  Learning the network structure underlying data is an important problem in
machine learning. This paper introduces a novel prior to study the inference of
scale-free networks, which are widely used to model social and biological
networks. The prior not only favors a desirable global node degree
distribution, but also takes into consideration the relative strength of all
the possible edges adjacent to the same node and the estimated degree of each
individual node.
  To fulfill this, ranking is incorporated into the prior, which makes the
problem challenging to solve. We employ an ADMM (alternating direction method
of multipliers) framework to solve the Gaussian Graphical model regularized by
this prior. Our experiments on both synthetic and real data show that our prior
not only yields a scale-free network, but also produces many more correctly
predicted edges than the others such as the scale-free inducing prior, the
hub-inducing prior and the $l_1$ norm.
