Jordan, J. and Wade, A.R. (2015) Phase transitions for random geometric preferential attachment graphs. Advances in Applied Probability, 47 (2). pp. 565-588. ISSN 0001-8678
Abstract
We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab\'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab\'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2015 Applied Probability Trust. This is an author produced version of a paper subsequently published in Advances in Applied Probability. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | math.PR; math.PR; 60D05 (Primary) 05C80, 90B15 (Secondary) |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 10 Jul 2015 10:34 |
Last Modified: | 21 Mar 2018 23:54 |
Published Version: | http://dx.doi.org/10.1239/aap/1435236988 |
Status: | Published |
Publisher: | Applied Probability Trust |
Refereed: | Yes |
Identification Number: | 10.1239/aap/1435236988 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:87912 |