# Topological Node2vec: Enhanced Graph Embedding via Persistent Homology

arXiv: https://arxiv.org/abs/2309.08241 · HTML: https://mlsanity.com/paper/2309.08241/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 68 references that are in the map.
- First posted on arXiv 15 Sep 2023.

## Builds on (most cited first, 30 of 35)

- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [node2vec: Scalable Feature Learning for Networks](https://mlsanity.com/paper/1607.00653/) (2016)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Deep Gaussian Embedding of Graphs: Unsupervised Inductive Learning via Ranking](https://mlsanity.com/paper/1707.03815/) (2017)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A Survey on Network Embedding](https://mlsanity.com/paper/1711.08752/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn's Algorithm](https://mlsanity.com/paper/1802.04367/) (2018)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Hyper-SAGNN: a self-attention based graph neural network for hypergraphs](https://mlsanity.com/paper/1911.02613/) (2019)
- [A Topological Regularizer for Classifiers via Persistent Homology](https://mlsanity.com/paper/1806.10714/) (2018)
- [On Unbalanced Optimal Transport: An Analysis of Sinkhorn Algorithm](https://mlsanity.com/paper/2002.03293/) (2020)
- [Sinkhorn Divergences for Unbalanced Optimal Transport](https://mlsanity.com/paper/1910.12958/) (2019)
- [Understanding graph embedding methods and their applications](https://mlsanity.com/paper/2012.08019/) (2020)
- [word2vec, node2vec, graph2vec, X2vec: Towards a Theory of Vector Embeddings of Structured Data](https://mlsanity.com/paper/2003.12590/) (2020)
- [Debiased Sinkhorn barycenters](https://mlsanity.com/paper/2006.02575/) (2020)
- [Large Scale computation of Means and Clusters for Persistence Diagrams using Optimal Transport](https://mlsanity.com/paper/1805.08331/) (2018)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [GPU-Accelerated Computation of Vietoris-Rips Persistence Barcodes](https://mlsanity.com/paper/2003.07989/) (2020)
- [Topology-Aware Surface Reconstruction for Point Clouds](https://mlsanity.com/paper/1811.12543/) (2018)
- [Understanding the Topology and the Geometry of the Space of Persistence Diagrams via Optimal Partial Transport](https://mlsanity.com/paper/1901.03048/) (2019)
- [Topological Optimization with Big Steps](https://mlsanity.com/paper/2203.16748/) (2022)
- [The Effects of Randomness on the Stability of Node Embeddings](https://mlsanity.com/paper/2005.10039/) (2020)
- [Continuation of Point Clouds via Persistence Diagrams](https://mlsanity.com/paper/1506.03147/) (2015)
- [Estimation and Quantization of Expected Persistence Diagrams](https://mlsanity.com/paper/2105.04852/) (2021)

## Built on by (most cited first, 5 of 5)

- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [TopER: Topological Embeddings in Graph Representation Learning](https://mlsanity.com/paper/2410.01778/) (2024)
- [T3former: Temporal Graph Classification with Topological Machine Learning](https://mlsanity.com/paper/2510.13789/) (2025)
- [Estimation of embedding vectors in high dimensions](https://mlsanity.com/paper/2312.07802/) (2023)
- [Scalable Topology-Preserving Graph Coarsening: Concepts and Algorithms](https://mlsanity.com/paper/2601.22943/) (2026)

## Abstract (arXiv)

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.