Electronic Journal of Probability

Preferential attachment with fitness: unfolding the condensate

Steffen Dereich

Full-text: Open access

Abstract

Preferential attachment models with fitness are a substantial extension of the classical preferential attachment model, where vertices have an independent fitness that has a linear impact on its attractiveness in the network formation. As observed by Bianconi and Barabási [4] such network models show different phases. In the condensation phase a small number of exceptionally fit vertices collects a finite fraction of all new links and hence forms a condensate. In this article, we analyse the formation of the condensate for a variant of the model with deterministic normalisation. We consider the regime where the fitness distribution is bounded and has polynomial tail behaviour in its upper end. The central result is a law of large numbers for an appropriately scaled version of the condensate. It follows that a $\Gamma$-distributed shape is formed and, in particular, that the number of vertices contributing to the condensate rises to infinity with increasing network size, in probability.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 3, 38 pp.

Dates
Received: 16 September 2014
Accepted: 29 August 2015
First available in Project Euclid: 3 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1454514663

Digital Object Identifier
doi:10.1214/16-EJP3801

Mathematical Reviews number (MathSciNet)
MR3485345

Zentralblatt MATH identifier
1338.05245

Subjects
Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 90B15: Network models, stochastic

Keywords
Barabási-Albert model preferential attachment fitness condensation

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dereich, Steffen. Preferential attachment with fitness: unfolding the condensate. Electron. J. Probab. 21 (2016), paper no. 3, 38 pp. doi:10.1214/16-EJP3801. https://projecteuclid.org/euclid.ejp/1454514663


Export citation

References

  • [1] Réka Albert and Albert-László Barabási, Statistical mechanics of complex networks, Rev. Modern Phys. 74 (2002), no. 1, 47–97.
  • [2] Albert-László Barabási and Réka Albert, Emergence of scaling in random networks, Science 286 (1999), no. 5439, 509–512.
  • [3] Shankar Bhamidi, Universal techniques to analyze preferential attachment trees: Global and local analysis, Available from http://www.unc.edu/~bhamidi/preferent.pdf, 2007.
  • [4] Ginestra Bianconi and Albert-László Barabási, Bose-Einstein condensation in complex networks, Phys. Rev. Lett. 86 (2001), 5632–35.
  • [5] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987.
  • [6] Béla Bollobás, Oliver Riordan, Joel Spencer, and Gábor Tusnády, The degree sequence of a scale-free random graph process, Random Structures Algorithms 18 (2001), no. 3, 279–290.
  • [7] Christian Borgs, Jennifer Chayes, Constantinos Daskalakis, and Sebastien Roch, First to market is not everything: an analysis of preferential attachment with fitness, STOC'07—Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, 2007, pp. 135–144.
  • [8] Steffen Dereich and Peter Mörters, Emergence of condensation in Kingman's model of selection and mutation, Acta Appl. Math. 127 (2013), 17–26.
  • [9] Steffen Dereich and Peter Mörters, Random networks with sublinear preferential attachment: the giant component, Ann. Probab. 41 (2013), no. 1, 329–384.
  • [10] Steffen Dereich and Peter Mörters, Cycle length distributions in random permutations with diverging cycle weights, Random Structures Algorithms 46 (2015), no. 4, 635–650.
  • [11] Steffen Dereich and Marcel Ortgiese, Robust analysis of preferential attachment models with fitness, Combin. Probab. Comput. 23 (2014), no. 3, 386–411.
  • [12] Sergey N. Dorogovtsev and José F. F. Mendes, Evolution of networks, Oxford University Press, Oxford, 2003, From biological nets to the Internet and WWW.
  • [13] Rick Durrett, Random graph dynamics, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010.
  • [14] Nicholas M. Ercolani and Daniel Ueltschi, Cycle structure of random permutations with cycle weights, Random Structures Algorithms 44 (2014), no. 1, 109–133.
  • [15] Remco van der Hofstad, Random graphs and complex networks, Eindhoven, 2014, Lecture Notes.
  • [16] Svante Janson, Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation, Probab. Surv. 9 (2012), 103–252.
  • [17] John F. C. Kingman, A simple model for the balance between selection and mutation, J. Appl. Probability 15 (1978), no. 1, 1–12.
  • [18] Tamás F. Móri, The maximum degree of the Barabási-Albert random tree, Combin. Probab. Comput. 14 (2005), no. 3, 339–348.
  • [19] Olle Nerman, On the convergence of supercritical general (C-M-J) branching processes, Z. Wahrsch. Verw. Gebiete 57 (1981), no. 3, 365–395.
  • [20] Mark E. J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003), no. 2, 167–256 (electronic).
  • [21] Roberto Oliveira and Joel Spencer, Connectivity transitions in networks with super-linear preferential attachment, Internet Math. 2 (2005), no. 2, 121–163.
  • [22] Su-Chan Park and Joachim Krug, Evolution in random fitness landscapes: the infinite sites model, J. Stat. Mech. (2008), P04014.
  • [23] Anna Rudas, Bálint Tóth, and Benedek Valkó, Random trees and general branching processes, Random Structures Algorithms 31 (2007), no. 2, 186–202.