# Multiresolution Equivariant Graph Variational Autoencoder

arXiv: https://arxiv.org/abs/2106.00967 · HTML: https://mlsanity.com/paper/2106.00967/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 98 references that are in the map.
- First posted on arXiv 2 Jun 2021.

## Builds on (most cited first, 30 of 50)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Categorical Reparameterization with Gumbel-Softmax](https://mlsanity.com/paper/1611.01144/) (2016)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [DeepWalk: Online Learning of Social Representations](https://mlsanity.com/paper/1403.6652/) (2014)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Interaction Networks for Learning about Objects, Relations and Physics](https://mlsanity.com/paper/1612.00222/) (2016)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [MolGAN: An implicit generative model for small molecular graphs](https://mlsanity.com/paper/1805.11973/) (2018)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoders](https://mlsanity.com/paper/1802.03480/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Learning Convolutional Neural Networks for Graphs](https://mlsanity.com/paper/1605.05273/) (2016)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [GraphAF: a Flow-based Autoregressive Model for Molecular Graph Generation](https://mlsanity.com/paper/2001.09382/) (2020)
- [Molecular Graph Convolutions: Moving Beyond Fingerprints](https://mlsanity.com/paper/1603.00856/) (2016)
- [Efficient Graph Generation with Graph Recurrent Attention Networks](https://mlsanity.com/paper/1910.00760/) (2019)
- [Learning Deep Generative Models of Graphs](https://mlsanity.com/paper/1803.03324/) (2018)

## Built on by (most cited first, 20 of 20)

- [Temporal Multiresolution Graph Neural Networks For Epidemic Prediction](https://mlsanity.com/paper/2205.14831/) (2022)
- [Predicting COVID-19 pandemic by spatio-temporal graph neural networks: A New Zealand's study](https://mlsanity.com/paper/2305.07731/) (2023)
- [Geometry of Linear Neural Networks: Equivariance and Invariance under Permutation Groups](https://mlsanity.com/paper/2309.13736/) (2023)
- [Multiresolution Graph Transformers and Wavelet Positional Encoding for Learning Hierarchical Structures](https://mlsanity.com/paper/2302.08647/) (2023)
- [EvoMesh: Adaptive Physical Simulation with Hierarchical Graph Evolutions](https://mlsanity.com/paper/2410.03779/) (2024)
- [Interpreting Equivariant Representations](https://mlsanity.com/paper/2401.12588/) (2024)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [LGDC: Latent Graph Diffusion via Spectrum-Preserving Coarsening](https://mlsanity.com/paper/2512.01190/) (2025)
- [ReGAE: Graph autoencoder based on recursive neural networks](https://mlsanity.com/paper/2201.12165/) (2022)
- [HDGL: A hierarchical dynamic graph representation learning model for brain disorder classification](https://mlsanity.com/paper/2311.02903/) (2023)
- [Thermodynamics-informed graph neural networks for real-time simulation of digital human twins](https://mlsanity.com/paper/2412.12034/) (2024)
- [Target-aware Variational Auto-encoders for Ligand Generation with Multimodal Protein Representation Learning](https://mlsanity.com/paper/2309.16685/) (2023)
- [Symmetry-preserving graph attention network to solve routing problems at multiple resolutions](https://mlsanity.com/paper/2310.15543/) (2023)
- [Range-aware Positional Encoding via High-order Pretraining: Theory and Practice](https://mlsanity.com/paper/2409.19117/) (2024)
- [MOB-GCN: A Novel Multiscale Object-Based Graph Neural Network for Hyperspectral Image Classification](https://mlsanity.com/paper/2502.16289/) (2025)
- [Binary Latent Protein Fitness Landscapes for Quantum Annealing Optimization](https://mlsanity.com/paper/2603.17247/) (2026)
- [Q-BIOLAT: Binary Latent Protein Fitness Landscapes for QUBO-Based Optimization](https://mlsanity.com/paper/2603.27526/) (2026)
- [PRIME: Protein Representation via Physics-Informed Multiscale Equivariant Hierarchies](https://mlsanity.com/paper/2605.01625/) (2026)
- [CARD: Coarse-to-fine Autoregressive Modeling with Radix-based Decomposition for Transferable Free Energy Estimation](https://mlsanity.com/paper/2605.02657/) (2026)
- [GeoGAE: Scalable Graph-Level Autoencoding via Hyperball Cloud Representations](https://mlsanity.com/paper/2609.35527/) (2026)

## Abstract (arXiv)

In this paper, we propose Multiresolution Equivariant Graph Variational Autoencoders (MGVAE), the first hierarchical generative model to learn and generate graphs in a multiresolution and equivariant manner. At each resolution level, MGVAE employs higher order message passing to encode the graph while learning to partition it into mutually exclusive clusters and coarsening into a lower resolution that eventually creates a hierarchy of latent distributions. MGVAE then constructs a hierarchical generative model to variationally decode into a hierarchy of coarsened graphs. Importantly, our proposed framework is end-to-end permutation equivariant with respect to node ordering. MGVAE achieves competitive results with several generative tasks including general graph generation, molecular generation, unsupervised molecular representation learning to predict molecular properties, link prediction on citation graphs, and graph-based image generation.