关于巨星的锁定概率
On the Fixation Probability of Superstars
摘要：
莫兰过程模拟了基因突变在种群中的传播。一个具有相对适应度$r$的突变体被引入种群，系统就会进化，要么达到固定（每个个体都是突变体），要么达到灭绝（没有个体是突变体）。在一篇被广泛引用的论文（Nature，2005）中，Lieberman，Hauert和Nowak将该模型推广到图的顶点上的种群。他们描述了一类图（称为“超级明星”），具有参数$k$。超级巨星被设计成随着$k$的增加而具有增加的注视概率。他们指出，固定的概率趋于$1-r^{-k}$的图形变得更大，但我们表明，这种说法是不真实的。具体而言，对于$k=5$，我们表明，真正的固定概率（在极限，随着图形变得更大）是最多$1-1/j(r)$，其中$j(r)=Θ(r^4)$，与声称的结果相反。我们确实相信Lieberman等人的定性主张。- 当$k$增加时，超级明星的注视概率趋向于1-是正确的，而且这可能可以沿着他们的草图得到证明。我们能够运行比他们论文中提出的更大的计算机模拟。然而，在大约40，000个顶点的图上的模拟并不支持他们的说法。也许这些图太小，无法显示极限行为。
英文摘要：
The Moran process models the spread of genetic mutations through a population. A mutant with relative fitness $r$ is introduced into a population and the system evolves, either reaching fixation (in which every individual is a mutant) or extinction (in which none is). In a widely cited paper (Nature, 2005), Lieberman, Hauert and Nowak generalize the model to populations on the vertices of graphs. They describe a class of graphs (called "superstars"), with a parameter $k$. Superstars are designed to have an increasing fixation probability as $k$ increases. They state that the probability of fixation tends to $1-r^{-k}$ as graphs get larger but we show that this claim is untrue as stated. Specifically, for $k=5$, we show that the true fixation probability (in the limit, as graphs get larger) is at most $1-1/j(r)$ where $j(r)=Θ(r^4)$, contrary to the claimed result. We do believe that the qualitative claim of Lieberman et al.\ --- that the fixation probability of superstars tends to 1 as $k$ increases --- is correct, and that it can probably be proved along the lines of their sketch. We were able to run larger computer simulations than the ones presented in their paper. However, simulations on graphs of around 40,000 vertices do not support their claim. Perhaps these graphs are too small to exhibit the limiting behaviour.
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。
