Fluid limit and gelation in the frozen Erd\H{o}s-R\'enyi random graph

TL;DR

This paper analyzes the fluid limit and gelation phenomena in a modified Erdős-Rényi graph model that prevents giant component formation by freezing components with a cycle, using differential equations to describe the process.

Contribution

It introduces a fluid limit analysis of the frozen Erdős-Rényi process and characterizes the asymptotic freezing time using differential equations.

Findings

Convergence of key statistics to differential equation solutions

Precise asymptotic description of the freezing time

Extension of Wormald's method to this process

Abstract

The frozen Erd\H{o}s-R\'enyi random graph is a variant of the standard dynamical Erd\H{o}s-R\'enyi random graph that prevents the creation of the giant component by freezing the evolution of connected components with a unique cycle. The formation of multicyclic components is forbidden, and the growth of components with a unique cycle is slowed down, depending on a parameter $p\in [0,1]$ that quantifies the slowdown. At the time when all connected components of the graph have a (necessary unique) cycle, the graph is entirely frozen and the process stops. In this paper we study the fluid limit of the main statistics of this process, that is their functional convergence as the number of vertices of the graph becomes large and after a proper rescaling, to the solution of a system of differential equations. Our proofs are based on an adaption of Wormald's differential equation method. We also obtain, as a main application, a precise description of the asymptotic behavior of the first time when the graph is entirely frozen.