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Degree sequences of geometric preferential attachment graphs

Part of: Graph theory Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Jonathan Jordan
Jonathan Jordan*
Affiliation:
University of Sheffield
*
∗ Postal address: Department of Probability and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK. Email address: [email protected]
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Abstract

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We investigate the degree sequence of the geometric preferential attachment model of Flaxman, Frieze and Vera (2006), (2007) in the case where the self-loop parameter α is set to 0. We show that, given certain conditions on the attractiveness function F, the degree sequence converges to the same sequence as found for standard preferential attachment in Bollobás et al. (2001). We also apply our method to the extended model introduced in van der Esker (2008) which allows for an initial attractiveness term, proving similar results.

Keywords

Geometric random graphpreferential attachment

MSC classification

Primary: 05C80: Random graphs
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 42 , Issue 2 , June 2010 , pp. 319 - 330
Copyright
Copyright © Applied Probability Trust 2010 

References

Barabási, A.-L., Albert, R. and Jeong, H. (1999). Mean-field theory for scale-free random networks. Physica A 272, 173–187.CrossRefGoogle Scholar
Barnett, L., Di Paolo, E., and Bullock, S. (2007). Spatially embedded random networks. Phys. Rev. E 76, 056115, 18 pp.CrossRefGoogle ScholarPubMed
Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.Google 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.Google Scholar
Deijfen, M., van den Esker, H., van der Hofstad, R. and Hooghiemstra, G. (2009). A preferential attachment model with random initial degrees. Ark. Mat. 47, 41–72.CrossRefGoogle Scholar
Dorogovtsev, S. N., Mendes, J. F. F. and Samukhin, A. N. (2000). Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636.Google Scholar
Flaxman, A. D., Frieze, A. M. and Vera, J. (2006). A geometric preferential attachment model of networks. Internet Math. 3, 187–206.CrossRefGoogle Scholar
Flaxman, A. D., Frieze, A. M. and Vera, J. (2007). A geometric preferential attachment model of networks II. Internet Math. 4, 87–112.Google Scholar
Jordan, J. (2006). The degree sequences and spectra of scale-free random graphs. Random Structures Algorithms 29, 226–242.CrossRefGoogle Scholar
Manna, S. S. and Sen, P. (2002). Modulated scale-free network in Euclidean space. Phys. Rev. E 66, 066114.CrossRefGoogle ScholarPubMed
Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 1–79.Google Scholar
Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.Google Scholar
Van der Esker, H. (2008). A geometric preferential attachment model with fitness. Preprint. Available at http://arxiv.org/abs/0801.1612.Google Scholar