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 $k^{-\gamma }$. 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 $\frac{1}{N+\zeta \left(\gamma \right){(k+1)}^{\gamma }}$ (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 $\gamma$. We generate viable realization with finite $N$ for $1\ll \gamma <2$ as well as $\gamma >2$. We observe an apparently discontinuous transition at $\gamma \approx 2$ between so-called superstar networks and more treelike realizations. Gradually increasing $\gamma$ 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.

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