# Uniform estimation in stochastic block models is slow

## TL;DR

This paper demonstrates that uniform estimation in stochastic block models is inherently slow, especially when classes are similar, with convergence rates depending on the number of vertices rather than edges.

## Contribution

It provides explicit nonasymptotic minimax bounds for estimation in SBMs, revealing slower uniform rates compared to pointwise estimation, and extends results to smooth graphons.

## Key findings

- Uniform estimation rate scales with vertices, not edges.
- Estimation is harder when classes are similar.
- Lower bounds are local around any SBM.

## Abstract

We explicitly quantify the empirically observed phenomenon that estimation under a stochastic block model (SBM) is hard if the model contains classes that are similar. More precisely, we consider estimation of certain functionals of random graphs generated by a SBM. The SBM may or may not be sparse, and the number of classes may be fixed or grow with the number of vertices. Minimax lower and upper bounds of estimation along specific submodels are derived. The results are nonasymptotic and imply that uniform estimation of a single connectivity parameter is much slower than the expected asymptotic pointwise rate. Specifically, the uniform quadratic rate does not scale as the number of edges, but only as the number of vertices. The lower bounds are local around any possible SBM. An analogous result is derived for functionals of a class of smooth graphons.