Growing optimal scale-free networks via likelihood

Michael Small, Y. Li, Thomas Stemler, Kevin Judd

    Research output: Contribution to journalArticlepeer-review

    22 Citations (Scopus)
    435 Downloads (Pure)

    Abstract

    © 2015 American Physical Society. Preferential attachment, by which new nodes attach to existing nodes with probability proportional to the existing nodes' degree, has become the standard growth model for scale-free networks, where the asymptotic probability of a node having degree k is proportional to k-γ. However, the motivation for this model is entirely ad hoc. We use exact likelihood arguments and show that the optimal way to build a scale-free network is to attach most new links to nodes of low degree. Curiously, this leads to a scale-free network with a single dominant hub: a starlike structure we call a superstar network. Asymptotically, the optimal strategy is to attach each new node to one of the nodes of degree k with probability proportional to 1N+ζ(γ)(k+1)γ (in a N node network): a stronger bias toward high degree nodes than exhibited by standard preferential attachment. Our algorithm generates optimally scale-free networks (the superstar networks) as well as randomly sampling the space of all scale-free networks with a given degree exponent γ. We generate viable realization with finite N for 1 γ2. We observe an apparently discontinuous transition at γ≈2 between so-called superstar networks and more treelike realizations. Gradually increasing γ further leads to reemergence of a superstar hub. To quantify these structural features, we derive a new analytic expression for the expected degree exponent of a pure preferential attachment process and introduce alternative measures of network entropy. Our approach is generic and can also be applied to an arbitrary degree distribution.
    Original languageEnglish
    Article number042801
    Number of pages11
    JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
    Volume91
    Issue number4
    DOIs
    Publication statusPublished - 7 Apr 2015

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