Local limit of the random degree constrained process

TL;DR

This paper establishes the local weak limit of a degree-constrained random graph process as a multi-type branching process, providing insights into the emergence of the giant component and resolving a problem for large degrees.

Contribution

It identifies the local limit of the degree-constrained process as a multi-type branching process and analyzes the critical time for giant component emergence.

Findings

The local weak limit is a multi-type branching process.

The critical time for giant component emergence is asymptotically characterized.

The results resolve a problem posed by Warnke and Wormald for large degrees.

Abstract

In this paper we show that the random degree constrained process (a time-evolving random graph model with degree constraints) has a local weak limit, provided that the underlying host graphs are high degree almost regular. We, moreover, identify the limit object as a multi-type branching process, by combining coupling arguments with the analysis of a certain recursive tree process. Using a spectral characterization, we also give an asymptotic expansion of the critical time when the giant component emerges in the so-called random $d$-process, resolving a problem of Warnke and Wormald for large $d$.