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Phase Transitions for Random Geometric Preferential Attachment Graphs

Part of: Operations research and management science Geometric probability and stochastic geometry Graph theory

Published online by Cambridge University Press:  22 February 2016

Jonathan Jordan  and
Andrew R. Wade
Jonathan Jordan*
Affiliation:
University of Sheffield
Andrew R. Wade*
Affiliation:
Durham University
*
∗ Postal address: School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Durham University, South Road, Durham DH1 2LE, UK.
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Abstract

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Vertices arrive sequentially in space and are joined to existing vertices at random according to a preferential rule combining degree and spatial proximity. We investigate phase transitions in the resulting graph as the relative strengths of these two components of the attachment rule are varied.

Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence mimics the standard Barabási-Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence mimics a purely geometric model, the on-line nearest-neighbour graph, for which we prove some extensions of known results. We also show the presence of an intermediate regime, with behaviour distinct from both the on-line nearest-neighbour graph and the Barabási-Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence.

Keywords

Random spatial networkpreferential attachmenton-line nearest-neighbour graphdegree sequence

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs 90B15: Network models, stochastic
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 565 - 588
Copyright
© Applied Probability Trust 

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