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December 2012 Maximizing the size of the giant
Tom Britton, Pieter Trapman
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J. Appl. Probab. 49(4): 1156-1165 (December 2012). DOI: 10.1239/jap/1354716664

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.

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Tom Britton. Pieter Trapman. "Maximizing the size of the giant." J. Appl. Probab. 49 (4) 1156 - 1165, December 2012. https://doi.org/10.1239/jap/1354716664

Information

Published: December 2012
First available in Project Euclid: 5 December 2012

zbMATH: 1257.05158
MathSciNet: MR3058995
Digital Object Identifier: 10.1239/jap/1354716664

Subjects:
Primary: 05C80 , 60J80 , 92D30
Secondary: 82B43

Keywords: branching process , epidemiology , random graph

Rights: Copyright © 2012 Applied Probability Trust

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Vol.49 • No. 4 • December 2012
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