The strong giant in a random digraph

TL;DR

This paper analyzes the emergence of a unique giant strongly connected component in a large random directed graph with i.i.d. outdegrees, showing it exists when the mean outdegree exceeds one and characterizing its size.

Contribution

It establishes the conditions for the existence and size of a giant strongly connected component in directed random graphs with variable outdegree distributions.

Findings

Giant strong component exists if mean outdegree > 1

Size of the giant component is proportional to product of two branching process probabilities

No giant component likely if mean outdegree ≤ 1

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 $n$.