Learning Networks from Wide-Sense Stationary Stochastic Processes

2024-12-04

arXiv.org

AI Summary

This paper addresses the problem of inferring network edge structure—specifically, recovering the support of a sparse graph Laplacian matrix (L^*)—from nodal potential observations of a wide-sense stationary stochastic process in the high-dimensional regime ((p gg n)). We propose the first convex optimization framework that combines the Whittle likelihood with (ell_1) regularization to model network dynamics governed by steady-state linear conservation laws. Theoretically, we introduce a novel incoherence condition and establish high-probability exact support recovery, along with convergence rate bounds in elementwise (ell_infty), Frobenius, and operator norms. The method is computationally tractable and statistically optimal. Empirical validation on real-world systems—including power grids, water distribution networks, and human brain functional connectivity—demonstrates its effectiveness in topology reconstruction.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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 $p imes p$ 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).

Problem

• Learn edge connectivity from node outputs in networked systems
• Estimate Laplacian sparsity pattern using Whittle's MLE
• Recover graph structure under high-dimensional settings

Innovation

• Uses Whittle's MLE for network learning
• Applies l1-regularization for sparsity
• Ensures strict convexity for unique solution