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

Nathan Ross

#### Abstract

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

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

Dates
First available in Project Euclid: 30 August 2013

https://projecteuclid.org/euclid.aap/1377868543

Digital Object Identifier
doi:10.1239/aap/1377868543

Mathematical Reviews number (MathSciNet)
MR3102476

Zentralblatt MATH identifier
1273.05205

#### Citation

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. https://projecteuclid.org/euclid.aap/1377868543

#### 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.
• 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 http://www.win.tue.nl/$\sim$rhofstad/NotesRGCN.pdf.