# The Map Equation Goes Neural: Mapping Network Flows with Graph Neural Networks

arXiv: https://arxiv.org/abs/2310.01144 · HTML: https://mlsanity.com/paper/2310.01144/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 85 references that are in the map.
- First posted on arXiv 2 Oct 2023.

## Builds on (most cited first, 30 of 48)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [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 Graph Infomax](https://mlsanity.com/paper/1809.10341/) (2018)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Self-Normalizing Neural Networks](https://mlsanity.com/paper/1706.02515/) (2017)
- [Cluster-GCN: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networks](https://mlsanity.com/paper/1905.07953/) (2019)
- [From Louvain to Leiden: guaranteeing well-connected communities](https://mlsanity.com/paper/1810.08473/) (2018)
- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [Graph Clustering with Graph Neural Networks](https://mlsanity.com/paper/2006.16904/) (2020)
- [Attributed Graph Clustering: A Deep Attentional Embedding Approach](https://mlsanity.com/paper/1906.06532/) (2019)
- [Structural Deep Clustering Network](https://mlsanity.com/paper/2002.01633/) (2020)
- [Attributed Graph Clustering via Adaptive Graph Convolution](https://mlsanity.com/paper/1906.01210/) (2019)
- [A Comprehensive Survey on Deep Clustering: Taxonomy, Challenges, and Future Directions](https://mlsanity.com/paper/2206.07579/) (2022)
- [The Description Length of Deep Learning Models](https://mlsanity.com/paper/1802.07044/) (2018)
- [A Comprehensive Survey on Community Detection with Deep Learning](https://mlsanity.com/paper/2105.12584/) (2021)
- [PageRank beyond the Web](https://mlsanity.com/paper/1407.5107/) (2014)
- [Overlapping Community Detection with Graph Neural Networks](https://mlsanity.com/paper/1909.12201/) (2019)
- [The ground truth about metadata and community detection in networks](https://mlsanity.com/paper/1608.05878/) (2016)
- [Hierarchical Block Structures and High-resolution Model Selection in Large Networks](https://mlsanity.com/paper/1310.4377/) (2013)
- [Rethinking Graph Auto-Encoder Models for Attributed Graph Clustering](https://mlsanity.com/paper/2107.08562/) (2021)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)

## Built on by (most cited first, 2 of 2)

- [Image Distillation for Safe Data Sharing in Histopathology](https://mlsanity.com/paper/2406.13536/) (2024)
- [InfoMotion: A Graph-Based Approach to Video Dataset Distillation for Echocardiography](https://mlsanity.com/paper/2512.09422/) (2025)

## Abstract (arXiv)

Community detection is an essential tool for unsupervised data exploration and revealing the organisational structure of networked systems. With a long history in network science, community detection typically relies on objective functions, optimised with custom-tailored search algorithms, but often without leveraging recent advances in deep learning. Recently, first works have started incorporating such objectives into loss functions for deep graph clustering and pooling. We consider the map equation, a popular information-theoretic objective function for unsupervised community detection, and express it in differentiable tensor form for optimisation through gradient descent. Our formulation turns the map equation compatible with any neural network architecture, enables end-to-end learning, incorporates node features, and chooses the optimal number of clusters automatically, all without requiring explicit regularisation. Applied to unsupervised graph clustering tasks, we achieve competitive performance against state-of-the-art deep graph clustering baselines in synthetic and real-world datasets.