  In the context of undirected Gaussian graphical models, we introduce three
estimators based on elastic net penalty for the underlying dependence graph.
Our goal is to estimate the sparse precision matrix, from which to retrieve
both the underlying conditional dependence graph and the partial correlation
graph. The first estimator is derived from the direct penalization of the
precision matrix in the likelihood function, while the second from using
conditional penalized regressions to estimate the precision matrix. Finally,
the third estimator relies on a 2-stages procedure that estimates the edge set
first and then the precision matrix elements. Through simulations we
investigate the performances of the proposed methods on a large set of
well-known network structures. Empirical results on simulated data show that
the 2-stages procedure outperforms all other estimators both w.r.t. estimating
the sparsity pattern in the graph and the edges' weights. Finally, using
real-world data on US economic sectors, we estimate dependencies and show the
impact of Covid-19 pandemic on the network strength.
