Phase transitions for modified Erd\"os-R\'enyi processes

TL;DR

This paper investigates phase transitions in modified Erd"os-Rényi processes, including initial graphs and the Bohman-Frieze process, showing they share similar universality properties despite different transition points.

Contribution

It extends the understanding of phase transitions to modified Erd"os-Rényi processes, highlighting their universal behavior in the growth of giant components.

Findings

Phase transitions occur at different points in modified processes.

Growth of the giant component is linear in the supercritical region.

Modified processes belong to the same universality class as classical Erd"os-Rényi.

Abstract

A fundamental and very well studied region of the Erd\"os-R\'enyi process is the phase transition at n/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman-Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.