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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

Information

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

[1]Achlioptas, D., D'Souza, R. M. and Spencer, J. (2009) Explosive percolation in random networks. Science 323 1453–1455.Google Scholar
[2]Aldous, D. J. (1999) Deterministic and stochastic models for coalescence (aggregation and coagulation): A review of the mean-field theory for probabilists. Bernoulli 5 3–48.CrossRefGoogle Scholar
[3]Bohman, T. and Frieze, A. (2001) Avoiding a giant component. Random Struct. Alg. 19 75–85.Google Scholar
[4]Cho, Y. S., Kahng, B. and Kim, D. (2010) Cluster aggregation model for discontinuous percolation transitions. Phys. Rev. E 81 030103.CrossRefGoogle ScholarPubMed
[5]Cho, Y. S., Kim, J. S., Park, J., Kahng, B. and Kim, D. (2009) Percolation transitions in scale-free networks under the Achlioptas process. Phys. Rev. Lett. 103 135702.CrossRefGoogle ScholarPubMed
[6]d ACosta, R. A., Dorogovtsev, S. N., Goltsev, A. V. and Mendes, J. F. F. (2010) Explosive percolation transition is actually continuous. Phys. Rev. Lett. 105 255701.Google Scholar
[7]Deaconu, M. and Tanré, E. (2000) Smoluchowski's coagulation equation: probabilistic interpretation of solutions for constant, additive and multiplicative kernels. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze-Serie IV 29 549–580.Google Scholar
[8]D'Souza, R. M. and Mitzenmacher, M. (2010) Local cluster aggregation models of explosive percolation. Phys. Rev. Lett. 104 195702.Google Scholar
[9]Dubhashi, D. and Panconesi, A. (2009) Concentration of Measure for the Analysis of Randomized Algorithms, Cambridge University Press.Google Scholar
[10]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[11]Friedman, E. J. and Landsberg, A. S. (2009) Construction and analysis of random networks with explosive percolation. Phys. Rev. Lett. 103 255701.Google Scholar
[12]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience.Google Scholar
[13]Kreer, M. and Penrose, O. (1994) Proof of dynamical scaling in Smoluchowski's coagulation equation with constant kernel. J. Statist. Phys. 75 389–407.Google Scholar
[14]Radicchi, F. and Fortunato, S. (2010) Explosive percolation: A numerical analysis. Phys. Rev. E 81 036110.CrossRefGoogle ScholarPubMed
[15]Riordan, O. and Warnke, L. (2012) Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22 1450–1464.Google Scholar
[16]Riordan, O. and Warnke, L. (2011) Explosive percolation is continuous. Science 333 322–324.Google Scholar
[17]Spencer, J. and Wormald, N. (2007) Birth control for giants. Combinatorica 27 587–628.Google Scholar