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Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

Elchanan Mossel, Allan Sly, Youngtak Sohn

TL;DR

This work establishes exact phase transitions for sparse stochastic block models with q∈{3,4} by linking SBM detection to reconstruction on Galton–Watson trees. The authors develop a coupled SBM–broadcast framework and perform a detailed magnetization analysis, including a high-degree normal approximation, to show KS-bound tightness for q=3 and non-tightness in certain q=4 antiferromagnetic regimes, with explicit d^* thresholds. They validate conjectures from physics and information theory, showing no computational–statistical gap above a universal degree for q=3,4 and identifying regimes where KS is not tight for q≥5. The results hinge on a novel tree–graph coupling and a refined expansion of the magnetization, enabling precise control of higher-order terms and their impact on detectability. Collectively, the paper advances the understanding of exact phase transitions in SBM and reconstruction on trees, with implications for sparse graph inference and related computational-statistical questions.

Abstract

In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborová (2011) regarding the block model, in particular in the case of $q=3$ and $q=4$ communities. We prove that for $q=3$ and $q=4$ there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. The proof is based on showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for $q=3$ and $q=4$ if the average degree is sufficiently large. This improves on the result of Sly (2009), who proved similar results for regular trees for $q=3$. Our analysis of the critical case $q=4$ provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic case depends on the average degree of the tree. We also prove that for $q\geq 5$, the Kestin-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborová (2011), Moore (2017), Abbe and Sandon (2018) and Ricci-Tersenghi, Semerjian, and Zdeborová (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.

Exact Phase Transitions for Stochastic Block Models and Reconstruction on Trees

TL;DR

This work establishes exact phase transitions for sparse stochastic block models with q∈{3,4} by linking SBM detection to reconstruction on Galton–Watson trees. The authors develop a coupled SBM–broadcast framework and perform a detailed magnetization analysis, including a high-degree normal approximation, to show KS-bound tightness for q=3 and non-tightness in certain q=4 antiferromagnetic regimes, with explicit d^* thresholds. They validate conjectures from physics and information theory, showing no computational–statistical gap above a universal degree for q=3,4 and identifying regimes where KS is not tight for q≥5. The results hinge on a novel tree–graph coupling and a refined expansion of the magnetization, enabling precise control of higher-order terms and their impact on detectability. Collectively, the paper advances the understanding of exact phase transitions in SBM and reconstruction on trees, with implications for sparse graph inference and related computational-statistical questions.

Abstract

In this paper we continue to rigorously establish the predictions in ground breaking work in statistical physics by Decelle, Krzakala, Moore, Zdeborová (2011) regarding the block model, in particular in the case of and communities. We prove that for and there is no computational-statistical gap if the average degree is above some constant by showing it is information theoretically impossible to detect below the Kesten-Stigum bound. The proof is based on showing that for the broadcast process on Galton-Watson trees, reconstruction is impossible for and if the average degree is sufficiently large. This improves on the result of Sly (2009), who proved similar results for regular trees for . Our analysis of the critical case provides a detailed picture showing that the tightness of the Kesten-Stigum bound in the antiferromagnetic case depends on the average degree of the tree. We also prove that for , the Kestin-Stigum bound is not sharp. Our results prove conjectures of Decelle, Krzakala, Moore, Zdeborová (2011), Moore (2017), Abbe and Sandon (2018) and Ricci-Tersenghi, Semerjian, and Zdeborová (2019). Our proofs are based on a new general coupling of the tree and graph processes and on a refined analysis of the broadcast process on the tree.
Paper Structure (17 sections, 38 theorems, 300 equations)

This paper contains 17 sections, 38 theorems, 300 equations.

Table of Contents

  1. Introduction
  2. Main Results for the block model
  3. Main results for the broadcast model
  4. Proof overview
  5. Coupling SBM to multiple broadcast processes
  6. Analysis of the broadcast model on Galton-Watson trees
  7. Apriori estimates of the magnetization
  8. Basic identities and the proof of Lemma \ref{['lem:equiv:xn:nonreconstruct']}
  9. Apriori estimates on $u_n, v_n, w_n$
  10. Full expansion of the magnetization
  11. Expansion for $\Xi_2$
  12. Finer estimates of $u_n, w_n$
  13. Proof of Theorem \ref{['thm:four:antiferro']}
  14. Normal approximation for large degrees
  15. Coupling SBM to broadcast processes
  16. ...and 2 more sections

Key Result

Theorem 1.5

For $q\in\{3,4\}$, there exists a universal constant $d_{0}<\infty$ such that if $d\geq d_0$ and $d\lambda^2\leq 1$, i.e. $(a-b)^2\leq q(a+(q-1)b)$, then it is information-theoretically impossible to detect for $G\sim \mathcal{G}(n,q,\frac{a}{n},\frac{b}{n})$. That is, for any estimator $\hat{\sigma

Theorems & Definitions (82)

  • Definition 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Definition 1.8
  • Definition 1.9
  • Theorem 1.12
  • ...and 72 more