graphs (DAGs), which enforce a strict hierarchy without closed directed paths, cyclic causal graphs capture complex dependencies characteristic of systems with equilibrium, regulation, or recursive interactions, such as in biology, economics, and social systems. Emerging research has established rigorous mathematical and algorithmic underpinnings for cyclic models—spanning discovery algorithms, statistical identifiability, intervention design, algebraic invariants, and new separation properties—thus extending the formalism of causal inference far beyond the DAG paradigm.

1. Foundational Assumptions and Representational Frameworks

Causal discovery and inference in cyclic graphs require extensions or modifications of standard assumptions. The Global Directed Markov property remains central: if sets $A$ and $B$ are d-separated by $C$ in the graph $G$, then $A \perp B \mid C$ in the distribution $P$ (Richardson, 2013). The Causal Faithfulness Assumption posits that the only conditional independencies present in $P$ are those entailed by the graph structure through d-separation.

Because many distinct cyclic structures may induce the same independence constraints, a unique graph is generally not identifiable. Instead, a Partial Ancestral Graph (PAG) is constructed to represent the Markov equivalence class—encoding invariant ancestral relationships, the presence of cycles (mutual ancestorship), and indeterminacies in edge orientation. In the case of relational and functional causal models, additional lifted representations, such as the $\sigma$-abstract ground graph, are used to summarize dependencies over all possible instantiations in relational domains (Ahsan et al., 2022).

2. Algorithmic and Statistical Methodologies for Structure Learning

Several lines of research have contributed sound and complete algorithms for discovering cyclic structures:

• CCD Algorithm: The Cyclic Causal Discovery (CCD) approach generalizes the PC algorithm to graphs with feedback, constructing a PAG by iteratively removing edges upon detection of conditional independence, followed by orientation via rules sensitive to colliders and ancestral relations (Richardson, 2013). Edge removals are based on independence tests over all subsets of adjacent nodes, and edge orientation exploits minimal separating sets for triples or quadruples. In the large-sample limit, CCD is both sound and d-separation complete, and exhibits polynomial complexity for sparse graphs.
• Matrix Diagonalization and Moment Methods: For linear cyclic models under shift interventions, methods such as backShift (Rothenhäusler et al., 2015) identify the causal structure by joint diagonalization of differences between environment-specific covariance matrices. This approach is efficient and requires only second moments, with identifiability guaranteed if intervention variances differ across at least three settings. In sparse graphs with bounded degree, the number of independence tests or diagonalizations remains polynomial.
• Maximum Likelihood and Penalized Estimators: Cyclic linear SEMs can be estimated using penalized likelihood approaches that jointly maximize the Gaussian likelihood and penalize edge cardinality. Two-step procedures—initialization followed by local refinement—achieve near-minimax optimal rates for sparse graphs under interventional designs that form a complete separating system (Hütter et al., 2019). Moment-based methods (LLC estimators) and maximum-likelihood refinements are compared, with the latter offering improved error rates and sensitivity to design conditioning.
• Score-Based and miGraphical Approaches: For both acyclic and cyclic settings, score-based local or global search (incorporating likelihood scores, model dimension penalties, and explicitly handling maximal strongly connected components for cyclic subgraphs) provides practical and theoretically consistent structure discovery (Ghassami et al., 2019, Améndola et al., 2020). The search space may be further compressed using characteristic imsets, a vector invariant that indexes the covariance equivalence class—substantially reducing redundancy compared to graph enumeration (Johnson et al., 16 Jun 2025).

3. Identifiability, Equivalence, and Separation Properties

Identifiability in cyclic models poses fundamental challenges:

• Distribution Equivalence and Covariance Imsets: In the Gaussian setting, two cyclic graphs are distributionally (covariance) equivalent if they induce the same precision matrix set. This can be characterized analytically via support rotations of matrix factorizations or, algebraically, via matching characteristic imsets (Ghassami et al., 2019, Johnson et al., 16 Jun 2025). For linear non-Gaussian models, equivalence classes may be determined by cycle-reversing permutations (with appropriate edge weight transformations) (Drton et al., 14 Jul 2025).
• Beyond d-Separation: Standard d-separation fails to capture all conditional independencies in cyclic graphs. Alternative graph separation criteria have been developed. Notably, p-separation (post-selection separation) is introduced as a sound and complete criterion for both quantum and classical cyclic causal models, generalizing d-separation via mappings to acyclic "teleportation" graphs in which independence properties are revealed after post-selecting on special vertices (Ferradini et al., 6 Feb 2025, Ferradini et al., 6 Feb 2025). In relational models, $\sigma$-separation enables correct reasoning about conditional independence in the presence of feedback, being both sound and complete under mild conditions (Ahsan et al., 2022).

