# Learning Networks from Wide-Sense Stationary Stochastic Processes

arXiv: https://arxiv.org/abs/2412.03768 · HTML: https://mlsanity.com/paper/2412.03768/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 33 of its 103 references that are in the map.
- First posted on arXiv 4 Dec 2024.

## Builds on (most cited first, 30 of 33)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Structure Learning in Graphical Modeling](https://mlsanity.com/paper/1606.02359/) (2016)
- [A Unified Framework for Structured Graph Learning via Spectral Constraints](https://mlsanity.com/paper/1904.09792/) (2019)
- [Regularized estimation in sparse high-dimensional time series models](https://mlsanity.com/paper/1311.4175/) (2013)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [Structured Graph Learning Via Laplacian Spectral Constraints](https://mlsanity.com/paper/1909.11594/) (2019)
- [Graphical LASSO Based Model Selection for Time Series](https://mlsanity.com/paper/1410.1184/) (2014)
- [Identifying the Topology of Undirected Networks from Diffused Non-stationary Graph Signals](https://mlsanity.com/paper/1801.03862/) (2018)
- [Estimating Time-varying Brain Connectivity Networks from Functional MRI Time Series](https://mlsanity.com/paper/1310.3863/) (2013)
- [Joint Gaussian Graphical Model Estimation: A Survey](https://mlsanity.com/paper/2110.10281/) (2021)
- [Tensor Decompositions for Identifying Directed Graph Topologies and Tracking Dynamic Networks](https://mlsanity.com/paper/1610.08189/) (2016)
- [Spectral Analysis of High-dimensional Time Series](https://mlsanity.com/paper/1810.11223/) (2018)
- [Frequency Domain Statistical Inference for High-Dimensional Time Series](https://mlsanity.com/paper/2206.02250/) (2022)
- [AR Identification of Latent-variable Graphical Models](https://mlsanity.com/paper/1405.0027/) (2014)
- [Identification of Sparse Reciprocal Graphical Models](https://mlsanity.com/paper/1806.04423/) (2018)
- [Large Spectral Density Matrix Estimation by Thresholding](https://mlsanity.com/paper/1812.00532/) (2018)
- [Graphical Models in Meshed Distribution Grids: Topology estimation, change detection and limitations](https://mlsanity.com/paper/1905.06550/) (2019)
- [Does the ℓ1-norm Learn a Sparse Graph under Laplacian Constrained Graphical Models?](https://mlsanity.com/paper/2006.14925/) (2020)
- [Efficient and passive learning of networked dynamical systems driven by non-white exogenous inputs](https://mlsanity.com/paper/2110.00852/) (2021)
- [Network Topology Inference with Sparsity and Laplacian Constraints](https://mlsanity.com/paper/2309.00960/) (2023)
- [Regularized Estimation of Sparse Spectral Precision Matrices](https://mlsanity.com/paper/2401.11128/) (2024)
- [Learning Distribution Grid Topologies: A Tutorial](https://mlsanity.com/paper/2206.10837/) (2022)
- [On the identification of ARMA graphical models](https://mlsanity.com/paper/2308.14384/) (2023)
- [Modeling of Physical Network Systems](https://mlsanity.com/paper/1510.04984/) (2015)
- [Convex Formulation for Regularized Estimation of Structural Equation Models](https://mlsanity.com/paper/1809.06156/) (2018)
- [Exact Topology Learning in a Network of Cyclostationary Processes](https://mlsanity.com/paper/1903.09210/) (2019)
- [Empirical Bayesian Learning in AR Graphical Models](https://mlsanity.com/paper/1907.03829/) (2019)
- [Fused graphical lasso for brain networks with symmetries](https://mlsanity.com/paper/2005.11785/) (2020)
- [Learning Conserved Networks from Flows](https://mlsanity.com/paper/1905.08716/) (2019)

## Abstract (arXiv)

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: Xt = LastYt, where Xt, Yt ∈ ℝp denote inputs and potentials, respectively, and the sparsity pattern of the p × p Laplacian Last encodes the edge structure. Assuming Xt to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of Last from temporally correlated samples of Yt via an ℓ1-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size p significantly exceeds the number of samples n. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on (n, p, d), we show that the ML estimate recovers the sparsity pattern of L^ast with high probability, where d is the maximum degree of the graph underlying Last. We provide recovery guarantees for L^ast in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).