Degree asymptotics with rates for preferential attachment random graphs

Annals of Applied Probability

Volumn 23, Issue 3, 2013, Pages 1188-1218

  Peköz, Erol A ^a   Röllin, Adrian ^b   Ross, Nathan ^c  

^a BOSTON UNIVERSITY   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Preferential attachment; Random graphs; Stein's method; Urn models

Indexed keywords

EID: 84879662286     PISSN: 10505164     EISSN: None     Source Type: Journal    
DOI: 10.1214/12-AAP868     Document Type: Article

References (29)

1. ABRAMOWITZ, M. and STEGUN, I. A., eds. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York.
2. 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 Unpublished manuscript.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].
3. BACKHAUSZ, Á. (2011). Limit distribution of degrees in random family trees. Electron. Commun. Probab. 16 29-37. MR2763526
4. BARABÁSI, A.-L. and ALBERT, R. (1999). Emergence of scaling in random networks. Science 286 509-512. MR2091634
5. 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. MR1824277
6. BROWN, M. (2006). Exploiting the waiting time paradox: Applications of the size-biasing transformation. Probab. Engrg. Inform. Sci. 20 195-230. MR2261286
7. BUSTOZ, J. and ISMAIL, M. E. H. (1986). On gamma function inequalities. Math. Comp. 47 659667. MR0856710
8. CHAMAYOU, J.-F. and LETAC, G. (1999). Additive properties of the Dufresne laws and their multivariate extension. J. Theoret. Probab. 12 1045-1066. MR1729469
9. CHEN, L. H. Y., GOLDSTEIN, L. and SHAO, Q.-M. (2011). Normal Approximation by Stein's Method. Springer, Heidelberg. MR2732624
10. DURRETT, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge. MR2271734
11. DURRETT, R. (2010). Probability: Theory and Examples, 4th ed. Cambridge Univ. Press, Cambridge. MR2722836
12. FORD, E. (2009). Barabási-Albertrandom graphs, scale-free distributions and bounds for approximation through Stein's method. Ph. D. thesis, Univ. Oxford.
13. GOLDSTEIN, L. (2007). L bounds in normal approximation. Ann. Probab. 35 1888-1930. MR2349578
14. GOLDSTEIN, L. (2009). Personal communication and unpublished notes. Stein workshop, January 2009, Singapore.
15. 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. MR1484792
16. GORDON, R. D. (1941). Values ofMills' ratio of area to bounding ordinate and of the normalprobability integral for large values of the argument. Ann. Math. Statistics 12 364-366. MR0005558
17. JANSON, S. (2006). Limit theorems for triangular urn schemes. Probab. Theory Related Fields 134 417-452. MR2226887
18. JANSON, S. (2010). Moments of gamma type and the Brownian supremum process area. Probab. Surv. 7 1-52. MR2645216
19. LYONS, R., PEMANTLE, R. and PERES, Y. (1995). Conceptual proofs of LlogL criteria for mean behavior ofbranching processes. Ann. Probab. 23 1125-1138. MR1349164
20. MÓRI, T. F. (2005). The maximum degree of the Barabási- Albert random tree. Combin. Probab. Comput. 14 339-348. MR2138118
21. NEWMAN, M. E. J. (2003). The structure and function of complex networks. SIAMRev. 45 167-256 (electronic). MR2010377
22. PAKES, A. G. and KHATTREE, R. (1992). Length-biasing, characterizations oflaws andthemoment problem. Austral. J. Statist. 34 307-322. MR1193781
23. PEKÖZ, E. A. and RÖLLIN, A. (2011). New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 587-608. MR2789507
24. 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].
25. PITMAN, J. and ROSS, N. (2012). Archimedes, Gauss, and Stein. Notices Amer. Math. Soc. 59 14161421.
26. 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. MR2235451
27. ROSS, N. (2011). Fundamentals of Stein's method. Probab. Surv. 8 210-293. MR2861132
28. ROSS, S. and PEKÖZ, E. (2007). A Second Course in Probability. www.Probability Bookstore.com, Boston, MA.
29. STEIN, C. (1986). Approximate Computation ofExpectations. IMS, Hayward, CA. MR0882007