Cluster Causal Graphs

• Cluster causal graphs are graphical abstractions where nodes represent groups of variables, encoding directed and bidirected causal relations across clusters.
• They lift standard causal inference techniques, such as do-calculus and d-separation, from individual variables to aggregated cluster representations.
• They streamline high-dimensional causal discovery by preserving identifiability and reducing search spaces while addressing challenges like latent confounding and measurement error.

Cluster causal graphs are graphical abstractions in which nodes denote groups, modules, or clusters of variables rather than individual variables. In the dominant Cluster DAG formalization, a cluster graph summarizes directed and bidirected relations between clusters while leaving intra-cluster relations unspecified, so that the graph denotes an equivalence class of compatible micro-level acyclic directed mixed graphs and supports probabilistic, interventional, and counterfactual reasoning at the macro level (Anand et al., 2022). Closely related lines of work use clustering to preserve identifiability under graph contraction, to partition high-dimensional causal search spaces, to encode background knowledge for constraint-based discovery, or to learn cluster-specific causal graphs in heterogeneous data (Tikka et al., 2021, Shah et al., 2024, Vargas et al., 10 Dec 2025, Du et al., 19 May 2026).

1. Formal object and semantic interpretations

A Cluster DAG, or C-DAG, is a directed acyclic mixed graph over a set of clusters $C=\{C_1,\ldots,C_k\}$ that partitions the observed variables $V$. Each node $C_i$ represents a cluster, or macro-variable, consisting of a set of micro-variables; edges can be directed, $C_i \to C_j$, or bidirected, $C_i \leftrightarrow C_j$, the latter representing latent confounding across clusters. The defining abstraction is that relationships within a cluster are left unspecified, whereas relationships across clusters are asserted at the level of existence: $C_i \to C_j$ means that there exists at least one directed cause from $C_i$ to $C_j$ in the underlying micro-level graph, and $C_i \leftrightarrow C_j$ means that some members of $C_i$ and $V$0 share latent confounding (Anand et al., 2022).

This semantics is equivalence-class based. A C-DAG represents all individual-variable ADMGs that share the same inter-cluster structure but allow any possible intra-cluster relations consistent with acyclicity at the cluster level. Inference on the cluster graph is therefore interpreted uniformly over all compatible individual-variable graphs. In this sense, the cluster graph is not merely a compression device; it is a formal language for valid statements that must hold simultaneously for an entire class of micro-level models (Anand et al., 2022).

The classical definition uses an admissible partition. If $V$1 is an ADMG and $V$2 is a partition of $V$3, then the induced graph $V$4 contains $V$5 if there exist $V$6 and $V$7 such that $V$8 in $V$9, and contains $C_i$0 if there exist $C_i$1 and $C_i$2 such that $C_i$3 in $C_i$4. If $C_i$5 is acyclic, the partition is admissible and $C_i$6 is called a C-DAG compatible with $C_i$7 (Anand et al., 2022).

Subsequent work relaxes this admissibility requirement. In the relaxed framework, cycles and self-loops are allowed at the cluster level, provided that there exists at least one compatible acyclic micro-level ADMG. Formally, if $C_i$8 denotes the class of compatible micro-level graphs, then $C_i$9 if and only if $C_i \to C_j$0 contains no directed cycle composed entirely of singleton clusters. This permits cyclic macro-level abstractions, including macro feedback such as $C_i \to C_j$1, without abandoning acyclic micro-level causal semantics (Yvernes et al., 3 Nov 2025).

A related but distinct viewpoint appears in local-to-cluster abstraction. There, the cluster causal graph is again a mixed graph over clusters with directed and bidirected edges, but cycles and self-loops are explicitly permitted so long as there exists at least one acyclic compatible micro-level ADMG or DMG. Macro-level conclusions are then defined over the compatibility class $C_i \to C_j$2, and macro effects are written as $C_i \to C_j$3 for disjoint cluster sets $C_i \to C_j$4 and $C_i \to C_j$5 (Li, 24 Apr 2026).

2. Graphical inference at the cluster level

The central theoretical result of the original C-DAG framework is that standard graphical inference machinery lifts from micro variables to clusters. At the associational layer, if the underlying ADMG induces a Bayesian network with latent variables so that

$C_i \to C_j$6

then the observational distribution over clusters also factorizes along the cluster graph: $C_i \to C_j$7 where $C_i \to C_j$8 are the parents of cluster $C_i \to C_j$9 in $C_i \leftrightarrow C_j$0, and $C_i \leftrightarrow C_j$1 indicate cluster-level bidirected adjacencies (Anand et al., 2022).

