Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-23T09:59:17.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonstandard regular variation of in-degree and out-degree in the preferential attachment model

Published online by Cambridge University Press:  24 March 2016

Gennady Samorodnitsky ,
Sidney Resnick ,
Don Towsley ,
Richard Davis ,
Amy Willis  and
Phyllis Wan
Gennady Samorodnitsky
Affiliation:
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA. Email address: gs18@cornell.edu
Sidney Resnick*
Affiliation:
School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, USA.
Don Towsley
Affiliation:
Department of Computer Science, University of Massachusetts, Amherst, MA 01003, USA. Email address: towsley@cs.umass.edu
Richard Davis
Affiliation:
Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: rdavis@stat.columbia.edu
Amy Willis
Affiliation:
Department of Statistical Science, Cornell University, Ithaca, NY 14853, USA. Email address: adw96@cornell.edu
Phyllis Wan
Affiliation:
Department of Statistics, Columbia University, New York, NY 10027, USA. Email address: phyllis@stat.columbia.edu
*
*** Email address: sir1@cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

For the directed edge preferential attachment network growth model studied by Bollobás et al. (2003) and Krapivsky and Redner (2001), we prove that the joint distribution of in-degree and out-degree has jointly regularly varying tails. Typically, the marginal tails of the in-degree distribution and the out-degree distribution have different regular variation indices and so the joint regular variation is nonstandard. Only marginal regular variation has been previously established for this distribution in the cases where the marginal tail indices are different.

Keywords

Multivariate heavy tailspreferential attachment modelscale-free networks
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 1 , March 2016 , pp. 146 - 161
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2003). Directed scale-free graphs. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003), ACM, New York, pp. 132–139. Google Scholar
John, F. (1971). Partial Differential Equations (Appl. Math. Sci. 1). Springer, New York. Google Scholar
Krapivsky, P. and Redner, S. (2001). Organization of growing random networks. Phys. Rev. E 63, 066123. Google Scholar
Lindskog, F., Resnick, S. I. and Roy, J. (2014). Regularly varying measures on metric spaces: hidden regular variation and hidden jumps. Prob. Surv. 11, 270–314. Google Scholar
Loève, M. (1977). Probability Theory, I, 4th edn. Springer, New York. Google Scholar
Maulik, K., Resnick, S. and Rootzen, H. (2002). Asymptotic independence and a network traffic model. J. Appl. Prob. 39, 671–699. Google Scholar
Resnick, S. I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York. Google Scholar
Resnick, S. and Samorodnitsky, G. (2015). Tauberian theory for multivariate regularly varying distributions with application to preferential attachment networks. Extremes 18, 349–367. Google Scholar
Ross, N. (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. Appl. Prob. 45, 876–893. Google Scholar