  Node2vec is a graph embedding method that learns a vector representation for
each node of a weighted graph while seeking to preserve relative proximity and
global structure. Numerical experiments suggest Node2vec struggles to recreate
the topology of the input graph. To resolve this we introduce a topological
loss term to be added to the training loss of Node2vec which tries to align the
persistence diagram (PD) of the resulting embedding as closely as possible to
that of the input graph. Following results in computational optimal transport,
we carefully adapt entropic regularization to PD metrics, allowing us to
measure the discrepancy between PDs in a differentiable way. Our modified loss
function can then be minimized through gradient descent to reconstruct both the
geometry and the topology of the input graph. We showcase the benefits of this
approach using demonstrative synthetic examples.
