Jordan, J. (2011) Randomised reproducing graphs. Electronic Journal of Probability, 16. pp. 15491562. ISSN 10836489

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Abstract
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random element, and there are three parameters, alpha, beta and gamma, which are the probabilities of edges appearing between different types of vertices. We show that as the probabilities associated with the model vary there are a number of phase transitions, in particular concerning the degree sequence. If (1 + alpha) (1 + gamma) < 1 then the degree distribution converges to a stationary distribution, which in most cases has an approximately power law tail with an index which depends on alpha and gamma. If (1 + alpha) (1 + gamma) > 1 then the degree of a typical vertex grows to infinity, and the proportion of vertices having any fixed degree d tends to zero. We also give some results on the number of edges and on the spectral gap.
Item Type:  Article 

Copyright, Publisher and Additional Information:  © 2011 EJP. This is an author produced version of a paper subsequently published in Electronic Journal of Probability. Uploaded in accordance with the publisher's selfarchiving policy. 
Keywords:  reproducing graphs; random graphs; degree distribution; phase transition 
Institution:  The University of Sheffield 
Academic Units:  The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) 
Depositing User:  Miss Anthea Tucker 
Date Deposited:  21 Oct 2011 13:30 
Last Modified:  08 Feb 2013 17:34 
Published Version:  http://128.208.128.142/~ejpecp/index.php 
Status:  Published 
Publisher:  Institute of Mathematical Statistics 
Refereed:  Yes 
URI:  http://eprints.whiterose.ac.uk/id/eprint/43352 