Can Graph Neural Networks Count Substructures?

Abstract

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.

arXiv commentImproved the descriptions of the Local Relational Pooling (LRP) model and its practical implementation; Added more experimental results on synthetic and molecular datasets; Added the LRP-l-1 model in the experiments

References in corpus (35)

• Adam: A Method for Stochastic Optimization
• Semi-Supervised Classification with Graph Convolutional Networks
• Deep Residual Learning for Image Recognition
• Inductive Representation Learning on Large Graphs
• Neural Message Passing for Quantum Chemistry
• Geometric deep learning: going beyond Euclidean data
• Instance Normalization: The Missing Ingredient for Fast Stylization
• Spectral Networks and Locally Connected Networks on Graphs
• Graph Neural Networks: A Review of Methods and Applications
• Fast Graph Representation Learning with PyTorch Geometric
• Representation Learning on Graphs with Jumping Knowledge Networks
• Junction Tree Variational Autoencoder for Molecular Graph Generation
• GNNExplainer: Generating Explanations for Graph Neural Networks
• Open Graph Benchmark: Datasets for Machine Learning on Graphs
• How Powerful are Graph Neural Networks?
• Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting
• Benchmarking Graph Neural Networks
• Strategies for Pre-training Graph Neural Networks
• Universal Invariant and Equivariant Graph Neural Networks
• Invariant and Equivariant Graph Networks
• A Survey on The Expressive Power of Graph Neural Networks
• Generalization and Representational Limits of Graph Neural Networks
• Neural Subgraph Matching
• Relational Pooling for Graph Representations
• Hierarchical Graph-to-Graph Translation for Molecules
• What graph neural networks cannot learn: depth vs width
• G2SAT: Learning to Generate SAT Formulas
• Approximation Ratios of Graph Neural Networks for Combinatorial Problems
• On the Universality of Invariant Networks
• Global Attention Improves Graph Networks Generalization
• Discovering Molecular Functional Groups Using Graph Convolutional Neural Networks
• On the Combinatorial Power of the Weisfeiler-Lehman Algorithm
• On Weisfeiler-Leman Invariance: Subgraph Counts and Related Graph Properties
• Structural Analysis of Viral Spreading Processes in Social and Communication Networks Using Egonets
• Neural Subgraph Isomorphism Counting

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• Building powerful and equivariant graph neural networks with structural message-passing
• Pointer Graph Networks
• How hard is to distinguish graphs with graph neural networks?
• Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs
• From Local Structures to Size Generalization in Graph Neural Networks
• On Graph Neural Networks versus Graph-Augmented MLPs
• Counting Substructures with Higher-Order Graph Neural Networks: Possibility and Impossibility Results
• Utilising Graph Machine Learning within Drug Discovery and Development
• Graph convolutions that can finally model local structure
• Connecting Graph Convolutional Networks and Graph-Regularized PCA
• Topological Graph Neural Networks
• Theoretically Improving Graph Neural Networks via Anonymous Walk Graph Kernels
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