# Exact Recovery for a Family of Community-Detection Generative Models

## TL;DR

This paper introduces a new toy model called the planted REM for community detection, analyzes the error probability and recovery thresholds, and provides the first consistency results for 2-WSBM on graphs and hypergraphs with unequal communities.

## Contribution

It proposes the planted REM model, derives asymptotic error probabilities, and establishes the first consistency results for 2-WSBM with unequal-sized communities.

## Key findings

- Asymptotic behavior of error probability for the planted REM
- Recovery thresholds for community detection in the model
- First consistency results for 2-WSBM with non-equal communities

## Abstract

Generative models for networks with communities have been studied extensively for being a fertile ground to establish information-theoretic and computational thresholds. In this paper we propose a new toy model for planted generative models called planted Random Energy Model (REM), inspired by Derrida's REM. For this model we provide the asymptotic behaviour of the probability of error for the maximum likelihood estimator and hence the exact recovery threshold. As an application, we further consider the 2 non-equally sized community Weighted Stochastic Block Model (2-WSBM) on $h$-uniform hypergraphs, that is equivalent to the P-REM on both sides of the spectrum, for high and low edge cardinality $h$. We provide upper and lower bounds for the exact recoverability for any $h$, mapping these problems to the aforementioned P-REM. To the best of our knowledge these are the first consistency results for the 2-WSBM on graphs and on hypergraphs with non-equally sized community.