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: $X_{t} = {L^{\ast }}Y_{t}$, where $X_{t}, Y_{t} \in \mathbb {R}^{p}$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast }$ 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 $L^{\ast }$ 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 $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast }$. 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).