4. Experiment Design and Interventional Strategies

Identifying cyclic structures from observational data alone is generally impossible; intervention design is therefore critical:

• Fundamental Barriers: In cyclic graphs, "virtual edges" may appear in the observational statistics, obscuring the skeleton. Moreover, hard interventions on a single node may not suffice to orient incident edges or "break" cycles, in contrast to DAGs (Mokhtarian et al., 2022).
• Order-Optimal Schemes: Two-stage experimental designs leverage colored and lifted separating systems. The first stage identifies descendants and decomposes the graph into strongly connected components. The second stage uses interventions tailored to individual SCCs to recover parent-child relations, with lower bounds showing that at least as many experiments as the size of the largest SCC are required (up to a logarithmic additive constant) (Mokhtarian et al., 2022).
• Interventional Data in Linear Models: The use of interventions tailored to separate specific variables (completely separating systems) is indispensable for minimax-optimal rate estimation in sparse cyclic structures; careful combinatorial design achieves optimality with a logarithmic number of experiments in the number of variables (Hütter et al., 2019).

5. Extensions: Latent Variables, Non-Linearities, and Quantum Causal Models

• Latent Confounders and Mixed Graphs: Extensions to mixed graphs incorporating bidirected edges allow cyclic models to represent both feedback and unmeasured confounding (Améndola et al., 2020). Completeness and model selection criteria generalize to these settings by linking the skeleton and collider structure to the space of covariance matrices.
• Non-Gaussianity and Nonlinearity: For linear non-Gaussian SEMs, independent subspace analysis (ISA) enables local identification of the entire directed structure around a target variable from the Markov blanket, even with cycles (Dai et al., 21 Mar 2024). Nonlinear models require contractive mappings for invertibility and tractable likelihoods; residual flows and normalizing flow techniques allow for scalable and flexible estimation over cyclic graphs (Sethuraman et al., 2023).
• Relational and Functional Extensions: Techniques for cyclic causal discovery in relational domains exploit lifted separation properties ($\sigma$-separation), relational acyclification, and algorithmic methods (e.g., RCD) ensuring sound and complete discovery under relational $\sigma$-faithfulness (Ahsan et al., 2022, Ahsan et al., 2022). Bayesian frameworks for cyclic SEMs over functional data further extend estimation and uncertainty quantification in brain or gene regulatory networks (Roy et al., 2023).
• Quantum Causality: Recent work generalizes cyclic causal modeling frameworks to quantum settings, introducing robust probability rules and graph-separation principles (p-separation) applicable to cyclic quantum causal networks. This mapping enables the transfer of algorithmic and conceptual methods from classical to quantum cyclic causal discovery (Ferradini et al., 6 Feb 2025).

6. Practical Challenges, Limitations, and Domains of Application

• Computational Complexity: While exhaustive search is infeasible for large graphs, constraint-based, score-based, or algebraic methods (using imset compressions or topological orderings among cycles) offer scalable alternatives for practical structure learning over cyclic graphs, especially when leveraging sparsity.
• Incomplete Data: Approaches based on expectation-maximization (e.g., MissNODAGS, MissNODAG) enable differentiable, likelihood-based learning of cyclic structure in the presence of missing data, including MNAR mechanisms (Sethuraman et al., 23 Feb 2024, Sethuraman et al., 24 Oct 2024). These frameworks alternate between imputation and likelihood maximization, ensuring practical applicability in settings such as single-cell or gene perturbation data.
• Root Cause Analysis and Outlier Propagation: In unknown cyclic graphs, anomalies propagate according to the underlying structural equations. Inverting the estimated precision matrix reveals a shortlist of plausible root causes, even without explicit knowledge of the graph. In cyclic settings, parents entangled in cycles with the root cause may also exhibit anomalous signatures (Schkoda et al., 8 Oct 2025).
• Domain-Specific Impact: Cyclic causal models play a critical role in fields where feedback, equilibrium, or mutual influence are essential. This includes molecular biology (gene regulation), neuroscience (functional connectivity), macroeconomics (virtually always with feedback), engineering control, and social science (reciprocal behavior).

7. Synthesis and Ongoing Research Directions

Cyclic causal graphs expand the theoretical and practical landscape of causal inference. Sound and complete discovery algorithms (e.g., CCD, backShift), statistical and algebraic invariants (characteristic imsets), new separation theorems (p-separation, $\sigma$-separation), and advanced experiment design collectively enable robust inference of feedback-rich systems. The breakdown of conditional independence-based identifiability in cyclic graphs motivates the development of distributional, algebraic, or interventional methodologies, especially in non-Gaussian, nonlinear, or incomplete data settings.

Open directions include the further generalization of separation criteria to continuous or high-cardinality settings (Ferradini et al., 6 Feb 2025), development of structure learning algorithms grounded in new separation properties, extension to quantum informatic causal modeling, and deeper integration into applications involving relational, time-evolving, and high-dimensional functional data. The flexibility and rigor of cyclic causal graph formalism now supports causal discovery and inference in complex interconnected domains previously inaccessible to classical DAG-based methodologies.