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Explosive Percolation in Erdős–Rényi-Like Random Graph Processes

Published online by Cambridge University Press:  03 October 2012

KONSTANTINOS PANAGIOTOU ,
RETO SPÖHEL ,
ANGELIKA STEGER  and
HENNING THOMAS
KONSTANTINOS PANAGIOTOU
Affiliation:
University of Munich, Mathematical Institute, Theresienstr. 39, 80333 Munich, Germany (e-mail: kpanagio@math.lmu.de)
RETO SPÖHEL
Affiliation:
Max Planck Institute for Informatics, 66123 Saarbrücken, Germany (e-mail: rspoehel@mpi-inf.mpg.de)
ANGELIKA STEGER
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: steger@inf.ethz.ch, hthomas@inf.ethz.ch)
HENNING THOMAS
Affiliation:
Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland (e-mail: steger@inf.ethz.ch, hthomas@inf.ethz.ch)
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Abstract

The study of the phase transition of random graph processes, and recently in particular Achlioptas processes, has attracted much attention. Achlioptas, D'Souza and Spencer (Science, 2009) gave strong numerical evidence that a variety of edge-selection rules in Achlioptas processes exhibit a discontinuous phase transition. However, Riordan and Warnke (Science, 2011) recently showed that all these processes have a continuous phase transition.

In this work we prove discontinuous phase transitions for three random graph processes: all three start with the empty graph on n vertices and, depending on the process, we connect in every step (i) one vertex chosen randomly from all vertices and one chosen randomly from a restricted set of vertices, (ii) two components chosen randomly from the set of all components, or (iii) a randomly chosen vertex and a randomly chosen component.

Keywords

Primary 03C9003-02
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 22 , Issue 1 , January 2013 , pp. 133 - 145
Copyright
Copyright © Cambridge University Press 2012

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References

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