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Journal of Statistical Physics

, Volume 162, Issue 6, pp 1608–1638 | Cite as

Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models

  • Angelica Pachon
  • Federico Polito
  • Laura Sacerdote
Article

Abstract

We give a common description of Simon, Barabási–Albert, II-PA and Price growth models, by introducing suitable random graph processes with preferential attachment mechanisms. Through the II-PA model, we prove the conditions for which the asymptotic degree distribution of the Barabási–Albert model coincides with the asymptotic in-degree distribution of the Simon model. Furthermore, we show that when the number of vertices in the Simon model (with parameter \(\alpha \)) goes to infinity, a portion of them behave as a Yule model with parameters \((\lambda ,\beta ) = (1-\alpha ,1)\), and through this relation we explain why asymptotic properties of a random vertex in Simon model, coincide with the asymptotic properties of a random genus in Yule model. As a by-product of our analysis, we prove the explicit expression of the in-degree distribution for the II-PA model, given without proof in Newman (Contemp Phys 46:323-351, 2005). References to traditional and recent applications of the these models are also discussed.

Keywords

Preferential attachment Random graph growth Discrete and continuous time models Stochastic processes 

Mathematics Subject Classification

05C80 60G55 90B15 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Angelica Pachon
    • 1
  • Federico Polito
    • 1
  • Laura Sacerdote
    • 1
  1. 1.Mathematics Department “G. Peano”University of TorinoTurinItaly

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