Advances in Applied Probability

Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution

Nathan Ross

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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.

Article information

Adv. in Appl. Probab. Volume 45, Number 3 (2013), 876-893.

First available in Project Euclid: 30 August 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60F05: Central limit and other weak theorems 60C05: Combinatorial probability

Stein's method negative binomial distribution random graph preferential attachment distributional transformation power law


Ross, Nathan. Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. in Appl. Probab. 45 (2013), no. 3, 876--893. doi:10.1239/aap/1377868543.

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