# Learning the Causal Structure of Networked Dynamical Systems under Latent Nodes and Structured Noise

arXiv: https://arxiv.org/abs/2312.05974 · HTML: https://mlsanity.com/paper/2312.05974/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 47 references that are in the map.
- First posted on arXiv 10 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Causal Transformer for Estimating Counterfactual Outcomes](https://mlsanity.com/paper/2204.07258/) (2022)
- [Optimal Errors and Phase Transitions in High-Dimensional Generalized Linear Models](https://mlsanity.com/paper/1708.03395/) (2017)
- [Interpretable Models for Granger Causality Using Self-explaining Neural Networks](https://mlsanity.com/paper/2101.07600/) (2021)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Estimating Treatment Effects from Irregular Time Series Observations with Hidden Confounders](https://mlsanity.com/paper/2303.02320/) (2023)
- [Generalized Network Dismantling](https://mlsanity.com/paper/1801.01357/) (2018)
- [Causal Inference by Identification of Vector Autoregressive Processes with Hidden Components](https://mlsanity.com/paper/1411.3972/) (2014)
- [SILVar: Single Index Latent Variable Models](https://mlsanity.com/paper/1705.03536/) (2017)
- [Local Tomography of Large Networks under the Low-Observability Regime](https://mlsanity.com/paper/1805.09081/) (2018)
- [Topology Identification under Spatially Correlated Noise](https://mlsanity.com/paper/2012.04175/) (2020)
- [An Unbiased Symmetric Matrix Estimator for Topology Inference under Partial Observability](https://mlsanity.com/paper/2203.15500/) (2022)
- [Data-Driven Linear Koopman Embedding for Networked Systems: Model-Predictive Grid Control](https://mlsanity.com/paper/2206.01272/) (2022)

## Built on by (most cited first, 6 of 6)

- [Identifying Drift, Diffusion, and Causal Structure from Temporal Snapshots](https://mlsanity.com/paper/2410.22729/) (2024)
- [Topological Schr"odinger Bridge Matching](https://mlsanity.com/paper/2504.04799/) (2025)
- [Inferring the Graph of Networked Dynamical Systems under Partial Observability and Spatially Colored Noise](https://mlsanity.com/paper/2312.11324/) (2023)
- [Learning Graph from Smooth Signals under Partial Observation: A Robustness Analysis](https://mlsanity.com/paper/2509.14887/) (2025)
- [Inferring the Graph Structure of Images for Graph Neural Networks](https://mlsanity.com/paper/2509.04677/) (2025)
- [The Powers of Precision: Structure-Informed Detection in Complex Systems -- From Customer Churn to Seizure Onset](https://mlsanity.com/paper/2601.21170/) (2026)

## Abstract (arXiv)

This paper considers learning the hidden causal network of a linear networked dynamical system (NDS) from the time series data at some of its nodes -- partial observability. The dynamics of the NDS are driven by colored noise that generates spurious associations across pairs of nodes, rendering the problem much harder. To address the challenge of noise correlation and partial observability, we assign to each pair of nodes a feature vector computed from the time series data of observed nodes. The feature embedding is engineered to yield structural consistency: there exists an affine hyperplane that consistently partitions the set of features, separating the feature vectors corresponding to connected pairs of nodes from those corresponding to disconnected pairs. The causal inference problem is thus addressed via clustering the designed features. We demonstrate with simple baseline supervised methods the competitive performance of the proposed causal inference mechanism under broad connectivity regimes and noise correlation levels, including a real world network. Further, we devise novel technical guarantees of structural consistency for linear NDS under the considered regime.