Low-Rank Graphon Estimation: Theory and Applications to Graphon Games

Abstract

We study low-rank estimation of an unknown sparse graphon from sampled network data under operator-norm loss, motivated by targeted interventions in graphon games. Starting from the observed adjacency matrix, we construct low-rank surrogates by singular value thresholding and, for smooth graphons, by block averaging followed by thresholding. We obtain non-asymptotic bounds on both the operator-norm error and the rank of the resulting estimator for stochastic block model, Hölder, and analytic graphons, and we complement these results with minimax lower bounds showing that the rates are essentially sharp for these classes. Our analysis highlights that low rank is valuable here primarily for computation: while it does not improve the minimax operator-norm rate, it yields operator-norm accurate surrogates with substantially smaller rank. We then apply these estimators to linear-quadratic graphon games and derive non-asymptotic stability bounds showing that the welfare loss incurred by using an estimated graphon is controlled by the operator-norm perturbation. This yields near-optimal guarantees for targeted interventions computed from the estimated graphon, together with substantial computational savings. For zero baseline heterogeneity and under a spectral-gap condition, we also establish matching lower bounds for intervention regret. Numerical experiments illustrate the trade-off between statistical accuracy, retained rank, and runtime.