Growing optimal scale-free networks via likelihood.

Abstract

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 1/N+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≪γ<2 as well as γ>2. 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.