arXiv 1503.070292015-03-24math.PR

随机图G（n，p）上的多数自助渗流

Majority bootstrap percolation on the random graph G(n,p)

Sigurdur Örn Stefánsson, Thomas Vallier

备注：: 27 pages, 1 figure

摘要：

随机图$G_{n,p}$上的多数自举逾渗是在具有给定数量的初始活跃节点的图的给定实现上“激活”的传播过程。在每一步中，具有比不活动邻居更多的活动邻居的那些顶点也变得活动。我们研究了最终活动集的大小$A^*$。该模型的参数除了$n$（趋于$\infty$）之外，还有初始活动集的大小$A(0)=A_0(n)$和图中边的概率$p=p(n)$。我们证明了该过程不能渗透$A(0) = o(n)$。我们研究了$A(0) = θn$和$p$的每个范围的过程，并表明该模型对不同的$p$范围表现出不同的行为。对于非常小的$p \ll \frac{1}{n}$，激活不会显著扩散。对于大的$p \gg \frac{1}{n}$，则我们在$A(0) \simeq \frac{1}{2}n$处看到相变。在$p= \frac{c}{n}$的情况下，激活传播到图的显著更大的部分，但是（该过程不渗透）图的正部分保持不活动。

英文摘要：

Majority bootstrap percolation on the random graph $G_{n,p}$ is a process of spread of "activation" on a given realisation of the graph with a given number of initially active nodes. At each step those vertices which have more active neighbours than inactive neighbours become active as well. We study the size $A^*$ of the final active set. The parameters of the model are, besides $n$ (tending to $\infty$), the size $A(0)=A_0(n)$ of the initially active set and the probability $p=p(n)$ of the edges in the graph. We prove that the process cannot percolate for $A(0) = o(n)$. We study the process for $A(0) = θn$ and every range of $p$ and show that the model exhibits different behaviours for different ranges of $p$. For very small $p \ll \frac{1}{n}$, the activation does not spread significantly. For large $p \gg \frac{1}{n}$ then we see a phase transition at $A(0) \simeq \frac{1}{2}n$. In the case $p= \frac{c}{n}$, the activation propagates to a significantly larger part of the graph but (the process does not percolate) a positive part of the graph remains inactive.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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