# The number of descendants in a random directed acyclic graph

## TL;DR

This paper analyzes the distribution of the number of descendants of the last vertex in a random directed acyclic graph, revealing a limit distribution related to Gamma and chi distributions, with detailed moment asymptotics.

## Contribution

It provides the first detailed distributional analysis of descendant counts in a well-known random DAG model, including convergence results and moment asymptotics.

## Key findings

- Normalized descendant count converges in distribution to a scaled Gamma root.
- For d=2, the distribution is a chi distribution with 4 degrees of freedom.
- The paper derives asymptotics for moments of the descendant count.

## Abstract

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.