# Optimal graphon estimation in cut distance

## TL;DR

This paper establishes minimax estimation rates for graphons and connection probability matrices in cut distance, revealing that the adjacency matrix alone is already optimally informative for this metric.

## Contribution

It proves that the adjacency matrix achieves optimal minimax rates in cut distance, showing no benefit from more complex estimation procedures.

## Key findings

- Raw adjacency matrix is minimax optimal in cut distance.
- Estimation rates are established for block constant matrices and step function graphons.
- Contrasts with classical distances where more complex methods improve results.

## Abstract

Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block constant matrices and step function graphons. Surprisingly, our results imply that, from the minimax point of view, the raw data, that is, the adjacency matrix of the observed graph, is already optimal and more involved procedures cannot improve the convergence rates for this metric. This phenomenon contrasts with optimal rates of convergence with respect to other classical distances for graphons such as the l 1 or l 2 metrics.