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

Augusto Santos, Diogo Rente, Rui Seabra, José M. F. Moura

TL;DR

The paper tackles learning the latent causal structure of linear networked dynamical systems (NDS) under partial observability and colored noise, where noise correlations create spurious node associations. It introduces a feature-based embedding for each node pair and shows there exists an affine separating hyperplane that distinguishes connected from disconnected pairs, enabling structure recovery via clustering. The authors derive a detailed error characterization of the estimators under noise structure, establish a sufficient identifiability condition, and propose novel composite features that improve separability; they validate the approach with simulations and a real brain connectome, demonstrating competitive performance against standard baselines. The work advances identifiability guarantees for LDIMs in challenging observation regimes and provides practical tools (feature construction and FFNN clustering) for robust causal structure learning in complex networks.

Abstract

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.

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

TL;DR

The paper tackles learning the latent causal structure of linear networked dynamical systems (NDS) under partial observability and colored noise, where noise correlations create spurious node associations. It introduces a feature-based embedding for each node pair and shows there exists an affine separating hyperplane that distinguishes connected from disconnected pairs, enabling structure recovery via clustering. The authors derive a detailed error characterization of the estimators under noise structure, establish a sufficient identifiability condition, and propose novel composite features that improve separability; they validate the approach with simulations and a real brain connectome, demonstrating competitive performance against standard baselines. The work advances identifiability guarantees for LDIMs in challenging observation regimes and provides practical tools (feature construction and FFNN clustering) for robust causal structure learning in complex networks.

Abstract

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.
Paper Structure (11 sections, 2 theorems, 19 equations, 3 figures)

This paper contains 11 sections, 2 theorems, 19 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Structural Consistency under Noise Structure and Partial Observability
  5. Structural Consistency: Error Characterization
  6. Identifiability under Noise Structure
  7. Novel Features
  8. Methodology
  9. Simulation Results
  10. Concluding Remarks
  11. Acknowledgments

Key Result

Theorem 1

Under Assumptions $1$, $2$, and $3$, for the NDS eq:model, we have with where the convergence holds in probability. Accordingly, under partial observability, we have

Figures (3)

  • Figure 1: Sufficient condition for structural consistency. The plots represent the off-diagonal entries of the corresponding matrices: the estimator, the ground truth and the error.
  • Figure 2: Impact of the correlation structure of the noise on the features. The overarching idea (Theorem $1$) is that the average $\beta$ across the off-diagonals of the covariance $\Sigma_x$ yields a drift of the features, while the oscillation across the off-diagonals of $\Sigma_x$ leads to a greater spread of the features.
  • Figure 3: Plots (a)-(f) contrast the accuracy of distinct methods as a function of the number of time series samples: $i)$ (a)-(c) undirected networks (graph learning); (d)-(e) directed networks (causal inference); and (f) refers to a directed Brain connectome matrix. The hyperparameters $N$, $S$, $\beta$ and $p$ displayed framed in the plots entail the underlying regime.

Theorems & Definitions (6)

  • Definition 1: Structural Consistency
  • Theorem 1: Error Characterization
  • proof
  • Theorem 2: Linear Separability & Stability
  • proof
  • Remark 1: Exogenous interventions