# Can Graph Neural Networks Count Substructures?

arXiv: https://arxiv.org/abs/2002.04025 · HTML: https://mlsanity.com/paper/2002.04025/

## Key facts

- Cited by 236 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Feb 2020.

## Built on by (most cited first, 30 of 236)

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- [From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness](https://mlsanity.com/paper/2110.03753/) (2021)
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- [Graph Neural Networks for Link Prediction with Subgraph Sketching](https://mlsanity.com/paper/2209.15486/) (2022)
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- [Discrete-state Continuous-time Diffusion for Graph Generation](https://mlsanity.com/paper/2405.11416/) (2024)
- [A Comprehensive Survey on Deep Graph Representation Learning](https://mlsanity.com/paper/2304.05055/) (2023)
- [A Survey on Graph Representation Learning Methods](https://mlsanity.com/paper/2204.01855/) (2022)
- [Reconstruction for Powerful Graph Representations](https://mlsanity.com/paper/2110.00577/) (2021)
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## Abstract (arXiv)

The ability to detect and count certain substructures in graphs is important for solving many tasks on graph-structured data, especially in the contexts of computational chemistry and biology as well as social network analysis. Inspired by this, we propose to study the expressive power of graph neural networks (GNNs) via their ability to count attributed graph substructures, extending recent works that examine their power in graph isomorphism testing and function approximation. We distinguish between two types of substructure counting: induced-subgraph-count and subgraph-count, and establish both positive and negative answers for popular GNN architectures. Specifically, we prove that Message Passing Neural Networks (MPNNs), 2-Weisfeiler-Lehman (2-WL) and 2-Invariant Graph Networks (2-IGNs) cannot perform induced-subgraph-count of substructures consisting of 3 or more nodes, while they can perform subgraph-count of star-shaped substructures. As an intermediary step, we prove that 2-WL and 2-IGNs are equivalent in distinguishing non-isomorphic graphs, partly answering an open problem raised in Maron et al. (2019). We also prove positive results for k-WL and k-IGNs as well as negative results for k-WL with a finite number of iterations. We then conduct experiments that support the theoretical results for MPNNs and 2-IGNs. Moreover, motivated by substructure counting and inspired by Murphy et al. (2019), we propose the Local Relational Pooling model and demonstrate that it is not only effective for substructure counting but also able to achieve competitive performance on molecular prediction tasks.