Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-16T20:19:19.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Power Laws in Preferential Attachment Graphs and Stein's Method for the Negative Binomial Distribution

Part of: Graph theory Combinatorial probability Limit theorems

Published online by Cambridge University Press:  04 January 2016

Nathan Ross
Nathan Ross*
Affiliation:
University of California, Berkeley
*
∗ Current address: Department of Mathematics and Statistics, Richard Berry Building, University of Melbourne, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au
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.

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to ∞. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

Keywords

Stein's methodnegative binomial distributionrandom graphpreferential attachmentdistributional transformationpower law

MSC classification

Primary: 05C80: Random graphs 60F05: Central limit and other weak theorems
Secondary: 60C05: Combinatorial probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 3 , September 2013 , pp. 876 - 893
Copyright
© Applied Probability Trust 

References

Adell, J. A. and Jodrá, P. (2006). Exact Kolmogorov and total variation distances between some familiar discrete distributions. J. Inequal. Appl. 2006, 64307, 8 pp.CrossRefGoogle Scholar
Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.CrossRefGoogle ScholarPubMed
Barbour, A. D., Holst, L. and Janson, S. (1992). Poisson Approximation (Oxford Stud. Prob. 2). The Clarendon Press, Oxford University Press, New York.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
Brown, T. C. and Phillips, M. J. (1999). Negative binomial approximation with Stein's method. Methodology Comput. Appl. Prob. 1, 407–421.CrossRefGoogle Scholar
Brown, T. C. and Xia, A. (2001). Stein's method and birth-death processes. Ann. Prob. 29, 1373–1403.CrossRefGoogle Scholar
Buckley, P. G. and Osthus, D. (2004). Popularity based random graph models leading to a scale-free degree sequence. Discrete Math. 282, 53–68.CrossRefGoogle Scholar
Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg.CrossRefGoogle Scholar
Goldstein, L. and Reinert, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Prob. 7, 935–952.CrossRefGoogle Scholar
Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. John Wiley, New York.Google Scholar
Jordan, J. (2006). The degree sequences and spectra of scale-free random graphs. Random Structures Algorithms 29, 226–242.CrossRefGoogle Scholar
Klar, B. (2000). Bounds on tail probabilities of discrete distributions. Prob. Eng. Inf. Sci. 14, 161–171.CrossRefGoogle Scholar
Móri, T. F. (2005). The maximum degree of the Barabási-Albert random tree. Combinatorics Prob. Comput. 14, 339–348.CrossRefGoogle Scholar
Peköz, E. A. (1996). Stein's method for geometric approximation. J. Appl. Prob. 33, 707–713.CrossRefGoogle Scholar
Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Prob. 39, 587–608.CrossRefGoogle Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2013). Total variation error bounds for geometric approximation. Bernoulli 19, 610–632.CrossRefGoogle Scholar
Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 1188–1218.CrossRefGoogle Scholar
Pitman, J. and Ross, N. (2012). Archimedes, Gauss, and Stein. Notices AMS 59, 1416–1421.Google Scholar
Ross, N. (2011). Fundamentals of Stein's method. Prob. Surveys 8, 210–293.CrossRefGoogle Scholar
Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186–202.CrossRefGoogle Scholar
Van Der Hofstad, R. (2012). Random graphs and complex networks. Lecture Notes, Eindhoven University of Technology. Available at http://www.win.tue.nl/∼rhofstad/NotesRGCN.pdf.Google Scholar