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Degree sequences of geometric preferential attachment graphs

  • Jonathan Jordan (a1)

Abstract

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.

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Copyright

Corresponding author

Postal address: Department of Probability and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK. Email address: jonathan.jordan@shef.ac.uk

References

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[1] Barabási, A.-L., Albert, R. and Jeong, H. (1999). Mean-field theory for scale-free random networks. Physica A 272, 173187.
[2] Barnett, L., Di Paolo, E., and Bullock, S. (2007). Spatially embedded random networks. Phys. Rev. E 76, 056115, 18 pp.
[3] Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 534.
[4] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290.
[5] 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, 4172.
[6] Dorogovtsev, S. N., Mendes, J. F. F. and Samukhin, A. N. (2000). Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 46334636.
[7] Flaxman, A. D., Frieze, A. M. and Vera, J. (2006). A geometric preferential attachment model of networks. Internet Math. 3, 187206.
[8] Flaxman, A. D., Frieze, A. M. and Vera, J. (2007). A geometric preferential attachment model of networks II. Internet Math. 4, 87112.
[9] Jordan, J. (2006). The degree sequences and spectra of scale-free random graphs. Random Structures Algorithms 29, 226242.
[10] Manna, S. S. and Sen, P. (2002). Modulated scale-free network in Euclidean space. Phys. Rev. E 66, 066114.
[11] Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 179.
[12] Penrose, M. (2003). Random Geometric Graphs. Oxford University Press.
[13] Van der Esker, H. (2008). A geometric preferential attachment model with fitness. Preprint. Available at http://arxiv.org/abs/0801.1612.

Keywords

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Degree sequences of geometric preferential attachment graphs

  • Jonathan Jordan (a1)

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