  We propose a new method to learn the structure of a Gaussian graphical model
with finite sample false discovery rate control. Our method builds on the
knockoff framework of Barber and Cand\`{e}s for linear models. We extend their
approach to the graphical model setting by using a local (node-based) and a
global (graph-based) step: we construct knockoffs and feature statistics for
each node locally, and then solve a global optimization problem to determine a
threshold for each node. We then estimate the neighborhood of each node, by
comparing its feature statistics to its threshold, resulting in our graph
estimate. Our proposed method is very flexible, in the sense that there is
freedom in the choice of knockoffs, feature statistics, and the way in which
the final graph estimate is obtained. For any given data set, it is not clear a
priori what choices of these hyperparameters are optimal. We therefore use a
sample-splitting-recycling procedure that first uses half of the samples to
select the hyperparameters, and then learns the graph using all samples, in
such a way that the finite sample FDR control still holds. We compare our
method to several competitors in simulations and on a real data set.
