When is Nontrivial Estimation Possible for Graphons and Stochastic Block Models?

TL;DR

This paper establishes fundamental lower bounds on the accuracy of estimators for block graphons, showing that nontrivial estimation is impossible under certain conditions, thereby characterizing the limits of graphon estimation.

Contribution

It provides the first comprehensive lower bound on estimation accuracy for block graphons with many blocks, clarifying when nontrivial estimation is feasible.

Findings

Lower bound on estimator error in the delta_2 metric for block graphons.

Nontrivial estimation impossible when number of blocks exceeds a certain threshold.

Characterization of minimax estimation accuracy up to logarithmic factors.

Abstract

Block graphons (also called stochastic block models) are an important and widely-studied class of models for random networks. We provide a lower bound on the accuracy of estimators for block graphons with a large number of blocks. We show that, given only the number $k$ of blocks and an upper bound $\rho$ on the values (connection probabilities) of the graphon, every estimator incurs error at least on the order of $\min(\rho, \sqrt{\rho k^2/n^2})$ in the $\delta_2$ metric with constant probability, in the worst case over graphons. In particular, our bound rules out any nontrivial estimation (that is, with $\delta_2$ error substantially less than $\rho$) when $k\geq n\sqrt{\rho}$. Combined with previous upper and lower bounds, our results characterize, up to logarithmic terms, the minimax accuracy of graphon estimation in the $\delta_2$ metric. A similar lower bound to ours was obtained independently by Klopp, Tsybakov and Verzelen (2016).