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

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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.

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Authors and Affiliations

  1. Mathematics Department “G. Peano”, University of Torino, Turin, Italy

    Angelica Pachon, Federico Polito & Laura Sacerdote

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  1. Angelica Pachon
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  2. Federico Polito
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  3. Laura Sacerdote
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Correspondence to Angelica Pachon.

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Pachon, A., Polito, F. & Sacerdote, L. Random Graphs Associated to Some Discrete and Continuous Time Preferential Attachment Models. J Stat Phys 162, 1608–1638 (2016). https://doi.org/10.1007/s10955-016-1462-7

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  • DOI: https://doi.org/10.1007/s10955-016-1462-7

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