Shifting the Phase Transition Threshold for Random Graphs Using Degree Set Constraints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


We show that by restricting the degrees of the vertices of a graph to an arbitrary set \( \varDelta \), the threshold point \( \alpha (\varDelta ) \) of the phase transition for a random graph with \( n \) vertices and \( m = \alpha (\varDelta ) n \) edges can be either accelerated (e.g., \( \alpha (\varDelta ) \approx 0.381 \) for \( \varDelta = \{0,1,4,5\} \)) or postponed (e.g., \( \alpha (\{ 2^0, 2^1, \cdots , 2^k, \cdots \}) \approx 0.795 \)) compared to a classical Erdős–Rényi random graph with \( \alpha (\mathbb Z_{\ge 0}) = \tfrac{1}{2} \). In particular, we prove that the probability of graph being nonplanar and the probability of having a complex component, goes from \( 0 \) to \( 1 \) as \( m \) passes \( \alpha (\varDelta ) n \). We investigate these probabilities and also different graph statistics inside the critical window of transition (diameter, longest path and circumference of a complex component).



We would like to thank Fedor Petrov for his help with a proof of technical condition for saddle-point analysis, Élie de Panafieu, Lutz Warnke, and several anonymous referees for their valuable remarks.


  1. 1.
    Bohman, T., Freize, A.: Avoiding a giant component. Random Struct. Algorithms 19(1), 75–85 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bollobás, B.: A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Eur. J. Comb. 1, 311–316 (1980)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    de Panafieu, É., Ramos, L.: Enumeration of graphs with degree constraints. In: Proceedings of the Meeting on Analytic Algorithmics and Combinatorics (2016)Google Scholar
  4. 4.
    Erdős, P., Rényi, A.: On the evolution of random graphs. A Magyar Tudományos Akadémia Matematikai Kutató Intézetének Közleményei 5, 17–61 (1960)MathSciNetMATHGoogle Scholar
  5. 5.
    Flajolet, P., Odlyzko, A.M.: The average height of binary trees and other simple trees. J. Comput. Syst. Sci. 25, 171–213 (1982)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge Press, Cambridge (2009)CrossRefMATHGoogle Scholar
  7. 7.
    Hatami, H., Molloy, M.: The scaling window for a random graph with a given degree sequence. Random Struct. Algorithms 41(1), 99–123 (2012)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Janson, S., Knuth, D.E., Łuczak, T., Pittel, B.: The birth of the giant component. Random Struct. Algorithms 4(3), 231–358 (1993)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Joos, F., Perarnau, G., Rautenbach, D., Reed, B.: How to determine if a random graph with a fixed degree sequence has a giant component. In: 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pp. 695–703 (2016)Google Scholar
  10. 10.
    Liebenau, A., Wormald, N.: Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph. arXiv preprint arXiv:1702.08373 (2017)
  11. 11.
    Molloy, M., Reed, B.A.: A critical point for random graphs with a given degree sequence. Random Struct. Algorithms 6(2/3), 161–180 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nachmias, A., Peres, Y.: Critical random graphs: diameter and mixing time. Ann. Probab. 36(4), 1267–1286 (2008)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Noy, M., Ravelomanana, V., Rué, J.: On the probability of planarity of a random graph near the critical point. Proc. Am. Math. Soc. 143(3), 925–936 (2015)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Riordan, O.: The phase transition in the configuration model. Comb. Probab. Comput. 21(1–2), 265–299 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Riordan, O., Warnke, L.: Achlioptas process phase transitions are continuous. Ann. Appl. Probab. 22(4), 1450–1464 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Riordan, O., Warnke, L.: The phase transition in bounded-size Achlioptas processes. arXiv preprint arXiv:1704.08714 (2017)

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LIPN – UMR CNRS 7030. Université Paris 13VilletaneuseFrance
  2. 2.IRIF – UMR CNRS 8243. Université Paris 7ParisFrance
  3. 3.Moscow Institute of Physics and TechnologyDolgoprudnyRussia

Personalised recommendations