  The couplings in a sparse asymmetric, asynchronous Ising network are
reconstructed using an exact learning algorithm. L$_1$ regularization is used
to remove the spurious weak connections that would otherwise be found by simply
minimizing the minus likelihood of a finite data set. In order to see how L$_1$
regularization works in detail, we perform the calculation in several ways
including (1) by iterative minimization of a cost function equal to minus the
log likelihood of the data plus an L$_1$ penalty term, and (2) an approximate
scheme based on a quadratic expansion of the cost function around its minimum.
In these schemes, we track how connections are pruned as the strength of the
L$_1$ penalty is increased from zero to large values. The performance of the
methods for various coupling strengths is quantified using ROC curves.
