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

  • Sergey Dovgal
  • Vlady Ravelomanana
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.


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© 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

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