Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A central limit theorem for the giant in a stochastic block model

David Clancy

TL;DR

The paper proves a central limit theorem for the size of the giant component in a supercritical stochastic block model (SBM) by adapting a breadth-first exploration framework. It extends the classical Erdős-Rényi CLT to multi-type graphs with edge intensities $\kappa_{i,j}$ under mild regularity on class sizes, and shows the one-type SBM reduces to Stepanov's ERCLT when $d=1$, thereby recovering known results. The fluctuation analysis relies on a degree-corrected SBM encoding (CKL) and a hitting-time representation of the exploration process, combining Donsker-type limits with a Delta-method argument around a nontrivial fixed point. The main result expresses the limiting fluctuations as a Gaussian vector given by a linear transform of a Gaussian driving term, with an explicitly invertible Jacobian $J$, enabling a closed-form covariance description and connecting to prior work on giant fluctuations in multi-type models. This provides a concise, tractable CLT for SBM giants and clarifies the relation between SBM fluctuations, ER fluctuations, and related functional limit theorems.

Abstract

We provide a simple proof for of the central limit theorem for the number of vertices in the giant for super-critical stochastic block model using the breadth-first walk of Konarovskyi, Limic and the author (2024). Our approach follows the recent work of Corujo, Limic and Lemaire (2024) and reduces to the classic central limit theorem for the Erdős-Rényi model obtained by Stepanov (1970).

A central limit theorem for the giant in a stochastic block model

TL;DR

The paper proves a central limit theorem for the size of the giant component in a supercritical stochastic block model (SBM) by adapting a breadth-first exploration framework. It extends the classical Erdős-Rényi CLT to multi-type graphs with edge intensities under mild regularity on class sizes, and shows the one-type SBM reduces to Stepanov's ERCLT when , thereby recovering known results. The fluctuation analysis relies on a degree-corrected SBM encoding (CKL) and a hitting-time representation of the exploration process, combining Donsker-type limits with a Delta-method argument around a nontrivial fixed point. The main result expresses the limiting fluctuations as a Gaussian vector given by a linear transform of a Gaussian driving term, with an explicitly invertible Jacobian , enabling a closed-form covariance description and connecting to prior work on giant fluctuations in multi-type models. This provides a concise, tractable CLT for SBM giants and clarifies the relation between SBM fluctuations, ER fluctuations, and related functional limit theorems.

Abstract

We provide a simple proof for of the central limit theorem for the number of vertices in the giant for super-critical stochastic block model using the breadth-first walk of Konarovskyi, Limic and the author (2024). Our approach follows the recent work of Corujo, Limic and Lemaire (2024) and reduces to the classic central limit theorem for the Erdős-Rényi model obtained by Stepanov (1970).
Paper Structure (12 sections, 12 theorems, 53 equations)

This paper contains 12 sections, 12 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Discussion
  3. Gaussian fluctuations for the giant
  4. Recovering \ref{['eqn:cltforer']}
  5. Size of the giant
  6. Fluctuations of the giant
  7. Preliminaries
  8. A change of variable
  9. Path behavior of $Z_{i,j}^{(n)}$
  10. Path properties of $\boldsymbol{\varphi}$
  11. Estimate for the long excursion interval
  12. Proof of the CLT

Key Result

Proposition 1.2

Let ${\mathcal{C}}_n(1)$ be largest connected component of ${\operatorname{SBM}}_n(n_1,\dotsm,n_d,P)$. Under Assumption ass:1, where $\boldsymbol{\rho}$ is as in eqn:rhodef. Moreover, $\max_{l\ge 2}\#{\mathcal{C}}_n(l) = O_{\mathbb{P}}(\log(n))$.

Theorems & Definitions (15)

  • Proposition 1.2: Bollobás, Janson and Riordan BJR.07
  • Theorem 1.3
  • Lemma 2.1: Lemmas 5.8, 5.10 in BJR.07
  • Lemma 2.2
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Lemma 3.5
  • proof
  • ...and 5 more