  Gaussian graphical models are recently used in economics to obtain networks
of dependence among agents. A widely-used estimator is the Graphical Lasso
(GLASSO), which amounts to a maximum likelihood estimation regularized using
the $L_{1,1}$ matrix norm on the precision matrix $\Omega$. The $L_{1,1}$ norm
is a lasso penalty that controls for sparsity, or the number of zeros in
$\Omega$. We propose a new estimator called Structured Graphical Lasso
(SGLASSO) that uses the $L_{1,2}$ mixed norm. The use of the $L_{1,2}$ penalty
controls for the structure of the sparsity in $\Omega$. We show that when the
network size is fixed, SGLASSO is asymptotically equivalent to an infeasible
GLASSO problem which prioritizes the sparsity-recovery of high-degree nodes.
Monte Carlo simulation shows that SGLASSO outperforms GLASSO in terms of
estimating the overall precision matrix and in terms of estimating the
structure of the graphical model. In an empirical illustration using a classic
firms' investment dataset, we obtain a network of firms' dependence that
exhibits the core-periphery structure, with General Motors, General Electric
and U.S. Steel forming the core group of firms.
