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

Topological Node2vec: Enhanced Graph Embedding via Persistent Homology