Electronic Journal of Statistics

Consistent estimation in general sublinear preferential attachment trees

Fengnan Gao, Aad van der Vaart, Rui Castro, and Remco van der Hofstad

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We propose an empirical estimator of the preferential attachment function $f$ in the setting of general sublinear preferential attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 3979-3999.

Received: June 2017
First available in Project Euclid: 18 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties

Preferential attachment model consistency branching process

Creative Commons Attribution 4.0 International License.


Gao, Fengnan; van der Vaart, Aad; Castro, Rui; van der Hofstad, Remco. Consistent estimation in general sublinear preferential attachment trees. Electron. J. Statist. 11 (2017), no. 2, 3979--3999. doi:10.1214/17-EJS1356. https://projecteuclid.org/euclid.ejs/1508292533

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  • A.-L. Barabási and R. Albert. Emergence of scaling in random networks., science, 286 (5439):509–512, 1999.
  • A.-L. Barabási, R. Albert, and H. Jeong. Mean-field theory for scale-free random networks., Physica A: Statistical Mechanics and its Applications, 272(1):173–187, 1999.
  • A.-L. Barabási, R. Albert, and H. Jeong. Scale-free characteristics of random networks: the topology of the world-wide web., Physica A: Statistical Mechanics and its Applications, 281(1):69–77, 2000.
  • S. Bhamidi. Universal techniques to analyze preferential attachment trees: Global and local analysis., preparation. Version August, 19, 2007.
  • B. Bollobás, O. Riordan, J. Spencer, G. Tusnády, et al. The degree sequence of a scale-free random graph process., Random Structures & Algorithms, 18(3):279–290, 2001.
  • S. Bubeck, E. Mossel, and M. Z. Rácz. On the influence of the seed graph in the preferential attachment model., IEEE Transactions on Network Science and Engineering, 2(1):30–39, Jan 2015. ISSN 2327-4697. 10.1109/TNSE.2015.2397592.
  • V. Chernozhukov, I. Fernández-val, and A. Galichon. Improving point and interval estimators of monotone functions by rearrangement., Biometrika, 96(3):559–575, 2009.
  • F. Gao and A. van der Vaart. On the asymptotic normality of estimating the affine preferential attachment network models with random initial degrees., Stochastic Processes and their Applications, pages –, 2017. ISSN 0304-4149. https://doi.org/10.1016/j.spa.2017.03.008.
  • P. Jagers., Branching processes with biological applications. Wiley, 1975.
  • P. L. Krapivsky and S. Redner. Organization of growing random networks., Physical Review E, 63(6) :066123, 2001.
  • T. Móri. On random trees., Studia Scientiarum Mathematicarum Hungarica, 39(1):143–155, 2002.
  • O. Nerman. On the convergence of supercritical general (C-M-J) branching processes., Probability Theory and Related Fields, 57(3):365–395, 1981.
  • R. Oliveira and J. Spencer. Connectivity transitions in networks with super-linear preferential attachment., Internet Mathematics, 2(2):121–163, 2005.
  • A. Rudas, B. Tóth, and B. Valkó. Random trees and general branching processes., Random Structures & Algorithms, 31(2):186–202, 2007.
  • R. van der Hofstad., Random graphs and complex networks, volume 1. Cambridge University Press, 2017.