Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T18:41:32.687Z Has data issue: false hasContentIssue false
  • English
  • Français

Phase Transitions for Random Geometric Preferential Attachment Graphs

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

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.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 

References

Berger, N. et al. (2007). Degree distribution of the FKP network model. Theoret. Comput. Sci. 379, 306–316.CrossRefGoogle Scholar
Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.CrossRefGoogle Scholar
Brockwell, P.J. and Brown, B. M. (1981). High-efficiency estimation for the positive stable laws. J. Amer. Statist. Assoc. 76, 626–631.CrossRefGoogle Scholar
Chalker, T. K. et al. (1999). On the size of a random sphere of influence graph, Adv. Appl. Prob. 31, 596–609.CrossRefGoogle Scholar
Fabrikant, A., Koutsoupias, E. and Papadimitriou, C. H. (2002). Heuristically optimized trade-offs: a new paradigm for power laws in the internet. In Automata, Languages and Programming (Lecture Notes Comput. Sci. 2380), Springer, Berlin, pp. 110–122.CrossRefGoogle Scholar
Ferretti, L. and Cortelezzi, M. (2011). Preferential attachment in growing spatial networks. Phys. Rev. E 84, 016103.CrossRefGoogle ScholarPubMed
Flaxman, A. D., Frieze, A. M. and Vera, J. (2006). A geometric preferential attachment model of networks. Internet Math. 3, 187–205.CrossRefGoogle Scholar
Flaxman, A. D., Frieze, A. M. and Vera, J. (2007). A geometric preferential attachment model of networks. II. Internet Math. 4, 87–111.CrossRefGoogle Scholar
Jacob, E. and Mörters, P. (2015). Spatial preferential attachment networks: power laws and clustering coefficients. Ann. Appl. Prob. 25, 632–662.CrossRefGoogle Scholar
Jordan, J. (2006). The degree sequences and spectra of scale-free random graphs. Random Structures Algorithms 29, 226–242.CrossRefGoogle Scholar
Jordan, J. (2010). Degree sequences of geometric preferential attachment graphs. Adv. Appl. Prob. 42, 319–330.CrossRefGoogle Scholar
Manna, S. S. and Sen, P. (2002). Modulated scale-free network in Euclidean space. Phys. Rev. E 66, 066114.CrossRefGoogle ScholarPubMed
McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics, Springer, Berlin, pp. 195–248.CrossRefGoogle Scholar
Molloy, M. and Reed, B. (2002). Graph Colouring and the Probabilistic Method. Springer, Berlin.CrossRefGoogle Scholar
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surv. 4, 1–79.CrossRefGoogle Scholar
Penrose, M. D. (2005). Multivariate spatial central limit theorems with applications to percolation and spatial graphs. Ann. Prob. 33, 1945–1945.CrossRefGoogle Scholar
Penrose, M. D. (2007). Laws of large numbers in stochastic geometry with statistical applications. Bernoulli 13, 1124–1150.CrossRefGoogle Scholar
Penrose, M. D. and Wade, A. R. (2010). Random directed and on-line networks. In New Perspectives in Stochastic Geometry, Oxford University Press, pp. 248–274.Google Scholar
Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.CrossRefGoogle Scholar