Arkiv för Matematik

  • Ark. Mat.
  • Volume 50, Number 2 (2012), 305-329.

Phase transitions for modified Erdős–Rényi processes

Svante Janson and Joel Spencer

Full-text: Open access

Abstract

A fundamental and very well studied region of the Erdős–Rényi process is the phase transition at mn/2 edges in which a giant component suddenly appears. We examine the process beginning with an initial graph. We further examine the Bohman–Frieze process in which edges between isolated vertices are more likely. While the positions of the phase transitions vary, the three processes belong, roughly speaking, to the same universality class. In particular, the growth of the giant component in the barely supercritical region is linear in all cases.

Note

This research was mainly done at Institute Mittag-Leffler, Djursholm, Sweden, during the program Discrete Probability, 2009. We thank other participants, in particular Oliver Riordan, for helpful comments. We thank Will Perkins for the numerical calculations in Remark 3.6.

Article information

Source
Ark. Mat., Volume 50, Number 2 (2012), 305-329.

Dates
Received: 25 May 2010
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907180

Digital Object Identifier
doi:10.1007/s11512-011-0157-1

Mathematical Reviews number (MathSciNet)
MR2961325

Zentralblatt MATH identifier
1255.05175

Rights
2011 © Institut Mittag-Leffler

Citation

Janson, Svante; Spencer, Joel. Phase transitions for modified Erdős–Rényi processes. Ark. Mat. 50 (2012), no. 2, 305--329. doi:10.1007/s11512-011-0157-1. https://projecteuclid.org/euclid.afm/1485907180


Export citation

References

  • Achlioptas, D., D’Souza, R. and Spencer, J., Explosive percolation in random networks, Science 323 (2009), 1453–1455.
  • Alon, N. and Spencer, J., The Probabilistic Method, Wiley, New York, 2008.
  • Bohman, T. and Frieze, A., Avoiding a giant component, Random Structures Algorithms 19 (2001), 75–85.
  • Bohman, T., Frieze, A. and Wormald, N. C., Avoidance of a giant component in half the edge set of a random graph, Random Structures Algorithms 25 (2004), 432–449.
  • Bohman, T. and Kravitz, D., Creating a giant component, Combin. Probab. Comput. 15 (2006), 489–511.
  • Bollobás, B., Random Graphs, Cambridge Univ. Press, Cambridge, 2001.
  • Bollobás, B., Janson, S. and Riordan, O., The phase transition in inhomogeneous random graphs, Random Structures Algorithms 31 (2007), 3–122.
  • Erdős, P. and Rényi, A., On random graphs. I, Publ. Math. Debrecen 6 (1959), 290–297.
  • Erdős, P. and Rényi, A., On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61.
  • Gut, A., Probability : A Graduate Course, Springer, New York, 2005.
  • Janson, S., Probability asymptotics: notes on notation, Preprint 31, Institut Mittag-Leffler, Djursholm, 2009.
  • Janson, S., Susceptibility of random graphs with given vertex degrees, J. Comb. 1 (2010), 357–387.
  • Janson, S. and Luczak, M., Susceptibility in subcritical random graphs, J. Math. Phys. 49 (2008), 125207.
  • Janson, S., Łuczak, T. and Ruciński, A., Random Graphs, Wiley, New York, 2000.
  • Janson, S. and Riordan, O., Susceptibility in inhomogeneous random graphs, Preprint, 2009.
  • Perkins, W., The Bohman–Frieze Process and the Forgetfulness of Balls and Bins, Ph.D. thesis, Courant Institute, New York University, 2011.
  • Spencer, J. and Wormald, N., Birth control for giants, Combinatorica 27 (2007), 587–628.
  • Wormald, N., The differential equation method for random graph processes and greedy algorithms, in Lectures on Approximation and Randomized Algorithms, pp. 73–155, PWN, Warsaw, 1999.