  We define a metric---the network Gromov-Wasserstein distance---on weighted,
directed networks that is sensitive to the presence of outliers. In addition to
proving its theoretical properties, we supply network invariants based on
optimal transport that approximate this distance by means of lower bounds. We
test these methods on a range of simulated network datasets and on a dataset of
real-world global bilateral migration. For our simulations, we define a network
generative model based on the stochastic block model. This may be of
independent interest for benchmarking purposes.
