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Critical point representation of the mutual information in the sparse stochastic block model

Tomas Dominguez, Jean-Christophe Mourrat

TL;DR

The problem of recovering the community structure in the stochastic block model is considered, and a representation of the limit of this quantity as an explicit functional evaluated at a critical point of that functional is presented.

Abstract

We consider the problem of recovering the community structure in the stochastic block model. We aim to describe the mutual information between the observed network and the actual community structure as the number of nodes diverges while the average degree of a given node remains bounded. Our main contribution is a representation of the limit of this quantity, assuming it exists, as an explicit functional evaluated at a critical point of that functional. While we mostly focus on the two-community setting for clarity, we expect our method to be robust to model generalizations. We also present an example involving four communities where we show the invalidity of a plausible candidate variational formula for this limit.

Critical point representation of the mutual information in the sparse stochastic block model

TL;DR

The problem of recovering the community structure in the stochastic block model is considered, and a representation of the limit of this quantity as an explicit functional evaluated at a critical point of that functional is presented.

Abstract

We consider the problem of recovering the community structure in the stochastic block model. We aim to describe the mutual information between the observed network and the actual community structure as the number of nodes diverges while the average degree of a given node remains bounded. Our main contribution is a representation of the limit of this quantity, assuming it exists, as an explicit functional evaluated at a critical point of that functional. While we mostly focus on the two-community setting for clarity, we expect our method to be robust to model generalizations. We also present an example involving four communities where we show the invalidity of a plausible candidate variational formula for this limit.
Paper Structure (13 sections, 31 theorems, 259 equations)

This paper contains 13 sections, 31 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Informal summary of the main results
  3. Precise statement of the main results
  4. Organization of the paper
  5. Regularity properties of the free energy
  6. Uniform Lipschitz continuity of the free energy
  7. Almost everywhere differentiability of the limit free energy
  8. Convergence of the derivative free energy
  9. Multioverlap concentration through perturbation
  10. Cavity computation
  11. Critical point representation
  12. The disassortative and small-time settings
  13. Critical point selection

Key Result

Theorem 1.1

Suppose that the sequence of enriched free energies $(\overline F_N)_{N\geq 1}$ converges pointwise to some limit $f:\mathbb{R}_{\geq 0}\times \mathcal{M}_+\to \mathbb{R}$. For every $(t,\mu)\in \mathbb{R}_{\geq 0}\times \mathcal{M}_+$, the map $\nu\mapsto \Gamma_{t,\mu}(\nu)$ admits a fixed point $

Theorems & Definitions (63)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Proposition 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Proposition \ref{['p.regularity.Lipschitz']}
  • ...and 53 more