# PieClam: A Universal Graph Autoencoder Based on Overlapping Inclusive and Exclusive Communities

arXiv: https://arxiv.org/abs/2409.11618 · HTML: https://mlsanity.com/paper/2409.11618/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 33 of its 63 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 30 of 33)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks](https://mlsanity.com/paper/1511.06434/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [Representation Learning on Graphs: Methods and Applications](https://mlsanity.com/paper/1709.05584/) (2017)
- [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)
- [A Comprehensive Survey on Graph Anomaly Detection with Deep Learning](https://mlsanity.com/paper/2106.07178/) (2021)
- [Recent Advances in Autoencoder-Based Representation Learning](https://mlsanity.com/paper/1812.05069/) (2018)
- [Data Augmentation for Deep Graph Learning: A Survey](https://mlsanity.com/paper/2202.08235/) (2022)
- [NetGAN: Generating Graphs via Random Walks](https://mlsanity.com/paper/1803.00816/) (2018)
- [Learning in Implicit Generative Models](https://mlsanity.com/paper/1610.03483/) (2016)
- [GraphGAN: Graph Representation Learning with Generative Adversarial Nets](https://mlsanity.com/paper/1711.08267/) (2017)
- [GraphNVP: An Invertible Flow Model for Generating Molecular Graphs](https://mlsanity.com/paper/1905.11600/) (2019)
- [Graphite: Iterative Generative Modeling of Graphs](https://mlsanity.com/paper/1803.10459/) (2018)
- [Community detection in networks: A user guide](https://mlsanity.com/paper/1608.00163/) (2016)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [A Systematic Survey on Deep Generative Models for Graph Generation](https://mlsanity.com/paper/2007.06686/) (2020)
- [AnomalyDAE: Dual autoencoder for anomaly detection on attributed networks](https://mlsanity.com/paper/2002.03665/) (2020)
- [Graph Representation Learning: A Survey](https://mlsanity.com/paper/1909.00958/) (2019)
- [NeVAE: A Deep Generative Model for Molecular Graphs](https://mlsanity.com/paper/1802.05283/) (2018)
- [A Review of Stochastic Block Models and Extensions for Graph Clustering](https://mlsanity.com/paper/1903.00114/) (2019)
- [Overlapping Community Detection with Graph Neural Networks](https://mlsanity.com/paper/1909.12201/) (2019)
- [Truncated Affinity Maximization: One-class Homophily Modeling for Graph Anomaly Detection](https://mlsanity.com/paper/2306.00006/) (2023)
- [Stochastic Blockmodels meet Graph Neural Networks](https://mlsanity.com/paper/1905.05738/) (2019)
- [A graphon-signal analysis of graph neural networks](https://mlsanity.com/paper/2305.15987/) (2023)
- [vGraph: A Generative Model for Joint Community Detection and Node Representation Learning](https://mlsanity.com/paper/1906.07159/) (2019)

## Built on by (most cited first, 1 of 1)

- [Efficient Learning on Large Graphs using a Densifying Regularity Lemma](https://mlsanity.com/paper/2504.18273/) (2025)

## Abstract (arXiv)

We propose PieClam (Prior Inclusive Exclusive Cluster Affiliation Model): a probabilistic graph model for representing any graph as overlapping generalized communities. Our method can be interpreted as a graph autoencoder: nodes are embedded into a code space by an algorithm that maximizes the log-likelihood of the decoded graph, given the input graph. PieClam is a community affiliation model that extends well-known methods like BigClam in two main manners. First, instead of the decoder being defined via pairwise interactions between the nodes in the code space, we also incorporate a learned prior on the distribution of nodes in the code space, turning our method into a graph generative model. Secondly, we generalize the notion of communities by allowing not only sets of nodes with strong connectivity, which we call inclusive communities, but also sets of nodes with strong disconnection, which we call exclusive communities. To model both types of communities, we propose a new type of decoder based the Lorentz inner product, which we prove to be much more expressive than standard decoders based on standard inner products or norm distances. By introducing a new graph similarity measure, that we call the log cut distance, we show that PieClam is a universal autoencoder, able to uniformly approximately reconstruct any graph. Our method is shown to obtain competitive performance in graph anomaly detection benchmarks.