The Bohman-Frieze Process Near Criticality

TL;DR

This paper investigates the phase transition behavior of the Bohman-Frieze process, a modification of Erdős-Rényi, showing it exhibits similar critical phenomena with detailed component size characterizations near the critical point.

Contribution

The paper provides new results on the phase transition of the Bohman-Frieze process, including component structure and size estimates near criticality, using combinatorial and differential equation methods.

Findings

Components are trees or unicyclic before the critical point.

Largest component size is (ps^{-2} \, \log n) near criticality.

Second-largest component is (ps^{-2} \, \log n) after the critical point.

Abstract

The Erd\H{o}s-R\'{e}nyi process begins with an empty graph on n vertices and edges are added randomly one at a time to a graph. A classical result of Erd\H{o}s and R\'{e}nyi states that the Erd\H{o}s-R\'{e}nyi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd\H{o}s and R\'{e}nyi, various random graph models have been introduced and studied. In this paper we study the so-called Bohman-Frieze process, a simple modification of the Erd\H{o}s-R\'{e}nyi process. The Bohman-Frieze process begins with an empty graph on n vertices. At each step two random edges are present and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman-Frieze random graph process. We show that the Bohman-Frieze process has a qualitatively similar phase transition to the Erd\H{o}s-R\'{e}nyi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c-\eps (that is, when the number of edges are (t_c-\eps) n/2) are trees or unicyclic components and that the largest component is of size \Omega(\eps^{-2} \log n). Further, at t_c + \eps, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is \Theta(\eps^{-2} \log n). Each of these results corresponds to an analogous well-known result for the Erd\H{o}s-R\'{e}nyi process. Our methods include combinatorial arguments and a combination of the differential equation method for random processes with singularity analysis of generating functions which satisfy quasi-linear partial differential equations.