We introduce a model for the spreading of fake news in a community of size n. There are $j_n = \alpha n - g_n$ active gullible persons who are willing to believe and spread the fake news, the rest do not react to it. We address the question ‘How long does it take for $r = \rho n - h_n$ persons to become spreaders?’ (The perturbation functions $g_n$ and $h_n$ are o(n), and $0\le \rho \le \alpha\le 1$ .) The setup has a straightforward representation as a convolution of geometric random variables with quadratic probabilities. However, asymptotic distributions require delicate analysis that gives a somewhat surprising outcome. Normalized appropriately, the waiting time has three main phases: (a) away from the depletion of active gullible persons, when $0< \rho < \alpha$ , the normalized variable converges in distribution to a Gumbel random variable; (b) near depletion, when $0< \rho = \alpha$ , with $h_n - g_n \to \infty$ , the normalized variable also converges in distribution to a Gumbel random variable, but the centering function gains weight with increasing perturbations; (c) at almost complete depletion, when $r = j -c$ , for integer $c\ge 0$ , the normalized variable converges in distribution to a convolution of two independent generalized Gumbel random variables. The influence of various perturbation functions endows the three main phases with an infinite number of phase transitions at the seam lines.

中文翻译：

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假新闻传播的典范

我们介绍了一个在规模较大的社区中传播假新闻的模型n. 有$j_n = \alpha n - g_n$主动轻信的人愿意相信和传播假新闻，其余的人不会对此做出反应。我们解决了“需要多长时间”的问题$r = \rho n - h_n$人成为传播者？（扰动函数$g_n$和$h_n$是○(n）， 和$0\le \rho \le \alpha\le 1$.) 该设置具有直接表示为几何随机变量与二次概率的卷积。然而，渐近分布需要精细的分析，这会产生一些令人惊讶的结果。适当归一化，等待时间分为三个主要阶段：（a）远离活跃的易受骗者的耗尽，当$0< \rho < \alpha$，归一化变量在分布上收敛到 Gumbel 随机变量；(b) 接近枯竭，当$0< \rho = \alpha$， 和$h_n - g_n \to \infty$，归一化变量在分布上也收敛到 Gumbel 随机变量，但中心化函数随着扰动的增加而增加权重；(c) 在几乎完全耗尽时，当$r = j -c$, 对于整数$c\ge 0$，归一化变量在分布上收敛到一个卷积二独立的广义 Gumbel 随机变量。各种微扰函数的影响使三个主要相在接缝线处具有无限数量的相变。