# On the Equivalence between Positional Node Embeddings and Structural Graph Representations

arXiv: https://arxiv.org/abs/1910.00452 · HTML: https://mlsanity.com/paper/1910.00452/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2019.

## Built on by (most cited first, 7 of 7)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Algorithm and System Co-design for Efficient Subgraph-based Graph Representation Learning](https://mlsanity.com/paper/2202.13538/) (2022)
- [Graph Neural Networks for Graph Drawing](https://mlsanity.com/paper/2109.10061/) (2021)
- [On the Generalization Capability of Temporal Graph Learning Algorithms: Theoretical Insights and a Simpler Method](https://mlsanity.com/paper/2402.16387/) (2024)
- [Graph Structure Refinement with Energy-based Contrastive Learning](https://mlsanity.com/paper/2412.17856/) (2024)
- [Navigating the Kaleidoscope of COVID-19 Misinformation Using Deep Learning](https://mlsanity.com/paper/2110.15703/) (2021)
- [ProjB: An Improved Bilinear Biased ProjE model for Knowledge Graph Completion](https://mlsanity.com/paper/2209.02390/) (2022)

## Abstract (arXiv)

This work provides the first unifying theoretical framework for node (positional) embeddings and structural graph representations, bridging methods like matrix factorization and graph neural networks. Using invariant theory, we show that the relationship between structural representations and node embeddings is analogous to that of a distribution and its samples. We prove that all tasks that can be performed by node embeddings can also be performed by structural representations and vice-versa. We also show that the concept of transductive and inductive learning is unrelated to node embeddings and graph representations, clearing another source of confusion in the literature. Finally, we introduce new practical guidelines to generating and using node embeddings, which fixes significant shortcomings of standard operating procedures used today.