On two biased graph processes

Abstract

In [Amir et al.], the authors consider the generalization $\Gor$ of the Erd\H{o}s-R\'enyi random graph process $G$, where instead of adding new edges uniformly, $\Gor$ gives a weight of size 1 to missing edges between pairs of isolated vertices, and a weight of size $K\in[0,\infty)$ otherwise. This can correspond to the linking of settlements or the spreading of an epidemic. The authors investigate $\tgor(K)$, the critical time for the appearance of a giant component as a function of $K$, and prove that $\tgor=(1+o(1))\frac{4}{\sqrt{3K}}$, using a proper timescale. In this work, we show that a natural variation of the model $\Gor$ has interesting properties. Define the process $\Gand$, where a weight of size $K$ is assigned to edges between pairs of non-isolated vertices, and a weight of size 1 otherwise. We prove that the asymptotical behavior of the giant component threshold is essentially the same for $\Gand$, and namely $\tgand / \tgor$ tends to $\frac{64\sqrt{6}}{\pi(24+\pi^2)}\approx 1.47$ as $K\to\infty$. However, the corresponding thresholds for connectivity satisfy $\tcand / \tcor=\max\{{1/2},K\}$ for every $K>0$. Following the methods of [Amir et al.], $\tgand$ is characterized as the singularity point to a system of differential equations, and computer simulations of both models agree with the analytical results as well as with the asymptotic analysis. In the process, we answer the following question: when does a giant component emerge in a graph process where edges are chosen uniformly out of all edges incident to isolated vertices, while such exist, and otherwise uniformly? This corresponds to the value of $\tgand(0)$, which we show to be ${3/2}+\frac{4}{3\mathrm{e}^2-1}$.