We consider a well known model of random directed acyclic graphs of order $n$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree $d\ge2$ and the endpoints of the $d$ edges from it are chosen uniformly at random among previously existing vertices. Our main results concern the number $X$ of vertices that are descendants of $n$. We show that $X/\sqrt n$ converges in distribution; the limit distribution is, up to a constant factor, given by the $d$th root of a Gamma distributed variable. $\Gamma(d/(d-1))$. When $d=2$, the limit distribution can also be described as a chi distribution $\chi(4)$. We also show convergence of moments, and find thus the asymptotics of the mean and higher moments.

翻译：我们考虑一个经典的$n$阶随机有向无环图模型，该模型通过递归添加顶点生成：每个新顶点具有固定的出度$d\ge2$，且其$d$条边所连接的终点均匀随机地选自已有顶点。我们的主要结果关注顶点$n$的后代节点数量$X$。我们证明$X/\sqrt n$依分布收敛；其极限分布（除常数因子外）由Gamma分布变量的$d$次根给出，即$\Gamma(d/(d-1))$。当$d=2$时，该极限分布也可描述为卡方分布$\chi(4)$。我们还证明了矩的收敛性，并由此得到均值及高阶矩的渐近性质。