The strong giant in a random digraph

Abstract

Consider a random directed graph on $n$ vertices with independent identically distributed outdegrees with distribution $F$ having mean $\mu$, and destinations of arcs selected uniformly at random. We show that if $\mu >1$ then for large $n$ there is very likely to be a unique giant strong component with proportionate size given as the product of two branching process survival probabilities, one with offspring distribution $F$ and the other with Poisson offspring distribution with mean $\mu$. If $\mu \leq 1$ there is very likely to be no giant strong component. We also extend this to allow for $F$ varying with $n$.