Creating a Giant Component

  • Authors:

  • Tom Bohman;David Kravitz

  • Affiliations:

  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15123, USA (e-mail: tbohman@moser.math.cmu.edu, kravitz@cmu.edu);Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15123, USA (e-mail: tbohman@moser.math.cmu.edu, kravitz@cmu.edu)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2006

Quantified Score

Hi-index 0.00

Visualization

Abstract

Let $c$ be a constant and $(e_1,f_1), (e_2,f_2), \dots, (e_{cn},f_{cn})$ be a sequence of ordered pairs of edges on vertex set $[n]$ chosen uniformly and independently at random. Let $A$ be an algorithm for the on-line choice of one edge from each presented pair, and for $i= 1,\hellip,cn$ let $G_A(i)$ be the graph on vertex set $[n]$ consisting of the first $i$ edges chosen by $A$. We prove that all algorithms in a certain class have a critical value $c_A$ for the emergence of a giant component in $G_A(cn) (ie$, if $c \gt c_A$, then with high probability the largest component in $G_A(cn)$ has $o(n)$ vertices, and if $c c_A$ then with high probability there is a component of size $\Omega(n)$ in $G_A(cn))$. We show that a particular algorithm in this class with high probability produces a giant component before $0.385 n$ steps in the process ($ie$, we exhibit an algorithm that creates a giant component relatively quickly). The fact that another specific algorithm that is in this class has a critical value resolves a conjecture of Spencer.In addition, we establish a lower bound on the time of emergence of a giant component in any process produced by an on-line algorithm and show that there is a phase transition for the off-line version of the problem of creating a giant component.