Core forging and local limit theorems for the k-core of random graphs

TL;DR

This paper introduces a new probabilistic approach to analyze the k-core of Erdős-Rényi graphs, providing precise local limit theorems for core parameters through a generative model inspired by message passing algorithms.

Contribution

It develops a novel generative model for the k-core, enabling direct analysis of core properties and establishing multivariate local limit theorems for core size and order.

Findings

Established multivariate local limit theorem for k-core parameters

Introduced a generative model inspired by Warning Propagation

Facilitated direct analysis of core and mantle properties

Abstract

We establish a multivariate local limit theorem for the order and size as well as several other parameters of the k-core of the Erdos-Renyi graph. The proof is based on a novel approach to the k-core problem that replaces the meticulous analysis of the peeling process by a generative model of graphs with a core of a given order and size. The generative model, which is inspired by the Warning Propagation message passing algorithm, facilitates the direct study of properties of the core and its connections with the mantle and should therefore be of interest in its own right.