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EnglishWe give a common description of Simon, Barab\'asi--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 Barab\'asi--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 \cite{Newman2005}. References to traditional and recent applications of the these models are also discussed.
| Titolo: | Random Graphs Associated to some Discrete and Continuous Time Preferential Attachment Models |
| Autori Riconosciuti: | |
| Autori: | Pachon, Angelica; Polito, Federico; Sacerdote, Laura |
| Data di pubblicazione: | 2016 |
| Abstract: | We give a common description of Simon, Barab\'asi--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 Barab\'asi--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 \cite{Newman2005}. References to traditional and recent applications of the these models are also discussed. |
| Volume: | 162 |
| Fascicolo: | 6 |
| Pagina iniziale: | 1608 |
| Pagina finale: | 1638 |
| Digital Object Identifier (DOI): | 10.1007/s10955-016-1462-7 |
| URL: | http://arxiv.org/pdf/1503.06150 |
| Parole Chiave: | Preferential attachment, Random graph growth, Discrete and continuous time models, Stochastic processes |
| Rivista: | JOURNAL OF STATISTICAL PHYSICS |
| Appare nelle tipologie: | 03A-Articolo su Rivista |
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http://hdl.handle.net/2318/1551766
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