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

• Published in: Journal of statistical physics Vol. 162; no. 6; pp. 1608 - 1638
• Main Authors: Pachon, Angelica, Polito, Federico, Sacerdote, Laura
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-03-2016; Springer
• Subjects: Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Physics; Statistical Physics and Dynamical Systems; Theoretical; Preferential attachment; 60G55; Discrete and continuous time models; 90B15; Stochastic processes; 05C80; Random graph growth
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-016-1462-7

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
α
) goes to infinity, a portion of them behave as a Yule model with parameters
(
λ
,
β
)
=
(
1
-
α
,
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.