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- W1656503067 abstract "In [Amir et al.], the authors consider the generalization $Gor$ of the Erdős-Renyi 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 $Kin[0,infty)$ 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 $frac{64sqrt{6}}{pi(24+pi^2)}approx 1.47$ as $Ktoinfty$. 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 ${3/2}+frac{4}{3mathrm{e}^2-1}$." @default.
- W1656503067 created "2016-06-24" @default.
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- W1656503067 date "2008-01-01" @default.
- W1656503067 modified "2023-09-27" @default.
- W1656503067 title "On two biased graph processes" @default.
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