  We use machine learning tools to model the line interaction of failure
cascading in power grid networks. We first collect data sets of simulated
trajectories of possible consecutive line failure following an initial random
failure and considering actual constraints in a model power network until the
system settles at a steady state. We use weighted $l_1$-regularized logistic
regression-based models to find static and dynamic models that capture pairwise
and latent higher-order lines' failure interactions using pairwise statistical
data. The static model captures the failures' interactions near the steady
states of the network, and the dynamic model captures the failure unfolding in
a time series of consecutive network states. We test models over independent
trajectories of failure unfolding in the network to evaluate their failure
predictive power. We observe asymmetric, strongly positive, and negative
interactions between different lines' states in the network. We use the static
interaction model to estimate the distribution of cascade size and identify
groups of lines that tend to fail together, and compare against the data. The
dynamic interaction model successfully predicts the network state for
long-lasting failure propagation trajectories after an initial failure.
