## Journal of Applied Probability

- J. Appl. Probab.
- Volume 53, Number 1 (2016), 57-70.

### The strong giant in a random digraph

#### Abstract

Consider a random directed graph on *n* vertices with independent and identically distributed outdegrees with distribution *F* having mean μ, and destinations of arcs selected uniformly at random. We show that if μ > 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 μ. If μ ≤ 1 there is very likely to be no giant strong component. We also extend this to allow for *F* varying with *n*.

#### Article information

**Source**

J. Appl. Probab., Volume 53, Number 1 (2016), 57-70.

**Dates**

First available in Project Euclid: 8 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1457470558

**Mathematical Reviews number (MathSciNet)**

MR3471946

**Zentralblatt MATH identifier**

1335.05164

**Subjects**

Primary: 05C80: Random graphs [See also 60B20]

Secondary: 60J85: Applications of branching processes [See also 92Dxx] 92D30: Epidemiology

**Keywords**

Semi-homogeneous random digraph giant component branching process

#### Citation

Penrose, Mathew D. The strong giant in a random digraph. J. Appl. Probab. 53 (2016), no. 1, 57--70. https://projecteuclid.org/euclid.jap/1457470558