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Joint degree distributions of preferential attachment random graphs

Published online by Cambridge University Press:  26 June 2017

Erol Peköz*
Boston University
Adrian Röllin*
National University of Singapore
Nathan Ross*
University of Melbourne
* Postal address: Questrom School of Business, Boston University, 595 Commonwealth Avenue, Room 607, Boston, MA 02215, USA. Email address:
** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore. Email address:
*** Postal address: School of Mathematics and Statistics, University of Melbourne, Richard Berry Building, VIC 3010, Australia. Email address:


We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p and also provide optimal rates of convergence of the finite-dimensional distributions. The results hold for models with any general initial seed graph and any fixed number of initial outgoing edges per vertex; we generate nontree graphs using both a lumping and a sequential rule. Convergence of the order statistics and optimal rates of convergence to the maximum of the degrees is also established.

Research Article
Copyright © Applied Probability Trust 2017 

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[1] Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 128. CrossRefGoogle Scholar
[2] Aldous, D. (1993). The continuum random tree. III. Ann. Prob. 21, 248289. Google Scholar
[3] Antunović, T., Mossel, E. and Rácz, M. Z. (2016). Coexistence in preferential attachment networks. Combinatorics Prob. Comput. 25, 797822. Google Scholar
[4] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509512. Google Scholar
[5] Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 140. Google Scholar
[6] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290. Google Scholar
[7] Bubeck, S., Mossel, E. and Rácz, M. Z. (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2, 3039. Google Scholar
[8] Collevecchio, A., Cotar, C. and LiCalzi, M. (2013). On a preferential attachment and generalized Polya's urn model. Ann. Appl. Prob. 23, 12191253. Google Scholar
[9] Curien, N., Duquesne, T., Kortchemski, I. and Manolescu, I. (2015). Scaling limits and influence of the seed graph in preferential attachment trees. J. Éc. Polytech. Math. 2, 134. Google Scholar
[10] Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edn. John Wiley, New York. Google Scholar
[11] Goldstein, L. and Reinert, G. (2013). Stein's method for the beta distribution and the Pólya–Eggenberger urn. J. Appl. Prob. 50, 11871205. Google Scholar
[12] James, L. F. (2015). Generalized Mittag Leffler distributions arising as limits in preferential attachment models. Preprint. Available at Google Scholar
[13] Janson, S. (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417452. Google Scholar
[14] Krapivsky, P. L., Redner, S. and Leyvraz, F. (2000). Connectivity of growing random networks. Phys. Rev. Lett. 85, 4629. Google Scholar
[15] Lieb, E. H. and Loss, M. (2001). Analysis (Graduate Stud. Math. 14), 2nd edn. American Mathematical Society, Providence, RI. Google Scholar
[16] Móri, T. F. (2005). The maximum degree of the Barabási–Albert random tree. Combinatorics Prob. Comput. 14, 339348. Google Scholar
[17] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167256. Google Scholar
[18] Newman, M., Barabási, A.-L. and Watts, D. J. (2006). The Structure and Dynamics of Networks. Princeton University Press. Google Scholar
[19] Peköz, E. A., Röllin, A. and Ross, N. (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 11881218. Google Scholar
[20] Peköz, E. A., Röllin, A. and Ross, N. (2013). Total variation error bounds for geometric approximation. Bernoulli 19, 610632. CrossRefGoogle Scholar
[21] Peköz, E. A., Röllin, A. and Ross, N. (2016). Generalized gamma approximation with rates for urns, walks and trees. Ann. Prob. 44, 17761816. Google Scholar
[22] Pemantle, R. (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 179. Google Scholar
[23] Pitman, J. (1999). Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Prob. 4, 33pp. Google Scholar
[24] Pitman, J. (2006). Combinatorial Stochastic Processes (Lecture Notes Math. 1875). Springer, Berlin. Google Scholar
[25] Ross, N. (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. Appl. Prob. 45, 876893. CrossRefGoogle Scholar
[26] Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186202. Google Scholar
[27] Suquet, C. (1999). Tightness in Schauder decomposable Banach spaces. In Proc. St. Petersburg Mathematical Society (Amer. Math. Soc. Transl. Ser. 193), Vol. V, American Mathematical Society, Providence, RI, pp. 201224. Google Scholar
[28] Van der Hofstad, R. (2016). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press. Google Scholar