Learning Networks from Wide-Sense Stationary Stochastic Processes

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: $X_t=L*Y_t$ , where $X_t,Y_t\in {ℝ}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p\times p$ Laplacian $L*$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L*$ from temporally correlated samples of $Y_t$ via an ${\ell }_1$ -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 $L*$ with high probability, where $d$ is the maximum degree of the graph underlying $L*$ . We provide recovery guarantees for $L*$ 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).