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

TL;DR

This paper extends the Kahn-Kalai conjecture to non-uniform measures by introducing 'spread', enabling threshold analysis for subgraph containment and perfect matchings in various inhomogeneous random graph models.

Contribution

It introduces the concept of 'spread' for non-uniform measures, providing a new framework for establishing thresholds in inhomogeneous random graphs and degree sequences.

Findings

Derived conditions for perfect matchings in stochastic block and Chung-Lu models.

Established thresholds for perfect matchings in $ ext{G}(n, extbf{d})$ for broad degree sequences.

Demonstrated that these conditions capture thresholds across many regimes.

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 $\textbf{d}$.