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

Abstract

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).

This work is partially supported by the French project MetACOnc, ANR-15-CE40-0014 and by the French project CNRS-PICS-22479.