Rate-optimal graphon estimation

TL;DR

This paper establishes the fundamental limits of graphon estimation, revealing the optimal convergence rates and demonstrating how they depend on the number of clusters and smoothness, with implications for network analysis.

Contribution

It provides the first minimax optimal rates for graphon estimation, introduces novel techniques for lower bounds, and clarifies the impact of cluster number and smoothness on estimation accuracy.

Findings

Optimal rate under mean squared error: $n^{-1}\log k + k^2/n^2$

For $k \\leq \\sqrt{n \\log n}$, the rate grows logarithmically with $k$

In Hölder class, the rate is $n^{-1}\\log n$ for smoothness $\\alpha \\geq 1$, and $n^{-2\\alpha/(\\alpha+1)}$ for $\\alpha \\in(0,1)$

Abstract

Network analysis is becoming one of the most active research areas in statistics. Significant advances have been made recently on developing theories, methodologies and algorithms for analyzing networks. However, there has been little fundamental study on optimal estimation. In this paper, we establish optimal rate of convergence for graphon estimation. For the stochastic block model with $k$ clusters, we show that the optimal rate under the mean squared error is $n^{-1}\log k+k^2/n^2$. The minimax upper bound improves the existing results in literature through a technique of solving a quadratic equation. When $k\leq\sqrt{n\log n}$, as the number of the cluster $k$ grows, the minimax rate grows slowly with only a logarithmic order $n^{-1}\log k$. A key step to establish the lower bound is to construct a novel subset of the parameter space and then apply Fano's lemma, from which we see a clear distinction of the nonparametric graphon estimation problem from classical nonparametric regression, due to the lack of identifiability of the order of nodes in exchangeable random graph models. As an immediate application, we consider nonparametric graphon estimation in a H\"{o}lder class with smoothness $\alpha$. When the smoothness $\alpha\geq1$, the optimal rate of convergence is $n^{-1}\log n$, independent of $\alpha$, while for $\alpha\in(0,1)$, the rate is $n^{-2\alpha/(\alpha+1)}$, which is, to our surprise, identical to the classical nonparametric rate.