arXiv math/06080972006-08-03math.COmath.APmath.PR

关于两个有偏图过程

On two biased graph processes

Gideon Amir, Eyal Lubetzky

摘要：

在[Amir等人]中，作者考虑Erdens-Rényi随机图过程$G$的推广$\Gor$，其中$\Gor$不是均匀地添加新的边，而是对孤立顶点对之间的缺失边给出大小为1的权重，否则给出大小为$K\in[0,\infty)$的权重。这可以对应于定居点的连接或流行病的传播。作者研究了$\tgor(K)$，作为$K$的函数的巨组分出现的临界时间，并证明了$\tgor=(1+o(1))\frac{4}{\sqrt{3K}}$，使用适当的时间尺度。在这项工作中，我们表明，自然变化的模型$\Gor$有有趣的属性。定义过程$\Gand$，其中大小为$K$的权重被分配给非孤立顶点对之间的边，否则大小为1的权重。我们证明了巨分量阈值的渐近行为对于$\Gand$基本上是相同的，即$\tgand / \tgor$趋向于$\frac{64\sqrt{6}}{π(24+π^2)}\approx 1.47$为$K\to\infty$。然而，对于每个$K>0$，对应的连通性阈值满足$\tcand / \tcor=\max\{{1/2},K\}$。按照[Amir等人]的方法，$\tgand$的特点是奇点的微分方程组，和计算机模拟的两个模型同意的分析结果，以及与渐近分析。在这个过程中，我们回答了以下问题：当一个巨大的组件出现在一个图形过程中，边缘被均匀地选择出所有的边缘事件孤立的顶点，而这样的存在，否则均匀？这对应于$\tgand(0)$的值，我们证明它是${3/2}+\frac{4}{3\mathrm{e}^2-1}$。

英文摘要：

In [Amir et al.], the authors consider the generalization $\Gor$ of the Erdős-Rényi 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 $\frac{64\sqrt{6}}{π(24+π^2)}\approx 1.47$ 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 ${3/2}+\frac{4}{3\mathrm{e}^2-1}$.

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