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Joint degree distributions of preferential attachment random graphs

  • Erol Peköz (a1), Adrian Röllin (a2) and Nathan Ross (a3)
Abstract
Abstract

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.

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Corresponding author
* Postal address: Questrom School of Business, Boston University, 595 Commonwealth Avenue, Room 607, Boston, MA 02215, USA. Email address: pekoz@bu.edu
** Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546, Singapore. Email address: adrian.roellin@nus.edu.sg
*** Postal address: School of Mathematics and Statistics, University of Melbourne, Richard Berry Building, VIC 3010, Australia. Email address: nathan.ross@unimelb.edu.au
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[1] D. Aldous (1991). The continuum random tree. I. Ann. Prob. 19, 128.

[2] D. Aldous (1993). The continuum random tree. III. Ann. Prob. 21, 248289.

[3] T. Antunović , E. Mossel and M. Z. Rácz (2016). Coexistence in preferential attachment networks. Combinatorics Prob. Comput. 25, 797822.

[5] N. Berger , C. Borgs , J. T. Chayes and A. Saberi (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 140.

[6] B. Bollobás , O. Riordan , J. Spencer and G. Tusnády (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279290.

[7] S. Bubeck , E. Mossel and M. Z. Rácz (2015). On the influence of the seed graph in the preferential attachment model. IEEE Trans. Network Sci. Eng. 2, 3039.

[8] A. Collevecchio , C. Cotar and M. LiCalzi (2013). On a preferential attachment and generalized Polya's urn model. Ann. Appl. Prob. 23, 12191253.

[9] N. Curien , T. Duquesne , I. Kortchemski and I. Manolescu (2015). Scaling limits and influence of the seed graph in preferential attachment trees. J. Éc. Polytech. Math. 2, 134.

[11] L. Goldstein and G. Reinert (2013). Stein's method for the beta distribution and the Pólya–Eggenberger urn. J. Appl. Prob. 50, 11871205.

[13] S. Janson (2006). Limit theorems for triangular urn schemes. Prob. Theory Relat. Fields 134, 417452.

[14] P. L. Krapivsky , S. Redner and F. Leyvraz (2000). Connectivity of growing random networks. Phys. Rev. Lett. 85, 4629.

[16] T. F. Móri (2005). The maximum degree of the Barabási–Albert random tree. Combinatorics Prob. Comput. 14, 339348.

[17] M. E. J. Newman (2003). The structure and function of complex networks. SIAM Rev. 45, 167256.

[19] E. A. Peköz , A. Röllin and N. Ross (2013). Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Prob. 23, 11881218.

[20] E. A. Peköz , A. Röllin and N. Ross (2013). Total variation error bounds for geometric approximation. Bernoulli 19, 610632.

[22] R. Pemantle (2007). A survey of random processes with reinforcement. Prob. Surveys 4, 179.

[23] J. Pitman (1999). Brownian motion, bridge, excursion, and meander characterized by sampling at independent uniform times. Electron. J. Prob. 4, 33pp.

[25] N. Ross (2013). Power laws in preferential attachment graphs and Stein's method for the negative binomial distribution. Adv. Appl. Prob. 45, 876893.

[26] A. Rudas , B. Tóth and B. Valkó (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186202.

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Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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