# Clustering and Structural Robustness in Causal Diagrams

arXiv: https://arxiv.org/abs/2111.04513 · HTML: https://mlsanity.com/paper/2111.04513/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 51 references that are in the map.
- First posted on arXiv 8 Nov 2021.

## Builds on (most cited first, 12 of 12)

- [Causal Discovery in the Presence of Missing Data](https://mlsanity.com/paper/1807.04010/) (2018)
- [Clustering and Community Detection in Directed Networks: A Survey](https://mlsanity.com/paper/1308.0971/) (2013)
- [Identifying Causal Effects via Context-specific Independence Relations](https://mlsanity.com/paper/2009.09768/) (2020)
- [Identifying Causal Effects with the R Package causaleffect](https://mlsanity.com/paper/1806.07161/) (2018)
- [Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approach](https://mlsanity.com/paper/1902.01073/) (2019)
- [Learning Structures of Bayesian Networks for Variable Groups](https://mlsanity.com/paper/1508.07753/) (2015)
- [Causal Discovery in the Presence of Measurement Error: Identifiability Conditions](https://mlsanity.com/paper/1706.03768/) (2017)
- [Enhancing Identification of Causal Effects by Pruning](https://mlsanity.com/paper/1806.07085/) (2018)
- [Simplifying Probabilistic Expressions in Causal Inference](https://mlsanity.com/paper/1806.07082/) (2018)
- [Improving Efficiency and Accuracy of Causal Discovery Using a Hierarchical Wrapper](https://mlsanity.com/paper/2107.05001/) (2021)
- [Estimation of causal effects with small data in the presence of trapdoor variables](https://mlsanity.com/paper/2003.03187/) (2020)
- [Identification Methods With Arbitrary Interventional Distributions as Inputs](https://mlsanity.com/paper/2004.01157/) (2020)

## Built on by (most cited first, 12 of 12)

- [Learning Causal Abstractions of Linear Structural Causal Models](https://mlsanity.com/paper/2406.00394/) (2024)
- [Identifying Macro Causal Effects in a C-DMG over ADMGs](https://mlsanity.com/paper/2504.01551/) (2025)
- [Coarsening Causal DAG Models](https://mlsanity.com/paper/2601.10531/) (2026)
- [Identifying Macro Causal Effects in C-DMGs over DMGs](https://mlsanity.com/paper/2506.19650/) (2025)
- [Causal DAG Summarization (Full Version)](https://mlsanity.com/paper/2504.14937/) (2025)
- [Clustering and Pruning in Causal Data Fusion](https://mlsanity.com/paper/2505.15215/) (2025)
- [Cluster-Dags as Powerful Background Knowledge For Causal Discovery](https://mlsanity.com/paper/2512.10032/) (2025)
- [Simulating counterfactuals](https://mlsanity.com/paper/2306.15328/) (2023)
- [Fairness under Graph Uncertainty: Achieving Interventional Fairness with Partially Known Causal Graphs over Clusters of Variables](https://mlsanity.com/paper/2602.23611/) (2026)
- [Invariance of Clustering Operations in Causal Effect Identification](https://mlsanity.com/paper/2610.03101/) (2026)
- [GO-Based Clustering for Learning Cluster-Level Causal Gene Regulatory Networks](https://mlsanity.com/paper/2610.06042/) (2026)
- [Score-Based Learning of Cluster DAGs from Interventions](https://mlsanity.com/paper/2610.11947/) (2026)

## Abstract (arXiv)

Graphs are commonly used to represent and visualize causal relations. For a small number of variables, this approach provides a succinct and clear view of the scenario at hand. As the number of variables under study increases, the graphical approach may become impractical, and the clarity of the representation is lost. Clustering of variables is a natural way to reduce the size of the causal diagram, but it may erroneously change the essential properties of the causal relations if implemented arbitrarily. We define a specific type of cluster, called transit cluster, that is guaranteed to preserve the identifiability properties of causal effects under certain conditions. We provide a sound and complete algorithm for finding all transit clusters in a given graph and demonstrate how clustering can simplify the identification of causal effects. We also study the inverse problem, where one starts with a clustered graph and looks for extended graphs where the identifiability properties of causal effects remain unchanged. We show that this kind of structural robustness is closely related to transit clusters.