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Convergence of Achlioptas Processes via Differential Equations with Unique Solutions

Part of: Graph theory Combinatorial probability Equations and systems with randomness

Published online by Cambridge University Press:  14 October 2015

OLIVER RIORDAN  and
LUTZ WARNKE
OLIVER RIORDAN
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK (e-mail: [email protected])
LUTZ WARNKE
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: [email protected])
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Abstract

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. The evolution of the rescaled size of the largest component in such variations of the Erdős–Rényi random graph process has recently received considerable attention, in particular for Bollobás's ‘product rule’. In this paper we establish the following result for rules such as the product rule: the limit of the rescaled size of the ‘giant’ component exists and is continuous provided that a certain system of differential equations has a unique solution. In fact, our result applies to a very large class of Achlioptas-like processes.

Our proof relies on a general idea which relates the evolution of stochastic processes to an associated system of differential equations. Provided that the latter has a unique solution, our approach shows that certain discrete quantities converge (after appropriate rescaling) to this solution.

MSC classification

Primary: 05C80: Random graphs
Secondary: 34F05: Equations and systems with randomness 60C05: Combinatorial probability
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 154 - 171
Copyright
Copyright © Cambridge University Press 2015 

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