Journal of Applied Probability

Maximizing the size of the giant

Tom Britton and Pieter Trapman

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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 Xi with E(Xi)=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


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