Asymptotic size of the Karp-Sipser Core in Configuration Model

TL;DR

This paper analyzes the asymptotic size of the Karp-Sipser core in large random graphs with arbitrary degree distributions, using fixed-point equations and local weak limit analysis.

Contribution

It establishes the convergence of the Karp-Sipser core size to an explicit fixed-point equation for general degree distributions.

Findings

Convergence of core size to a fixed-point equation

Explicit characterization under general degree assumptions

Analysis via local weak limit and Warning Propagation

Abstract

We study the asymptotic size of the Karp-Sipser core in the configuration model with arbitrary degree distributions. The Karp-Sipser core is the induced subgraph obtained by iteratively removing all leaves and their neighbors through the leaf-removal process, and finally discarding any isolated vertices \cite{BCC}. Our main result establishes the convergence of the Karp-Sipser core size to an explicit fixed-point equation under general degree assumptions.The approach is based on analyzing the corresponding local weak limit of the configuration model - a unimodular Galton-Watson tree and tracing the evolution process of all vertex states under leaf-removal dynamics by use of the working mechanism of an enhanced version of Warning Propagation along with Node Labeling Propagation.