# Bures-Wasserstein Means of Graphs

arXiv: https://arxiv.org/abs/2305.19738 · HTML: https://mlsanity.com/paper/2305.19738/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 59 references that are in the map.
- First posted on arXiv 31 May 2023.

## Builds on (most cited first, 23 of 23)

- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Optimal Transport for structured data with application on graphs](https://mlsanity.com/paper/1805.09114/) (2018)
- [How to learn a graph from smooth signals](https://mlsanity.com/paper/1601.02513/) (2016)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Connecting the Dots: Identifying Network Structure via Graph Signal Processing](https://mlsanity.com/paper/1810.13066/) (2018)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Graph signal processing for machine learning: A review and new perspectives](https://mlsanity.com/paper/2007.16061/) (2020)
- [GOT: An Optimal Transport framework for Graph comparison](https://mlsanity.com/paper/1906.02085/) (2019)
- [Optimal transport for Gaussian mixture models](https://mlsanity.com/paper/1710.07876/) (2017)
- [Metrics for Graph Comparison: A Practitioner's Guide](https://mlsanity.com/paper/1904.07414/) (2019)
- [Multi-graph Fusion for Multi-view Spectral Clustering](https://mlsanity.com/paper/1909.06940/) (2019)
- [Measure-valued spline curves: an optimal transport viewpoint](https://mlsanity.com/paper/1801.03186/) (2018)
- [Modeling Network Populations via Graph Distances](https://mlsanity.com/paper/1904.07367/) (2019)
- [Spectral Clustering of Signed Graphs via Matrix Power Means](https://mlsanity.com/paper/1905.06230/) (2019)
- [Wasserstein-based Graph Alignment](https://mlsanity.com/paper/2003.06048/) (2020)
- [FGOT: Graph Distances based on Filters and Optimal Transport](https://mlsanity.com/paper/2109.04442/) (2021)
- [Fast Topological Clustering with Wasserstein Distance](https://mlsanity.com/paper/2112.00101/) (2021)
- [OrthoNet: Multilayer Network Data Clustering](https://mlsanity.com/paper/1811.00821/) (2018)
- [Generalized Matrix Means for Semi-Supervised Learning with Multilayer Graphs](https://mlsanity.com/paper/1910.13951/) (2019)
- [Statistical learning in Wasserstein space](https://mlsanity.com/paper/2004.07875/) (2020)

## Built on by (most cited first, 6 of 6)

- [Bures-Wasserstein Flow Matching for Graph Generation](https://mlsanity.com/paper/2506.14020/) (2025)
- [An Optimal Transport Approach for Network Regression](https://mlsanity.com/paper/2406.12204/) (2024)
- [The Spectral Barycentre of a Set of Graphs with Community Structure](https://mlsanity.com/paper/2502.00038/) (2025)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)
- [Geodesics of Dynamic Graphs for Regime Change Detection](https://mlsanity.com/paper/2606.07151/) (2026)
- [A dictionary learning framework for graphs via filters and optimal transport](https://mlsanity.com/paper/2609.05919/) (2026)

## Abstract (arXiv)

Finding the mean of sampled data is a fundamental task in machine learning and statistics. However, in cases where the data samples are graph objects, defining a mean is an inherently difficult task. We propose a novel framework for defining a graph mean via embeddings in the space of smooth graph signal distributions, where graph similarity can be measured using the Wasserstein metric. By finding a mean in this embedding space, we can recover a mean graph that preserves structural information. We establish the existence and uniqueness of the novel graph mean, and provide an iterative algorithm for computing it. To highlight the potential of our framework as a valuable tool for practical applications in machine learning, it is evaluated on various tasks, including k-means clustering of structured aligned graphs, classification of functional brain networks, and semi-supervised node classification in multi-layer graphs. Our experimental results demonstrate that our approach achieves consistent performance, outperforms existing baseline approaches, and improves the performance of state-of-the-art methods.