# DEMO-Net: Degree-specific Graph Neural Networks for Node and Graph Classification

arXiv: https://arxiv.org/abs/1906.02319 · HTML: https://mlsanity.com/paper/1906.02319/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2019.

## Built on by (most cited first, 30 of 59)

- [Graph Unlearning](https://mlsanity.com/paper/2103.14991/) (2021)
- [Breaking the Limit of Graph Neural Networks by Improving the Assortativity of Graphs with Local Mixing Patterns](https://mlsanity.com/paper/2106.06586/) (2021)
- [Understanding Pooling in Graph Neural Networks](https://mlsanity.com/paper/2110.05292/) (2021)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [Non-IID Transfer Learning on Graphs](https://mlsanity.com/paper/2212.08174/) (2022)
- [Rethinking Graph Auto-Encoder Models for Attributed Graph Clustering](https://mlsanity.com/paper/2107.08562/) (2021)
- [Counterfactual Graphs for Explainable Classification of Brain Networks](https://mlsanity.com/paper/2106.08640/) (2021)
- [LTE4G: Long-Tail Experts for Graph Neural Networks](https://mlsanity.com/paper/2208.10205/) (2022)
- [A Survey on Graph Counterfactual Explanations: Definitions, Methods, Evaluation, and Research Challenges](https://mlsanity.com/paper/2210.12089/) (2022)
- [Uncovering the Structural Fairness in Graph Contrastive Learning](https://mlsanity.com/paper/2210.03011/) (2022)
- [Robust Training of Graph Neural Networks via Noise Governance](https://mlsanity.com/paper/2211.06614/) (2022)
- [On Generalized Degree Fairness in Graph Neural Networks](https://mlsanity.com/paper/2302.03881/) (2023)
- [RawlsGCN: Towards Rawlsian Difference Principle on Graph Convolutional Network](https://mlsanity.com/paper/2202.13547/) (2022)
- [Privacy-Preserving Representation Learning on Graphs: A Mutual Information Perspective](https://mlsanity.com/paper/2107.01475/) (2021)
- [Graph Transformer Networks: Learning Meta-path Graphs to Improve GNNs](https://mlsanity.com/paper/2106.06218/) (2021)
- [Fairness Amidst Non-IID Graph Data: A Literature Review](https://mlsanity.com/paper/2202.07170/) (2022)
- [Self-Supervised Graph Structure Refinement for Graph Neural Networks](https://mlsanity.com/paper/2211.06545/) (2022)
- [Network Representation Learning: From Preprocessing, Feature Extraction to Node Embedding](https://mlsanity.com/paper/2110.07582/) (2021)
- [Self-supervised Contrastive Attributed Graph Clustering](https://mlsanity.com/paper/2110.08264/) (2021)
- [Efficient Graph Deep Learning in TensorFlow with tf_geometric](https://mlsanity.com/paper/2101.11552/) (2021)
- [Structure-Preserving Graph Representation Learning](https://mlsanity.com/paper/2209.00793/) (2022)
- [Self-supervised Consensus Representation Learning for Attributed Graph](https://mlsanity.com/paper/2108.04822/) (2021)
- [ResNorm: Tackling Long-tailed Degree Distribution Issue in Graph Neural Networks via Normalization](https://mlsanity.com/paper/2206.08181/) (2022)
- [A Survey of Imbalanced Learning on Graphs: Problems, Techniques, and Future Directions](https://mlsanity.com/paper/2308.13821/) (2023)
- [A Generalized Neural Diffusion Framework on Graphs](https://mlsanity.com/paper/2312.08616/) (2023)
- [OPORP: One Permutation + One Random Projection](https://mlsanity.com/paper/2302.03505/) (2023)
- [Residual2Vec: Debiasing graph embedding with random graphs](https://mlsanity.com/paper/2110.07654/) (2021)
- [Revisiting Graph Contrastive Learning on Anomaly Detection: A Structural Imbalance Perspective](https://mlsanity.com/paper/2507.14677/) (2025)
- [Data Augmentation for Graph Convolutional Network on Semi-Supervised Classification](https://mlsanity.com/paper/2106.08848/) (2021)
- [SAILOR: Structural Augmentation Based Tail Node Representation Learning](https://mlsanity.com/paper/2308.06801/) (2023)

## Abstract (arXiv)

Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations: (1) there is limited analysis regarding the graph convolution properties, such as seed-oriented, degree-aware and order-free; (2) the node's degree-specific graph structure is not explicitly expressed in graph convolution for distinguishing structure-aware node neighborhoods; (3) the theoretical explanation regarding the graph-level pooling schemes is unclear. To address these problems, we propose a generic degree-specific graph neural network named DEMO-Net motivated by Weisfeiler-Lehman graph isomorphism test that recursively identifies 1-hop neighborhood structures. In order to explicitly capture the graph topology integrated with node attributes, we argue that graph convolution should have three properties: seed-oriented, degree-aware, order-free. To this end, we propose multi-task graph convolution where each task represents node representation learning for nodes with a specific degree value, thus leading to preserving the degree-specific graph structure. In particular, we design two multi-task learning methods: degree-specific weight and hashing functions for graph convolution. In addition, we propose a novel graph-level pooling/readout scheme for learning graph representation provably lying in a degree-specific Hilbert kernel space. The experimental results on several node and graph classification benchmark data sets demonstrate the effectiveness and efficiency of our proposed DEMO-Net over state-of-the-art graph neural network models.