# Node Proximity Is All You Need: Unified Structural and Positional Node and Graph Embedding

arXiv: https://arxiv.org/abs/2102.13582 · HTML: https://mlsanity.com/paper/2102.13582/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 28 references that are in the map.
- First posted on arXiv 26 Feb 2021.

## Builds on (most cited first, 11 of 11)

- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [struc2vec: Learning Node Representations from Structural Identity](https://mlsanity.com/paper/1704.03165/) (2017)
- [Network Embedding as Matrix Factorization: Unifying DeepWalk, LINE, PTE, and node2vec](https://mlsanity.com/paper/1710.02971/) (2017)
- [Graph Embedding Techniques, Applications, and Performance: A Survey](https://mlsanity.com/paper/1705.02801/) (2017)
- [Characteristic Functions on Graphs: Birds of a Feather, from Statistical Descriptors to Parametric Models](https://mlsanity.com/paper/2005.07959/) (2020)
- [Learning Structural Node Embeddings Via Diffusion Wavelets](https://mlsanity.com/paper/1710.10321/) (2017)
- [NetLSD: Hearing the Shape of a Graph](https://mlsanity.com/paper/1805.10712/) (2018)
- [REGAL: Representation Learning-based Graph Alignment](https://mlsanity.com/paper/1802.06257/) (2018)
- [RetGK: Graph Kernels based on Return Probabilities of Random Walks](https://mlsanity.com/paper/1809.02670/) (2018)
- [InfiniteWalk: Deep Network Embeddings as Laplacian Embeddings with a Nonlinearity](https://mlsanity.com/paper/2006.00094/) (2020)
- [DELTACON: A Principled Massive-Graph Similarity Function](https://mlsanity.com/paper/1304.4657/) (2013)

## Built on by (most cited first, 9 of 9)

- [Geodesic Graph Neural Network for Efficient Graph Representation Learning](https://mlsanity.com/paper/2210.02636/) (2022)
- [Causal Lifting and Link Prediction](https://mlsanity.com/paper/2302.01198/) (2023)
- [Augmenting Textual Generation via Topology Aware Retrieval](https://mlsanity.com/paper/2405.17602/) (2024)
- [Generating Human Understandable Explanations for Node Embeddings](https://mlsanity.com/paper/2406.07642/) (2024)
- [Efficient Identity and Position Graph Embedding via Spectral-Based Random Feature Aggregation](https://mlsanity.com/paper/2505.20992/) (2025)
- [Digraphwave: Scalable Extraction of Structural Node Embeddings via Diffusion on Directed Graphs](https://mlsanity.com/paper/2207.10149/) (2022)
- [Fisher Information Embedding for Node and Graph Learning](https://mlsanity.com/paper/2305.07580/) (2023)
- [Associative Learning for Network Embedding](https://mlsanity.com/paper/2208.14376/) (2022)
- [Topology-induced Operators Reveal Complementary Graph Representations without Training](https://mlsanity.com/paper/2609.08152/) (2026)

## Abstract (arXiv)

While most network embedding techniques model the relative positions of nodes in a network, recently there has been significant interest in structural embeddings that model node role equivalences, irrespective of their distances to any specific nodes. We present PhUSION, a proximity-based unified framework for computing structural and positional node embeddings, which leverages well-established methods for calculating node proximity scores. Clarifying a point of contention in the literature, we show which step of PhUSION produces the different kinds of embeddings and what steps can be used by both. Moreover, by aggregating the PhUSION node embeddings, we obtain graph-level features that model information lost by previous graph feature learning and kernel methods. In a comprehensive empirical study with over 10 datasets, 4 tasks, and 35 methods, we systematically reveal successful design choices for node and graph-level machine learning with embeddings.