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: 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).