## Journal of Applied Probability

- J. Appl. Probab.
- Volume 49, Number 4 (2012), 1156-1165.

### Maximizing the size of the giant

Tom Britton and Pieter Trapman

#### Abstract

Consider a random graph where the mean degree is given and fixed. In
this paper we derive the maximal size of the largest connected
component in the graph. We also study the related question of the
largest possible outbreak size of an epidemic occurring `on' the random
graph (the graph describing the social structure in the community).
More precisely, we look at two different classes of random graphs.
First, the Poissonian random graph in which each node *i* is given
an independent and identically distributed (i.i.d.) random weight
*X*_{i} with E(X_{i})=µ, and
where there is an edge between *i* and *j* with probability
1-e^{-XiXj/(µ n)},
independently of other edges. The second model is the thinned
configuration model in which the *n* vertices of the ground graph
have i.i.d. ground degrees, distributed as *D*, with E(*D*) =
µ. The graph of interest is obtained by deleting edges
independently with probability 1-*p*. In both models the fraction
of vertices in the largest connected component converges in probability
to a constant 1-*q*, where *q* depends on *X* or
*D* and *p*. We investigate for which distributions *X*
and *D* with given µ and *p*, 1-*q* is maximized.
We show that in the class of Poissonian random graphs, *X* should
have all its mass at 0 and one other real, which can be explicitly
determined. For the thinned configuration model, *D* should have
all its mass at 0 and two subsequent positive integers.

#### Article information

**Source**

J. Appl. Probab., Volume 49, Number 4 (2012), 1156-1165.

**Dates**

First available in Project Euclid: 5 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.jap/1354716664

**Digital Object Identifier**

doi:10.1239/jap/1354716664

**Mathematical Reviews number (MathSciNet)**

MR3058995

**Zentralblatt MATH identifier**

1257.05158

**Subjects**

Primary: 05C80: Random graphs [See also 60B20] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Random graph branching process epidemiology

#### Citation

Britton, Tom; Trapman, Pieter. Maximizing the size of the giant. J. Appl. Probab. 49 (2012), no. 4, 1156--1165. doi:10.1239/jap/1354716664. https://projecteuclid.org/euclid.jap/1354716664