Annals of Applied Probability

Degree asymptotics with rates for preferential attachment random graphs

Erol A. Peköz, Adrian Röllin, and Nathan Ross

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We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Stein’s method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.

Article information

Ann. Appl. Probab., Volume 23, Number 3 (2013), 1188-1218.

First available in Project Euclid: 7 March 2013

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Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems 05C08

Random graphs preferential attachment Stein’s method urn models


Peköz, Erol A.; Röllin, Adrian; Ross, Nathan. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1188--1218. doi:10.1214/12-AAP868.

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  • Abramowitz, M. and Stegun, I. A., eds. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
  • Arratia, R. and Goldstein, L. (2010). Size bias, sampling, the waiting time paradox, and infinite divisibility: When is the increment independent? Unpublished manuscript. Available at arXiv:1007.3910 [math.PR].
  • Backhausz, Á. (2011). Limit distribution of degrees in random family trees. Electron. Commun. Probab. 16 29–37.
  • Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • 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, M. (2006). Exploiting the waiting time paradox: Applications of the size-biasing transformation. Probab. Engrg. Inform. Sci. 20 195–230.
  • Bustoz, J. and Ismail, M. E. H. (1986). On gamma function inequalities. Math. Comp. 47 659–667.
  • Chamayou, J.-F. and Letac, G. (1999). Additive properties of the Dufresne laws and their multivariate extension. J. Theoret. Probab. 12 1045–1066.
  • Chen, L. H. Y., Goldstein, L. and Shao, Q.-M. (2011). Normal Approximation by Stein’s Method. Springer, Heidelberg.
  • Durrett, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge.
  • Durrett, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge.
  • Ford, E. (2009). Barabási–Albert random graphs, scale-free distributions and bounds for approximation through Stein’s method. Ph.D. thesis, Univ. Oxford.
  • Goldstein, L. (2007). $L^{1}$ bounds in normal approximation. Ann. Probab. 35 1888–1930.
  • Goldstein, L. (2009). Personal communication and unpublished notes. Stein workshop, January 2009, Singapore.
  • Goldstein, L. and Reinert, G. (1997). Stein’s method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 935–952.
  • Gordon, R. D. (1941). Values of Mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument. Ann. Math. Statistics 12 364–366.
  • Janson, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417–452.
  • Janson, S. (2010). Moments of gamma type and the Brownian supremum process area. Probab. Surv. 7 1–52.
  • Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • Móri, T. F. (2005). The maximum degree of the Barabási–Albert random tree. Combin. Probab. Comput. 14 339–348.
  • Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
  • Pakes, A. G. and Khattree, R. (1992). Length-biasing, characterizations of laws and the moment problem. Austral. J. Statist. 34 307–322.
  • Peköz, E. A. and Röllin, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587–608.
  • Peköz, E., Röllin, A. and Ross, N. (2012). Total variation error bounds for geometric approximation. Bernoulli. To appear. Available at arXiv:1005.2774 [math.PR].
  • Pitman, J. and Ross, N. (2012). Archimedes, Gauss, and Stein. Notices Amer. Math. Soc. 59 1416–1421.
  • Reinert, G. (2005). Three general approaches to Stein’s method. In An Introduction to Stein’s Method. Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 4 183–221. Singapore Univ. Press, Singapore.
  • Ross, N. (2011). Fundamentals of Stein’s method. Probab. Surv. 8 210–293.
  • Ross, S. and Peköz, E. (2007). A Second Course in Probability., Boston, MA.
  • Stein, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA.