The Bohman-Frieze process near criticality

Mihyun Kang, Will Perkins, Joel Spencer

Research output: Contribution to journalArticle

Abstract

The Erdos-Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erdos and Rényi states that the Erdos-Rényi 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 Erdos and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman-Frieze process, a simple modification of the Erdos-Rényi process. The Bohman-Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, 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 process. We show that it has a qualitatively similar phase transition to the Erdos-Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time tc - ε{lunate} (that is, when the number of edges are (tc - ε{lunate})n/2) are trees or unicyclic components and that the largest component is of size Ω(ε{lunate}-2log n). Further, at tc + ε{lunate}, all components apart from the giant component are trees or unicyclic and the size of the second-largest component is Θ(ε{lunate}-2log n). Each of these results corresponds to an analogous well-known result for the Erdos-Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi-linear partial differential equation.

Original languageEnglish (US)
Pages (from-to)221-250
Number of pages30
JournalRandom Structures and Algorithms
Volume43
Issue number2
DOIs
StatePublished - Sep 2013

Fingerprint

Criticality
Erdös
Phase transitions
Random processes
Giant Component
Partial differential equations
Phase Transition
Differential equations
Graph in graph theory
Combinatorial argument
Singularity Analysis
Moment generating function
Linear partial differential equation
Graph Model
Random process
Random Graphs
Susceptibility
Join
Critical point
Differential equation

Keywords

  • Achlioptas process
  • Differential equation method
  • Phase transition
  • Random graphs

ASJC Scopus subject areas

  • Computer Graphics and Computer-Aided Design
  • Software
  • Mathematics(all)
  • Applied Mathematics

Cite this

The Bohman-Frieze process near criticality. / Kang, Mihyun; Perkins, Will; Spencer, Joel.

In: Random Structures and Algorithms, Vol. 43, No. 2, 09.2013, p. 221-250.

Research output: Contribution to journalArticle

Kang, Mihyun ; Perkins, Will ; Spencer, Joel. / The Bohman-Frieze process near criticality. In: Random Structures and Algorithms. 2013 ; Vol. 43, No. 2. pp. 221-250.
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