Cluster-level d-separation is defined exactly as a path-blocking criterion on the cluster graph. A path is blocked by $C_i \leftrightarrow C_j$2 if it contains a non-collider triplet with the middle cluster in $C_i \leftrightarrow C_j$3, or a collider triplet whose middle cluster and descendants are absent from $C_i \leftrightarrow C_j$4. The corresponding theorem is both sound and complete: if $C_i \leftrightarrow C_j$5, then for any compatible underlying ADMG $C_i \leftrightarrow C_j$6, the corresponding sets of micro-variables are d-separated in $C_i \leftrightarrow C_j$7; conversely, if the cluster-level d-separation does not hold, there exists some compatible $C_i \leftrightarrow C_j$8 in which the micro-level separation also fails. Only d-separations, not d-connections, have this uniform interpretation across the entire equivalence class (Anand et al., 2022).

At the interventional layer, do-calculus also lifts to clusters. A key compatibility lemma states that mutilating the cluster graph by removing incoming edges to $C_i \leftrightarrow C_j$9 and outgoing edges from $C_i \to C_j$0 yields a graph compatible with the corresponding mutilation of any compatible micro-level graph. Consequently, Pearl’s three do-calculus rules hold on clusters, with the same syntactic form as in ordinary causal graphs but evaluated in the appropriate mutilated cluster graph. The truncated factorization likewise carries over: $C_i \to C_j$1 Within this semantics, the standard ID algorithm is sound and complete on $C_i \to C_j$2: if ID succeeds, the resulting formula is valid in all compatible ADMGs; if ID fails, there exists a compatible graph in which the effect is not identifiable (Anand et al., 2022).

At the counterfactual layer, if the underlying ADMG is induced by an SCM $C_i \to C_j$3, then there exists an SCM $C_i \to C_j$4 over macro-variables $C_i \to C_j$5 that induces $C_i \to C_j$6 and is equivalent to $C_i \to C_j$7 on all counterfactual queries whose variables are macros in $C_i \to C_j$8. Twin-network constructions and identification procedures such as CTFID can therefore be applied directly at the cluster level (Anand et al., 2022).

For cyclic cluster abstractions, the separation criterion changes. The relaxed calculus over arbitrary clusterings uses “structures of interest,” an unfolded graph $C_i \to C_j$9, and a canonical compatible graph $C_i$0 to test whether d-connecting structures can be realized without creating micro-level cycles. On top of this device, cluster-level versions of Rules $C_i$1–$C_i$2 are proved sound and atomically complete with respect to the do-calculus: every valid primitive interventional step at the macro level is captured, and every failure of a rule is witnessed by some compatible micro-level graph (Yvernes et al., 3 Nov 2025). In the L2C framework, the analogous role is played by a $C_i$3-separation-based cluster-level calculus on mutilated cluster graphs, again used to eliminate $C_i$4 operators at the macro level (Li, 24 Apr 2026).

3. Clustering, contraction, and identifiability preservation

Cluster causal graphs are not merely abstractions chosen for convenience; their validity depends on how variables are grouped. A general warning from the structure-learning literature is that aggregation preserves causal direction only under strong forms of within-cluster cohesion, and that aggregating variables with diverging causes or effects risks ecological bias, including Simpson’s paradox. Measurement error, heterogeneous aggregation, latent cluster confounding, and within-cluster cycles can all distort cluster-level conditional independences and orientation decisions. This suggests that a cluster causal graph should not be interpreted as an arbitrary coarsening of a variable-level DAG (Heinze-Deml et al., 2017).

A precise identifiability-preserving clustering notion is provided by transit clusters. In a DAG $C_i$5, a nonempty set $C_i$6 is a transit cluster if it satisfies five graphical conditions: equal outside-parents for all receivers, equal outside-children for all emitters, internal connectivity to a receiver or emitter in a modified induced subgraph $C_i$7, receiver-to-emitter flow, and emitter-from-receiver flow. If $C_i$8 is a transit cluster, then contracting $C_i$9 into a super-node $C_j$0 produces an induced graph whose interface is exactly

$C_j$1

so that no new parent-to-child paths are created across the boundary and no original ones are lost (Tikka et al., 2021).

For causal identifiability, two subclasses are singled out. A transit cluster is plain if $C_j$2, and congested if all members of $C_j$3 lie in the same c-component and $C_j$4. If $C_j$5 is a restricted transit cluster with respect to $C_j$6 and is plain or congested, then replacing $C_j$7 by a single node preserves identifiability: $C_j$8 The proof proceeds through the hedge criterion, showing that clustering neither creates nor destroys hedges under these conditions (Tikka et al., 2021).

This contraction-based perspective complements C-DAG semantics. The original C-DAG framework is conservative because a cluster-level conclusion must hold in all compatible micro-level ADMGs. Transit clustering, by contrast, identifies cases where contraction preserves the identifiability properties of a given graph exactly. A plausible implication is that the two frameworks address different abstraction regimes: one supports partial specification over an equivalence class, while the other certifies when a specific contraction does not alter the causal-identification landscape (Anand et al., 2022, Tikka et al., 2021).

