有偏图过程中的巨分支
Giant Components in Biased Graph Processes
摘要：
随机图过程$\Gorg[1](n)$是$\Gorg[1](n)$顶点上的一系列图，从无边图开始，并且在每一步中，根据缺失边的均匀分布添加一条边。众所周知，在这样的过程中，（线性尺寸的）巨分量通常出现在$(1+o(1))\frac{n}{2}$边缘之后（称为"双跳“的现象），即，当在每个步骤中使用$n/2$边缘的时间尺度时，在时间$t=1$。我们考虑这个过程的推广，$\Gorg[K](n)$，它给出了一个大小为1的权重，以孤立的顶点对之间的丢失的边缘，否则大小为$K \in [0,\infty)$的权重。这对应于在$n$初始隔离的定居点之间添加链路的情况，其中每个步骤中的新链路的概率根据其两个端点定居点是否仍然隔离而偏置。结合方法的\cite{SpencerWormald}与分析技术，我们描述了一个巨大的组件在这个过程中，$t_c(K)$，作为一组微分方程的解的奇点的典型出现时间。我们继续分析这些微分方程，并获得性质的$\Gorg$，特别是，我们表明，$t_c(K)$严格减少从3/2到0的$K$增加从0到$\infty$，和$t_c(K) = \frac{4}{\sqrt{3K}}(1 + o(1))$。微分方程的数值近似与过程$\Gorg(n)$的计算机模拟和分析结果一致。
英文摘要：
A random graph process, $\Gorg[1](n)$, is a sequence of graphs on $n$ vertices which begins with the edgeless graph, and where at each step a single edge is added according to a uniform distribution on the missing edges. It is well known that in such a process a giant component (of linear size) typically emerges after $(1+o(1))\frac{n}{2}$ edges (a phenomenon known as ``the double jump''), i.e., at time $t=1$ when using a timescale of $n/2$ edges in each step. We consider a generalization of this process, $\Gorg[K](n)$, which 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 corresponds to a case where links are added between $n$ initially isolated settlements, where the probability of a new link in each step is biased according to whether or not its two endpoint settlements are still isolated. Combining methods of \cite{SpencerWormald} with analytical techniques, we describe the typical emerging time of a giant component in this process, $t_c(K)$, as the singularity point of a solution to a set of differential equations. We proceed to analyze these differential equations and obtain properties of $\Gorg$, and in particular, we show that $t_c(K)$ strictly decreases from 3/2 to 0 as $K$ increases from 0 to $\infty$, and that $t_c(K) = \frac{4}{\sqrt{3K}}(1 + o(1))$. Numerical approximations of the differential equations agree both with computer simulations of the process $\Gorg(n)$ and with the analytical results.
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