arXiv 1111.13392011-11math.PRcond-mat.stat-mechmath.CO

幂律随机图的Bootstrap渗流

Bootstrap percolation in power-law random graphs

Hamed Amini, Nikolaos Fountoulakis

备注：: 23 pages

摘要：

图$G$上的自举逾渗过程是一个逐轮演化的“感染”过程。最初，有一个受感染节点的子集，在随后的每一轮中，至少有$r$感染邻居的每个未感染节点都被感染，并永远保持感染状态。参数$r\geq 2$是固定的。这些过程被用作在个人网络中传播思想或趋势的模型。我们分析的情况下，底层图是一个非均匀的随机图，它表现出幂律度分布，最初有$a(n)$随机感染节点的自举渗流过程。本文的主要焦点是在过程结束时将被感染的顶点的数量。这项工作的主要结果是，如果随机图的度序列遵循幂律指数$β$，其中$2 < β< 3$，那么一个次线性数量的初始感染的顶点是足够的随机图的节点的线性分数，以高概率传播感染。更具体地，我们明确地确定临界函数$a_c(n)$，使得$a_c(n)=o(n)$具有以下性质。假设$n$是基础随机图的顶点数，如果是$a(n) \ll a_c(n)$，则该过程根本不演化，随着$n$的增长具有高概率，而如果是$a(n)\gg a_c(n)$，则存在常数$\eps>0$，使得最终的感染顶点集具有至少$\eps n$的大小的高概率。结果表明，当最大度为$o(n^{1/(β-1)})$时，则$a_c(n)$也依赖于$r$。但当最大度为$Θ(n^{1/(β-1)})$时，则为$a_c (n)=n^{β-2 \over β-1}$。

英文摘要：

A bootstrap percolation process on a graph $G$ is an "infection" process which evolves in rounds. Initially, there is a subset of infected nodes and in each subsequent round each uninfected node which has at least $r$ infected neighbours becomes infected and remains so forever. The parameter $r\geq 2$ is fixed. Such processes have been used as models for the spread of ideas or trends within a network of individuals. We analyse bootstrap percolation process in the case where the underlying graph is an inhomogeneous random graph, which exhibits a power-law degree distribution, and initially there are $a(n)$ randomly infected nodes. The main focus of this paper is the number of vertices that will have been infected by the end of the process. The main result of this work is that if the degree sequence of the random graph follows a power law with exponent $β$, where $2 < β< 3$, then a sublinear number of initially infected vertices is enough to spread the infection over a linear fraction of the nodes of the random graph, with high probability. More specifically, we determine explicitly a critical function $a_c(n)$ such that $a_c(n)=o(n)$ with the following property. Assuming that $n$ is the number of vertices of the underlying random graph, if $a(n) \ll a_c(n)$, then the process does not evolve at all, with high probability as $n$ grows, whereas if $a(n)\gg a_c(n)$, then there is a constant $\eps>0$ such that, with high probability, the final set of infected vertices has size at least $\eps n$. It turns out that when the maximum degree is $o(n^{1/(β-1)})$, then $a_c(n)$ depends also on $r$. But when the maximum degree is $Θ(n^{1/(β-1)})$, then $a_c (n)=n^{β-2 \over β-1}$.

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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