The same issue appears in practical examples. In the C-DAG literature, clustering $C_j$9 and $C_i \leftrightarrow C_j$0 into a single $C_i \leftrightarrow C_j$1 can destroy a back-door adjustment that is available when $C_i \leftrightarrow C_j$2 and $C_i \leftrightarrow C_j$3 are kept separate, and a macro-level bow graph $C_i \leftrightarrow C_j$4 together with $C_i \leftrightarrow C_j$5 can render $C_i \leftrightarrow C_j$6 non-identifiable at the cluster level even if some compatible micro-level graphs admit identification. Overly coarse clustering therefore destroys separations and identifiability, while finer clusterings can preserve blocking structure (Anand et al., 2022).

4. Role in causal discovery and search-space reduction

Cluster causal graphs also serve as discovery targets and as structured priors for variable-level discovery. In one line of work, the graph over clusters is itself the target of estimation. If clusters are defined as $C_i \leftrightarrow C_j$7, then constraint-based methods such as PC and rankPC, score-based methods such as GES and rankGES, hybrid methods such as MMHC, interventional methods such as GIES, latent-variable methods such as FCI/PAGs, non-Gaussian methods such as LiNGAM, and multi-environment methods such as BACKSHIFT can all be applied to cluster variables, provided the relevant assumptions hold at the cluster level. However, aggregation can violate faithfulness, alter functional form, and dampen non-Gaussianity, so cluster-level learning is not a mechanically safe substitute for micro-level discovery (Heinze-Deml et al., 2017).

A different strategy is to use clustering to partition the hypothesis space of causal discovery. The causal graph partitioning framework begins from a superstructure $C_i \leftrightarrow C_j$8 satisfying $C_i \leftrightarrow C_j$9, computes a disjoint graph clustering $C_i$0, expands each cluster to $C_i$1, runs a consistent PAG learner such as FCI or RFCI on each $C_i$2, and merges the local outputs with a screening rule that keeps only edges present across all relevant subsets and orients unshielded colliders. Under Markov, faithfulness, correct superstructure coverage, and a consistent PAG learner, the resulting graph recovers the MEC of the true DAG. This makes clustering a device for divide-and-conquer discovery rather than a semantic abstraction of causal variables themselves (Shah et al., 2024).

C-DAGs can also be injected as background knowledge into constraint-based discovery. In the fully observed setting, Cluster-PC starts from a fully connected undirected micro graph, orients all cross-cluster edges implied by $C_i$3, deletes all forbidden cross-cluster adjacencies, and then performs adjacency search cluster-by-cluster in a topological order of the C-DAG. In the partially observed setting, Cluster-FCI transforms a cluster-level mixed graph into a partial mixed micro graph and then applies FCI-style skeleton discovery and orientation rules under cluster-imposed constraints. The paper proves that Cluster-PC with a CI oracle is sound and complete relative to standard PC plus the same constraints, whereas Cluster-FCI is sound but not complete in general (Vargas et al., 10 Dec 2025).

The same work establishes a precise expressivity comparison with tiered background knowledge. Tiered knowledge partitions variables into ordered tiers and allows edges only “forward” across tiers, but cannot express patterns such as $C_i$4 together with a strict absence of edges between $C_i$5 and $C_i$6, nor can it accommodate bidirected inter-cluster relations in the latent-variable setting. C-DAGs are therefore strictly more flexible as a background-knowledge language for discovery (Vargas et al., 10 Dec 2025).

A further development is automatic partition learning under latent confounding. L2C first performs local causal discovery on micro variables using MMB-by-MMB, then builds a similarity graph from shared parents, children, and $C_i$7-structures, clusters that graph, reduces each resulting cluster to at most three representative nodes, and finally performs macro-level inference on the derived cluster graph. The reduction theorem states that for any cluster $C_i$8 with $C_i$9, there exists a reduced cluster graph $V$00 such that

$V$01

This provides an automatic bridge from local latent-variable discovery to cluster-level causal reasoning (Li, 24 Apr 2026).

5. Applications and neighboring uses

One application area is fairness under graph uncertainty. In that framework, a cluster DAG $V$02 is learned over an admissible partition of observed variables, together with independence arcs and connection or separation marks that refine path activation and blocking. The cluster CPDAG learned by CLOC supports enumeration of adjustment cluster sets for a sensitive cluster $V$03, and the resulting interventional fairness objective minimizes the worst-case discrepancy between interventional prediction distributions across candidate adjustment sets. The fairness penalty is implemented with a barycenter kernel MMD, using the identity

$V$04

for uniform weights, which reduces $V$05 pairwise computations to $V$06. Here the cluster causal graph is neither only a discovery target nor only a semantic abstraction; it is an operational middle ground between full variable-level causal knowledge and purely associational prediction (Chikahara, 27 Feb 2026).

