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Maximizing the Size of the Giant

Part of: Equilibrium statistical mechanics Markov processes Graph theory

Published online by Cambridge University Press:  30 January 2018

Tom Britton  and
Pieter Trapman
Tom Britton*
Affiliation:
Stockholm University
Pieter Trapman*
Affiliation:
Stockholm University
*
∗ Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
∗ Postal address: Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden.
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Abstract

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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.

Keywords

Random graphbranching processepidemiology

MSC classification

Primary: 05C80: Random graphs 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 92D30: Epidemiology
Secondary: 82B43: Percolation
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 4 , December 2012 , pp. 1156 - 1165
Copyright
© Applied Probability Trust 

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