# Mutual Information for the Stochastic Block Model by the Adaptive Interpolation Method

## TL;DR

This paper derives an exact formula for the mutual information in the asymmetric two-groups stochastic block model using a novel direct adaptive interpolation method, simplifying previous indirect approaches.

## Contribution

It introduces a self-contained, direct proof for the mutual information of the stochastic block model, avoiding complex mappings to matrix estimation problems.

## Key findings

- Provides a single-letter variational expression for mutual information.
- Simplifies the proof technique using adaptive interpolation.
- Eliminates the need for indirect mappings and multiple existing methods.

## Abstract

We rigorously derive a single-letter variational expression for the mutual information of the asymmetric two-groups stochastic block model in the dense graph regime. Existing proofs in the literature are indirect, as they involve mapping the model to a rank-one matrix estimation problem whose mutual information is then determined by a combination of methods (e.g., interpolation, cavity, algorithmic, spatial coupling). In this contribution we provide a self-contained direct method using only the recently introduced adaptive interpolation method.