Another neighboring use concerns heterogeneous populations. The DAG-DC-ADMM framework jointly clusters subjects and learns cluster-specific dependency structures by assigning each subject a matrix $V$07, enforcing acyclicity through the NOTEARS-style constraint

$V$08

and introducing a groupwise truncated Lasso fusion penalty

$V$09

The output is a partition of subjects and a set of cluster-specific DAGs, rather than a graph whose nodes are variable clusters. The connection to cluster causal graphs is therefore structural rather than semantic: clustering is performed over subjects, while the causal graph remains at the variable level within each latent subgroup (Du et al., 19 May 2026).

A third neighboring meaning appears in causal inference on interference graphs. Randomized Graph Cluster Randomization chooses a random clustering $V$10 of the interference graph and then randomizes treatment at the cluster level. Exposure probabilities are integrated over the clustering distribution,

$V$11

which “washes out” extremely small exposure probabilities produced by fixed graph cluster randomization. Under restricted growth, the resulting Horvitz–Thompson variance bounds become polynomial rather than exponential in graph-metric parameters. This is a cluster-based causal design on graphs, not a C-DAG over grouped variables, but it shows that the phrase “cluster causal graph” also appears in design-based interference settings where clustering controls variance rather than encodes causal abstraction (Ugander et al., 2020).

These neighboring uses indicate that the topic spans at least three technical regimes: macro-level abstraction over variable groups, computational decomposition of high-dimensional discovery, and cluster-based experimental design on network graphs. This suggests that “cluster causal graph” is best understood as a family of structure-aware causal representations rather than a single universally fixed formalism.

6. Assumptions, limitations, and current directions

Across formulations, the dominant assumptions are acyclicity at the micro level, Markov and faithfulness properties, partition-based clustering, and explicit handling of latent confounding through bidirected edges or PAG semantics. In the original C-DAG framework, the layer assumptions are explicit: for $V$12, the underlying graph represents a BN; for $V$13, it represents a CBN; and for $V$14, it is induced by an SCM. Cross-cluster confounding must be represented via bidirected edges, clusters must form a partition of $V$15, and classical admissibility requires acyclicity at the cluster level (Anand et al., 2022).

A first limitation is conservatism. Because a C-DAG conclusion must hold in all compatible ADMGs, identification may fail on the cluster graph even if it succeeds in some members of the equivalence class. Likewise, d-connections in the cluster graph do not have the same uniform meaning as d-separations. Negative results are therefore safest interpreted as class-wise non-guarantees rather than as impossibility in every compatible micro model (Anand et al., 2022).

A second limitation concerns clustering itself. Overly coarse clusters can destroy blocking structure, while mis-specified clusters or cluster edges can force incorrect micro-level orientations or forbid true adjacencies in discovery procedures. In the fairness framework, refinement may be required when connection marks produce failures in adjustment completion. In the relaxed cyclic framework, overestimating cluster sizes preserves soundness but may lose completeness, whereas underestimating cluster sizes preserves completeness but can break soundness (Chikahara, 27 Feb 2026, Yvernes et al., 3 Nov 2025).

A third limitation is representational mismatch. Cluster-FCI is sound but not complete in general because some cluster-level priors encode non-ancestral information that PAGs cannot fully retain. The relaxed cyclic calculus is only atomically complete, not globally complete for all identifiable queries. And high-dimensional partitioning schemes require correct superstructure coverage $V$16; if the superstructure is too sparse, the partition may cease to be causal and recall degrades (Vargas et al., 10 Dec 2025, Yvernes et al., 3 Nov 2025, Shah et al., 2024).

A fourth issue is scalability. Although cluster-level reasoning avoids naïve enumeration of all compatible intra-cluster structures, worst-case identification remains exponential, as in ordinary ADMGs. Search-space partitioning and local-to-cluster reductions mitigate this by reducing the effective dimension from $V$17 to the number or size of clusters, and the “Infinity is at most three” theorem in the relaxed cyclic setting, together with the reduction-to-$V$18-representatives theorem in L2C, formalizes this compression principle (Yvernes et al., 3 Nov 2025, Li, 24 Apr 2026).

Current directions therefore move along two axes. One axis extends semantics, as in cyclic C-DAGs, $V$19-separation-based macro inference, and latent-variable-aware local-to-cluster abstraction. The other axis integrates cluster structure into discovery, fairness, and heterogeneous learning pipelines. Taken together, these developments position cluster causal graphs as a principled response to a recurring problem in causal analysis: the need to reason soundly when fine-grained causal knowledge is unavailable, computationally intractable, or scientifically less meaningful than relations among modules, pathways, sectors, or subsystems (Anand et al., 2022, Li, 24 Apr 2026).