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Learning the Causal Structure of Networked Dynamical Systems under Latent Nodes and Structured Noise
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Learns the hidden causal network of a linear networked dynamical system from partially observed nodes under spatially correlated (colored) noise by clustering engineered pair features with a feed-forward neural network, with a new identifiability condition on the noise covariance.
Problem
Causal structure recovery of a linear networked dynamical system from time series at a subset of nodes (partial observability) when the excitation noise is spatially correlated, which creates spurious associations between node pairs.
Approach
Assign each node pair a feature vector built from lagged empirical covariances and inverses of observed-node lag moments, show an affine hyperplane separates connected from disconnected pairs when the off-diagonal noise covariance oscillation is small relative to the smallest nonzero interaction, and classify pair features with a supervised FFNN instead of thresholding.
Key results
- The proposed feature-based FFNN classifier shows competitive or superior accuracy versus Granger, R1, NIG, Graphical Lasso and the prior feature approach across connectivity regimes and noise correlation levels.
- Theorem 2 gives a sufficient condition: Osc(Off(Sigma_x))/sigma_gap^2 <= A_min^+(1-rho^2)/(2rho(rho^2+1)) guarantees linear separability of centered features.
- Exogenous interventions with sufficiently large variance make the features separable regardless of the input noise covariance (Remark).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the proposed normalized combined features with an FFNN instead of the benchmark methods | Proposed causal inference method · Colored-noise regimes in the simulation experiments | accuracy | Helped |
| Combine lag-moment features F with inverse lag-moment features T (ablation) | Proposed feature set K · Under the sufficient condition in Theorem 2 | identifiability gap | Helped |
| Add standard-scaler normalization to the features (ablation) | Proposed causal inference method · Features under average noise cross-correlation | feature drift | Helped |
| Incorporate novel matrix-valued estimators into the feature vector (ablation) | Proposed features · Simulation experiments across connectivity and noise-correlation regimes | feature drift and oscillation | Helped |
| Use Granger estimation under colored noise and partial observability | Granger estimator · Colored noise with partial observability | performance | Hurt |
| Increase the average absolute off-diagonal noise covariance | Feature-based method in [cite] · Colored-noise regimes | performance | Hurt |
Surprising findings
- Structural information is preserved in partially observed time series irrespective of noise correlation level, as long as cross-correlations are sufficiently homogeneous across pairs.
Limitations
- Separability guarantee holds only under a sufficient condition on noise correlation oscillation; heterogeneous correlations need exogenous interventions.
- Benchmarks are simple baseline supervised methods; training uses a single graph realization and generalization is shown empirically.
Experimental setup
Datasets: synthetic linear NDS time series on Erdos-Renyi and Binomial random graphs human brain connectome (tractography) directed network
Benchmarks: synthetic directed and undirected random graphs (N=50) human brain connectome
Metrics: edge classification accuracy vs number of time series sample
Baselines: feature-based approach (prior work) Granger estimator one-lag covariance estimator R1 NIG estimator R1-R3 Graphical Lasso / precision matrix
Builds on: Granger causality feature-based network identification under partial observability Gaussian mixture graph recovery
Closest papers by meaning
- Recovering the Graph Underlying Networked Dynamical Systems under Partial Observability: A Deep Learning Approach (AAAI 2023 · similarity 0.93)
- CausalDynamics: A large‐scale benchmark for structural discovery of dynamical causal models (NeurIPS 2025 · similarity 0.88)
- Causal Abstraction Learning based on the Semantic Embedding Principle (ICML 2025 · similarity 0.86)
- Disentangling Continuous-Time Latent Dynamics: Identifiability of Latent SDEs via Diffusion Shifts (NeurIPS 2026 · similarity 0.86)
- Learning the Topology and Behavior of Discrete Dynamical Systems (AAAI 2024 · similarity 0.86)
- Coarsening Linear Non-Gaussian Causal Models with Cycles (NeurIPS 2026 · similarity 0.86)
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