IDEAS home Printed from
   My bibliography  Save this paper

On a preferential attachment and generalized Pólya's urn model


  • Andrea Collevecchio

    () (Department of Management, Università Ca' Foscari Venezia)

  • Codina Cotar

    () (Lehrstuhl für Mathematische Statistik, Technische Universität München)

  • Marco LiCalzi

    () (Department of Management, Università Ca' Foscari Venezia)


We study a general preferential attachment and Pólya's urn model. At each step a new vertex is introduced, which can be connected to at most one existing vertex. If it is disconnected, it becomes a pioneer vertex. Given that it is not disconnected, it joins an existing pioneer vertex with probability proportional to a function of the degree of that vertex. This function is allowed to be vertex-dependent, and is called the reinforcement function. We prove that there can be at most three phases in this model, depending on the behavior of the reinforcement function. Consider the set whose elements are the vertices with cardinality tending a.s. to infinity. We prove that this set either is empty, or it has exactly one element, or it contains all the pioneer vertices. Moreover, we describe the phase transition in the case where the reinforcement function is the same for all vertices. Our results are general, and in particular we are not assuming monotonicity of the reinforcement function. Finally, consider the regime where exactly one vertex has a degree diverging to infinity. We give a lower bound for the probability that a given vertex ends up being the leading one, i.e. its degree diverges to infinity. Our proofs rely on a generalization of the Rubin construction given for edge-reinforced random walks, and on a Brownian motion embedding.

Suggested Citation

  • Andrea Collevecchio & Codina Cotar & Marco LiCalzi, 2011. "On a preferential attachment and generalized Pólya's urn model," Working Papers 8, Department of Management, Università Ca' Foscari Venezia, revised Oct 2012.
  • Handle: RePEc:vnm:wpdman:8

    Download full text from publisher

    File URL:
    File Function: Revised version, 2012
    Download Restriction: no

    References listed on IDEAS

    1. Hansen, Ben & Pitman, Jim, 2000. "Prediction rules for exchangeable sequences related to species sampling," Statistics & Probability Letters, Elsevier, vol. 46(3), pages 251-256, February.
    Full references (including those not matched with items on IDEAS)

    More about this item


    Preferential attachment; Reinforcement processes; Species sampling sequence; Pólya's urn process;

    JEL classification:

    • D85 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Network Formation
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:vnm:wpdman:8. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Marco LiCalzi). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.