Open Access
2012 Vertices of high degree in the preferential attachment tree
Graham Brightwell, Malwina Luczak
Author Affiliations +
Electron. J. Probab. 17: 1-43 (2012). DOI: 10.1214/EJP.v17-1803


We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number $D_t(\ell)$ of vertices of each degree $\ell$ at each time $t$, focussing particularly on the case where $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ is concentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell \le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.


Download Citation

Graham Brightwell. Malwina Luczak. "Vertices of high degree in the preferential attachment tree." Electron. J. Probab. 17 1 - 43, 2012.


Accepted: 11 February 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1244.05197
MathSciNet: MR2892327
Digital Object Identifier: 10.1214/EJP.v17-1803

Primary: 05C80
Secondary: 60G42 , 60J10

Keywords: concentration of measure , Martingales , preferential attachment , Random graphs , web graphs

Vol.17 • 2012
Back to Top