Cluster-Directed Mixed Graphs

• C-DMGs are graphical models that aggregate micro-variables into clusters with directed and bidirected edges to encode causal and associational relationships.
• They extend classical do-calculus using σ-separation, allowing the identification of macro-level causal effects even in systems with cycles and feedback.
• Applications span biology, epidemiology, economics, and social sciences, providing a scalable framework for abstracted causal reasoning in complex domains.

Cluster-Directed Mixed Graphs (C-DMGs) are graphical models in which variables are aggregated into clusters, and the edges—directed and/or bidirected—encode causal or associational relationships between these clusters. Unlike classical directed mixed graphs (DMGs) or acyclic directed mixed graphs (ADMGs), C-DMGs can systematically incorporate cycles and feedback, and they are equipped to model partial causal knowledge at varying granularity levels. The development of C-DMGs responds to the practical difficulty of specifying complete micro-level causal structures in complex domains, enabling both abstracted causal reasoning and scalable identification theory.

1. Foundations and Motivation

C-DMGs are defined over a partition of a set of micro-variables into clusters, with the graph's vertices representing clusters themselves. An edge $C_i \rightarrow C_j$ (directed) is included if any variable in cluster $C_i$ causally influences any variable in $C_j$, and a bidirected edge $C_i \leftrightarrow C_j$ indicates the existence of latent confounding between some pair of micro-variables from these clusters. The transformation from fully specified micro-level graphs (ADMGs or DMGs) into C-DMGs, via overlaying or projection, results in a coarser, partially specified model in which macro-structure becomes the central object of analysis.

This abstraction is particularly valuable in high-dimensional settings, where direct estimation and identification in fully specified graphs may be computationally or statistically intractable, and when feedback or equilibrium behaviors must be represented.

2. Causal Identification and Do-Calculus

A major theoretical advance in the paper of C-DMGs lies in the extension of causal effect identification—specifically, the identification of macro causal effects—by adapting Pearl's do-calculus to this more general context. For C-DMGs over acyclic graphs (C-ADMGs), recent work has shown that do-calculus is both sound and complete for identifying effects of interventions on clusters (macro-level effects), provided that clusters are not of size one or that the mapping from micro- to macro-structure is not fully specified (2504.01551).

In the more general setting of C-DMGs over possibly cyclic DMGs (allowing feedback), do-calculus remains sound and complete without additional conditions, owing to the adoption of the σ-separation criterion that generalizes d-separation for cyclic settings (2506.19650). This ensures that even when cyclic (equilibrium) causal dynamics are present, the identification of macro-level cluster interventions is governed by graphical criteria that are computationally tractable and faithful to the abstraction.

The rules in C-DMGs take the same mathematical form as in classical do-calculus, with replacements of d-separation by σ-separation in the mutilated (intervened) graph. The conclusion is that for any partition into clusters, and any macro-level intervention, the identifiability question is fully captured by do-calculus in conjunction with the relevant separation criterion.

3. Macro vs. Micro Causal Effects

In C-DMGs, a key distinction emerges between macro causal effects—defined as the effect of interventions on the states of entire clusters—and micro causal effects, which consider interventions at the level of individual micro-variables. The identification theory developed for C-DMGs is chiefly situated at the macro level: one seeks to identify distributions like

$$\Pr(\mathbb{C}_Y = \mathbb{c}_Y \mid \operatorname{do}(\mathbb{C}_X = \mathbb{c}_X))$$

where $\mathbb{C}_X, \mathbb{C}_Y$ are disjoint collections (possibly singleton or multi-element) of clusters.

Micro causal effect identification is generally more elusive in the C-DMG abstraction, especially in the presence of cycles, since the detailed mapping of interventions at the micro-variable level to their aggregated macro-level graphical representation becomes nontrivial and can lead to ambiguity that is not addressable by cluster-level analysis alone.

4. Graphical Characterizations and Non-Identifiability

A significant challenge in C-DMGs is the characterization of cases where macro causal effects are not identifiable from observational data. The classical "hedge" structure, which detects non-identifiability in ADMGs, fails to exhaustively diagnose such instances in the cluster setting, particularly in the presence of cycles. To address this, the SC-hedge ("strongly connected hedge") criterion is introduced (2504.01551, 2506.19650): clusters within strongly connected components are checked for structures in the SC-projected C-DMG that guarantee non-identifiability of some macro effects.

This process involves augmenting the graph with bidirected edges within strongly connected clusters and searching for pairs of forests (collections of trees rooted at clusters) that satisfy the SC-hedge conditions. The existence of such a structure is a sufficient condition for non-identifiability; while completeness of this criterion is demonstrated in certain settings, it remains an open problem in full generality.

5. Theoretical Advances: Cyclic vs. Acyclic Systems

The completeness of do-calculus for macro-level effect identification in C-DMGs over DMGs (possibly with cycles) is a key conceptual leap. Unlike C-ADMGs, where completeness requires additional assumptions such as cluster sizes greater than one, C-DMGs over DMGs allow cycles and feedback without sacrificing identification power at the macro level. This unifies causal abstraction and equilibrium modeling both conceptually and technically.

Sigma-separation plays a foundational role in this extension, capturing conditional independence relations in the presence of cycles, equilibrium, or feedback—a common structure in complex systems from economic models to biology and social sciences.

The following table underscores the distinctions:

Aspect	C-ADMGs	C-DMGs over DMGs (with cycles)
Cycles allowed	No	Yes
Completeness of do-calculus	Only under restrictions	Always (macro effects)
Required separation criterion	d-separation	σ-separation
Cluster size restriction	Yes (for completeness)	None (macro effects)

6. Applications and Modeling Strategies

C-DMGs have broad application potential in domains requiring abstraction and scalable causal inference. They are particularly well suited to:

• High-dimensional biological systems (e.g., gene regulatory networks), where gene sets or modules function as clusters and underlying cyclic behavior is prevalent.
• Epidemiology and public health, enabling causal reasoning about interventions on population-level features (clusters such as symptoms, risk factors).
• Economy and engineering, in which feedback, complex intervention scenarios, and partial specification are the norm.
• Social sciences and systems with pronounced modularity, where only inter-group relations are practically available.

Analysis pipelines using C-DMGs typically involve the following:

1. Form definition of clusters based on domain or data-driven modularity.
2. Construct cluster-level edges (directed and/or bidirected) from available causal knowledge or coarsened micro-structure.
3. Apply do-calculus with appropriate separation criterion (d-separation for acyclic, σ-separation for cyclic) to mechanistically relate interventions and outcomes at the cluster level.
4. Evaluate identifiability of macro causal effects, consulting the SC-hedge criterion for obstruction to identification.

7. Open Problems and Future Directions

Key avenues for future investigation include:

• Completing the SC-hedge characterization for non-identifiability in all settings, especially when clusters may contain only one variable or under varying levels of cluster size knowledge.
• Algorithmic development for efficient identification and SC-hedge checking in large, cyclic C-DMGs.
• Extension of the framework to hybrid models encompassing both macro and micro-level effect identification, as well as to learning problems under partial or noisy cluster definitions.
• Extension to dynamically evolving cluster structures or contexts where clusters emerge endogenously.

• (2504.01551) Identifying Macro Causal Effects in C-DMGs
• (2506.19650) Identifying Macro Causal Effects in C-DMGs over DMGs

These works establish the theoretical framework for macro causal effect identification in C-DMGs under both acyclic and cyclic contexts and provide graphical criteria and methodological guidance for their use in practical, high-dimensional, and cyclic systems.