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Structure of lower tails in sparse random graphs

Byron Chin

TL;DR

The paper addresses the typical structure of sparse Erdős--Rényi graphs conditioned on lower-tail subgraph-count events. It develops a framework that connects the tail probability to a sparse variational problem $\Phi_p(H,\eta)$ via an entropy-increment approach and proves a new stability result for minimizers in the sparse regime. By combining conditioning arguments with a second-moment analysis, it shows that conditioned graphs concentrate around the entropy-minimizing profile $q=\eta^{1/e(H)}p$, yielding precise asymptotics for subgraph counts and a corresponding cut-norm concentration. The results generalize the mean-field-type description from dense graphs to sparse regimes and illuminate a broader hypergraph-container perspective for typical conditioned behavior. This advances understanding of large deviations and typical structure in sparse random graphs and suggests a universal framework for multi-graph conditioning via variational stability.

Abstract

We study the typical structure of a sparse Erdős--Rényi random graph conditioned on the lower tail subgraph count event. We show that in certain regimes, a typical graph sampled from the conditional distribution resembles the entropy minimizer of the mean field approximation in the sense of both subgraph counts and cut norm. The main ingredients are an adaptation of an entropy increment scheme of Kozma and Samotij, and a new stability for the solution of the associated entropy variational problem. The proof can be interpreted as a structural application of the new probabilistic hypergraph container lemma for sparser than average sets, and suggests a more general framework for establishing such typical behavior statements.

Structure of lower tails in sparse random graphs

TL;DR

The paper addresses the typical structure of sparse Erdős--Rényi graphs conditioned on lower-tail subgraph-count events. It develops a framework that connects the tail probability to a sparse variational problem via an entropy-increment approach and proves a new stability result for minimizers in the sparse regime. By combining conditioning arguments with a second-moment analysis, it shows that conditioned graphs concentrate around the entropy-minimizing profile , yielding precise asymptotics for subgraph counts and a corresponding cut-norm concentration. The results generalize the mean-field-type description from dense graphs to sparse regimes and illuminate a broader hypergraph-container perspective for typical conditioned behavior. This advances understanding of large deviations and typical structure in sparse random graphs and suggests a universal framework for multi-graph conditioning via variational stability.

Abstract

We study the typical structure of a sparse Erdős--Rényi random graph conditioned on the lower tail subgraph count event. We show that in certain regimes, a typical graph sampled from the conditional distribution resembles the entropy minimizer of the mean field approximation in the sense of both subgraph counts and cut norm. The main ingredients are an adaptation of an entropy increment scheme of Kozma and Samotij, and a new stability for the solution of the associated entropy variational problem. The proof can be interpreted as a structural application of the new probabilistic hypergraph container lemma for sparser than average sets, and suggests a more general framework for establishing such typical behavior statements.
Paper Structure (9 sections, 11 theorems, 62 equations)

This paper contains 9 sections, 11 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Proof Overview
  4. Organization
  5. Preliminaries
  6. Entropy Increment
  7. Stability of the variational problem
  8. Proof of the main theorem
  9. Discussion: extension to general hypergraphs

Key Result

Theorem 1.1

Fix any graphs $H'$ and $H$. Let $G_\mathcal{L} \sim G(n, p)$ conditioned on $\mathcal{L}_p(H', \eta)$, where $\eta > \eta_{H'}$. Let $q = \eta^{1/e(H')}p$ and $m = \max\{m_2(H), m_2(H')\}$. For any $\varepsilon > 0$ there exists an $L(\varepsilon, \eta, H, H')$ such that if $p \geq Ln^{-1/m}$,

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark
  • Corollary 1.2
  • Definition 1: Graphs
  • Definition 2: Random graphs
  • Definition 3: Lower tail event
  • Definition 4: Cut norm
  • Definition 5: Hypergraph of copies of $H$
  • Definition 6: Vector representation
  • Definition 7: Relative entropy
  • ...and 18 more