# Boosting the Cycle Counting Power of Graph Neural Networks with I2-GNNs

arXiv: https://arxiv.org/abs/2210.13978 · HTML: https://mlsanity.com/paper/2210.13978/

## Key facts

- Cited by 56 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 83 references that are in the map.
- First posted on arXiv 22 Oct 2022.

## Builds on (most cited first, 30 of 57)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (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)
- [Modeling Relational Data with Graph Convolutional Networks](https://mlsanity.com/paper/1703.06103/) (2017)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Link Prediction Based on Graph Neural Networks](https://mlsanity.com/paper/1802.09691/) (2018)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [Convolutional Networks on Graphs for Learning Molecular Fingerprints](https://mlsanity.com/paper/1509.09292/) (2015)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [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)
- [Learning Combinatorial Optimization Algorithms over Graphs](https://mlsanity.com/paper/1704.01665/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing](https://mlsanity.com/paper/1905.00067/) (2019)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Graph Convolutional Policy Network for Goal-Directed Molecular Graph Generation](https://mlsanity.com/paper/1806.02473/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Graph Neural Networks with Learnable Structural and Positional Representations](https://mlsanity.com/paper/2110.07875/) (2021)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [The Surprising Power of Graph Neural Networks with Random Node Initialization](https://mlsanity.com/paper/2010.01179/) (2020)

## Built on by (most cited first, 30 of 56)

- [A Complete Expressiveness Hierarchy for Subgraph GNNs via Subgraph Weisfeiler-Lehman Tests](https://mlsanity.com/paper/2302.07090/) (2023)
- [GOFA: A Generative One-For-All Model for Joint Graph Language Modeling](https://mlsanity.com/paper/2407.09709/) (2024)
- [An Empirical Study of Realized GNN Expressiveness](https://mlsanity.com/paper/2304.07702/) (2023)
- [On the Stability of Expressive Positional Encodings for Graphs](https://mlsanity.com/paper/2310.02579/) (2023)
- [WL meet VC](https://mlsanity.com/paper/2301.11039/) (2023)
- [Provably Powerful Graph Neural Networks for Directed Multigraphs](https://mlsanity.com/paper/2306.11586/) (2023)
- [Unifying Generation and Prediction on Graphs with Latent Graph Diffusion](https://mlsanity.com/paper/2402.02518/) (2024)
- [Is Distance Matrix Enough for Geometric Deep Learning?](https://mlsanity.com/paper/2302.05743/) (2023)
- [From Relational Pooling to Subgraph GNNs: A Universal Framework for More Expressive Graph Neural Networks](https://mlsanity.com/paper/2305.04963/) (2023)
- [Efficient Subgraph GNNs by Learning Effective Selection Policies](https://mlsanity.com/paper/2310.20082/) (2023)
- [Extending the Design Space of Graph Neural Networks by Rethinking Folklore Weisfeiler-Lehman](https://mlsanity.com/paper/2306.03266/) (2023)
- [Weisfeiler-Leman at the margin: When more expressivity matters](https://mlsanity.com/paper/2402.07568/) (2024)
- [What Can We Learn from State Space Models for Machine Learning on Graphs?](https://mlsanity.com/paper/2406.05815/) (2024)
- [Distance-Restricted Folklore Weisfeiler-Leman GNNs with Provable Cycle Counting Power](https://mlsanity.com/paper/2309.04941/) (2023)
- [MAG-GNN: Reinforcement Learning Boosted Graph Neural Network](https://mlsanity.com/paper/2310.19142/) (2023)
- [Boosting Graph Pooling with Persistent Homology](https://mlsanity.com/paper/2402.16346/) (2024)
- [Expressivity and Generalization: Fragment-Biases for Molecular GNNs](https://mlsanity.com/paper/2406.08210/) (2024)
- [Subgraphormer: Unifying Subgraph GNNs and Graph Transformers via Graph Products](https://mlsanity.com/paper/2402.08450/) (2024)
- [A Flexible, Equivariant Framework for Subgraph GNNs via Graph Products and Graph Coarsening](https://mlsanity.com/paper/2406.09291/) (2024)
- [Simplifying Subgraph Representation Learning for Scalable Link Prediction](https://mlsanity.com/paper/2301.12562/) (2023)
- [Rethinking Diffusion Models with Symmetries through Canonicalization with Applications to Molecular Graph Generation](https://mlsanity.com/paper/2602.15022/) (2026)
- [Union Subgraph Neural Networks](https://mlsanity.com/paper/2305.15747/) (2023)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [Balancing Efficiency and Expressiveness: Subgraph GNNs with Walk-Based Centrality](https://mlsanity.com/paper/2501.03113/) (2025)
- [Graph Representational Learning: When Does More Expressivity Hurt Generalization?](https://mlsanity.com/paper/2505.11298/) (2025)
- [PyTorch Geometric High Order: A Unified Library for High Order Graph Neural Network](https://mlsanity.com/paper/2311.16670/) (2023)
- [Graph as Point Set](https://mlsanity.com/paper/2405.02795/) (2024)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Towards Graph Foundation Models: A Study on the Generalization of Positional and Structural Encodings](https://mlsanity.com/paper/2412.07407/) (2024)
- [Homomorphism Expressivity of Spectral Invariant Graph Neural Networks](https://mlsanity.com/paper/2503.00485/) (2025)

## Abstract (arXiv)

Message Passing Neural Networks (MPNNs) are a widely used class of Graph Neural Networks (GNNs). The limited representational power of MPNNs inspires the study of provably powerful GNN architectures. However, knowing one model is more powerful than another gives little insight about what functions they can or cannot express. It is still unclear whether these models are able to approximate specific functions such as counting certain graph substructures, which is essential for applications in biology, chemistry and social network analysis. Motivated by this, we propose to study the counting power of Subgraph MPNNs, a recent and popular class of powerful GNN models that extract rooted subgraphs for each node, assign the root node a unique identifier and encode the root node's representation within its rooted subgraph. Specifically, we prove that Subgraph MPNNs fail to count more-than-4-cycles at node level, implying that node representations cannot correctly encode the surrounding substructures like ring systems with more than four atoms. To overcome this limitation, we propose I2-GNNs to extend Subgraph MPNNs by assigning different identifiers for the root node and its neighbors in each subgraph. I2-GNNs' discriminative power is shown to be strictly stronger than Subgraph MPNNs and partially stronger than the 3-WL test. More importantly, I2-GNNs are proven capable of counting all 3, 4, 5 and 6-cycles, covering common substructures like benzene rings in organic chemistry, while still keeping linear complexity. To the best of our knowledge, it is the first linear-time GNN model that can count 6-cycles with theoretical guarantees. We validate its counting power in cycle counting tasks and demonstrate its competitive performance in molecular prediction benchmarks.