  We introduce a theoretical framework for performing statistical
tasks---including, but not limited to, averaging and principal component
analysis---on the space of (possibly asymmetric) matrices with arbitrary
entries and sizes. This is carried out under the lens of the Gromov-Wasserstein
(GW) distance, and our methods translate the Riemannian framework of GW
distances developed by Sturm into practical, implementable tools for network
data analysis. Our methods are illustrated on datasets of letter graphs,
asymmetric stochastic blockmodel networks, and planar shapes viewed as metric
spaces. On the theoretical front, we supplement the work of Sturm by producing
additional results on the tangent structure of this "space of spaces", as well
as on the gradient flow of the Fr\'{e}chet functional on this space.
