Exceptional times of the critical dynamical Erd\H{o}s-R\'enyi graph

TL;DR

This paper studies a dynamic Erdős-Rényi graph evolving over time, revealing that the largest component during the process is significantly larger than in the static case, with size scaling as n^{2/3} log^{1/3} n.

Contribution

It introduces a time-evolving critical Erdős-Rényi graph model and characterizes the size of the largest component during the evolution, highlighting a new scaling behavior.

Findings

Largest component size during evolution is of order n^{2/3} log^{1/3} n

Contrast with static critical Erdős-Rényi graph where size is n^{2/3}

Dynamic process leads to larger maximum component sizes

Abstract

In this paper we introduce a network model which evolves in time, and study its largest connected component. We consider a process of graphs $(G_t:t\in [0,1])$, where initially we start with a critical Erd\H{o}s-R\'enyi graph ER(n, 1/n), and then evolve forwards in time by resampling each edge independently at rate 1. We show that the size of the largest connected component that appears during the time interval $[0, 1]$ is of order $n^{2/3} log^{1/3} n$ with high probability. This is in contrast to the largest component in the static critical Erd\H{o}s-R\'enyi graph, which is of order $n^{2/3}$.