Two critical periods in the evolution of random planar graphs

TL;DR

This paper investigates the evolution of random planar graphs, identifying two critical phases where the structure of components changes significantly, using combinatorial and analytic methods.

Contribution

It reveals two distinct critical periods in the evolution of random planar graphs, detailing their widths and the structural changes during these phases.

Findings

First critical period at M=n/2+O(n^{2/3}) with formation of the largest complex component.

Second critical period at M=n+O(n^{3/5}) where complex components dominate.

Post second phase, increasing edges mainly affects component density, not size.

Abstract

Let $P(n,M)$ be a graph chosen uniformly at random from the family of all labeled planar graphs with $n$ vertices and $M$ edges. In the paper we study the component structure of $P(n,M)$. Combining counting arguments with analytic techniques, we show that there are two critical periods in the evolution of $P(n,M)$. The first one, of width $\Theta(n^{2/3})$, is analogous to the phase transition observed in the standard random graph models and takes place for $M=n/2+O(n^{2/3})$, when the largest complex component is formed. Then, for $M=n+O(n^{3/5})$, when the complex components cover nearly all vertices, the second critical period of width $n^{3/5}$ occurs. Starting from that moment increasing of $M$ mostly affects the density of the complex components, not its size.