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Non-uniform Kahn-Kalai, spread, variants, and applications

Thinula De Silva, Pu Gao

Abstract

Building on B.Park and Vondrak's recent generalization of the J.Park-Pham Theorem (formerly known as Kahn-Kalai conjecture) to non-uniform probability measures, this paper introduces the notion of "spread" for the non-uniform setting. This provides a framework to establish 1-statements for subgraph containment in inhomogeneous random graphs with or without a set of forced edges. Using this approach, we derived conditions for the emergence of perfect matchings in the Stochastic Block Model and the Chung-Lu model, and verified that these conditions are in general not tight, but they capture thresholds across a broad range of regimes. Finally, we bridge this non-uniform framework with $\mathcal{G}(n,\textbf{d})$, utilizing a coupling argument to demonstrate thresholds for perfect matchings in $\mathcal{G}(n,\textbf{d})$ for a broad range of degree sequences $\textbf{d}$.

Non-uniform Kahn-Kalai, spread, variants, and applications

Abstract

Building on B.Park and Vondrak's recent generalization of the J.Park-Pham Theorem (formerly known as Kahn-Kalai conjecture) to non-uniform probability measures, this paper introduces the notion of "spread" for the non-uniform setting. This provides a framework to establish 1-statements for subgraph containment in inhomogeneous random graphs with or without a set of forced edges. Using this approach, we derived conditions for the emergence of perfect matchings in the Stochastic Block Model and the Chung-Lu model, and verified that these conditions are in general not tight, but they capture thresholds across a broad range of regimes. Finally, we bridge this non-uniform framework with , utilizing a coupling argument to demonstrate thresholds for perfect matchings in for a broad range of degree sequences .
Paper Structure (24 sections, 18 theorems, 59 equations)

This paper contains 24 sections, 18 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Inhomogeneous random graph models
  3. Test case: Perfect matchings
  4. Between non-uniform Kahn-Kalai and
  5. Threshold, integer program, and spread
  6. Non-uniform spread
  7. Towards Non-Uniform Kahn-Kalai
  8. Proofs of Theorem \ref{['Analogue of Observation 7.5']} and Theorem \ref{['Hamilton Spread Permutation Theorem']}
  9. Proof of Theorem \ref{['theorem: Non-Uniform Friedgut']}
  10. Proof of Lemma \ref{['lemma: Analogue of Observation 7.4']}
  11. Proof of Theorem \ref{['theorem: General Stochastic Perfect Matching Improved']}
  12. Perfect matchings in the Chung-Lu model
  13. Proof of Theorem \ref{['theorem: Phase Transition for PM in Bi-Valued Families']}
  14. Case 1:
  15. Case 2:
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\mathcal{F}\neq\emptyset, 2^X$ be an increasing family such that $\mathcal{F}\subseteq 2^X$ and $0<\varepsilon<1$ be fixed. If $\textbf{p}^*\in [0,1]^X$ such that $\varepsilon<\mu_{\textbf{p}^*}(\mathcal{F})<1-\varepsilon$ and $\alpha = \omega(1)$, then

Theorems & Definitions (49)

  • Theorem 1.1
  • Remark
  • Definition 1.2
  • Theorem 1.3
  • Definition
  • Remark
  • Definition
  • Remark
  • Definition
  • Definition 1.4
  • ...and 39 more