# Directed Cyclic Graphical Representations of Feedback Models

arXiv: https://arxiv.org/abs/1302.4982 · HTML: https://mlsanity.com/paper/1302.4982/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Feb 2013.

## Built on by (most cited first, 28 of 28)

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- [Establishing Markov Equivalence in Cyclic Directed Graphs](https://mlsanity.com/paper/2309.03092/) (2023)
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- [Differentiable Cyclic Causal Discovery Under Unmeasured Confounders](https://mlsanity.com/paper/2508.08450/) (2025)
- [Causal Discovery via Quantile Partial Effect](https://mlsanity.com/paper/2509.12981/) (2025)
- [Procedural Fairness Through Decoupling Objectionable Data Generating Components](https://mlsanity.com/paper/2311.14688/) (2023)
- [Interventional Causal Discovery in a Mixture of DAGs](https://mlsanity.com/paper/2406.08666/) (2024)
- [On the Foundations of Cycles in Bayesian Networks](https://mlsanity.com/paper/2301.08608/) (2023)
- [Coarsening Linear Non-Gaussian Causal Models with Cycles](https://mlsanity.com/paper/2605.10163/) (2026)
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- [DIGIC: Domain Generalizable Imitation Learning by Causal Discovery](https://mlsanity.com/paper/2402.18910/) (2024)
- [Spatio-Temporal Graphical Counterfactuals: An Overview](https://mlsanity.com/paper/2407.01875/) (2024)
- [Robust Causal Analysis of Linear Cyclic Systems With Hidden Confounders](https://mlsanity.com/paper/2411.11590/) (2024)
- [σ-Maximal Ancestral Graphs](https://mlsanity.com/paper/2507.00093/) (2025)
- [SCOUT: Cyclic Causal Discovery Under Soft Interventions with Unknown Targets](https://mlsanity.com/paper/2605.16620/) (2026)
- [Characterizing and Identifying Separable Graphical Models](https://mlsanity.com/paper/2607.01057/) (2026)
- [Beyond Directed Acyclic Graphs: Causal Zeros and Causal Differential Equations](https://mlsanity.com/paper/2607.22910/) (2026)
- [Causal Reasoning with Bipartite Graphical Causal Models](https://mlsanity.com/paper/2608.19831/) (2026)

## Abstract (arXiv)

The use of directed acyclic graphs (DAGs) to represent conditional independence relations among random variables has proved fruitful in a variety of ways. Recursive structural equation models are one kind of DAG model. However, non-recursive structural equation models of the kinds used to model economic processes are naturally represented by directed cyclic graphs with independent errors, a characterization of conditional independence errors, a characterization of conditional independence constraints is obtained, and it is shown that the result generalizes in a natural way to systems in which the error variables or noises are statistically dependent. For non-linear systems with independent errors a sufficient condition for conditional independence of variables in associated distributions is obtained.