Learning Networks from Gaussian Graphical Models and Gaussian Free Fields

TL;DR

This paper introduces a novel estimator for recovering the structure of weighted networks from Gaussian Free Field data, providing theoretical guarantees and sample complexity bounds, especially for large and random networks.

Contribution

The work proposes a new Fourier-based estimator for network structure from GFF measurements, with proven recovery guarantees and parametric rate estimation for fixed networks.

Findings

Achieves parametric rate of estimation for fixed network size.

Network recovery succeeds with high probability for Erdős-Rényi graphs when sample size exceeds a threshold.

Provides concrete bounds on sample complexity for network reconstruction.

Abstract

We investigate the problem of estimating the structure of a weighted network from repeated measurements of a Gaussian Graphical Model (GGM) on the network. In this vein, we consider GGMs whose covariance structures align with the geometry of the weighted network on which they are based. Such GGMs have been of longstanding interest in statistical physics, and are referred to as the Gaussian Free Field (GFF). In recent years, they have attracted considerable interest in the machine learning and theoretical computer science. In this work, we propose a novel estimator for the weighted network (equivalently, its Laplacian) from repeated measurements of a GFF on the network, based on the Fourier analytic properties of the Gaussian distribution. In this pursuit, our approach exploits complex-valued statistics constructed from observed data, that are of interest on their own right. We demonstrate the effectiveness of our estimator with concrete recovery guarantees and bounds on the required sample complexity. In particular, we show that the proposed statistic achieves the parametric rate of estimation for fixed network size. In the setting of networks growing with sample size, our results show that for Erdos-Renyi random graphs $G(d,p)$ above the connectivity threshold, we demonstrate that network recovery takes place with high probability as soon as the sample size $n$ satisfies $n \gg d^4 \log d \cdot p^{-2}$.