Path Integral Based Convolution and Pooling for Graph Neural Networks

Abstract

Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path integral based graph neural networks (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix we call maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus providing a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics we propose to boost applications of GNN in physical sciences.

arXiv comment15 pages, 4 figures, 6 tables. arXiv admin note: text overlap with arXiv:1904.10996

References in corpus (23)

• A Comprehensive Survey on Graph Neural Networks
• DeepWalk: Online Learning of Social Representations
• Inductive Representation Learning on Large Graphs
• Neural Message Passing for Quantum Chemistry
• Geometric deep learning: going beyond Euclidean data
• Fast and Accurate Modeling of Molecular Atomization Energies with Machine Learning
• Graph Neural Networks: A Review of Methods and Applications
• Subgraph Centrality in Complex Networks
• Fast Graph Representation Learning with PyTorch Geometric
• Simplifying Graph Convolutional Networks
• Open Graph Benchmark: Datasets for Machine Learning on Graphs
• Massively Multitask Networks for Drug Discovery
• MixHop: Higher-Order Graph Convolutional Architectures via Sparsified Neighborhood Mixing
• Towards Sparse Hierarchical Graph Classifiers
• Localization of maximal entropy random walk
• LanczosNet: Multi-Scale Deep Graph Convolutional Networks
• Robust Spatial Filtering with Graph Convolutional Neural Networks
• Watch Your Step: Learning Node Embeddings via Graph Attention
• Graph Wavelet Neural Network
• Maximal entropy random walk in community finding
• PAN: Path Integral Based Convolution for Deep Graph Neural Networks
• Towards Interpretable Sparse Graph Representation Learning with Laplacian Pooling
• Graph Convolutional Networks with EigenPooling

Cited by in corpus (7)

• High-Level Synthesis Performance Prediction using GNNs: Benchmarking, Modeling, and Advancing
• Decimated Framelet System on Graphs and Fast G-Framelet Transforms
• On Positional and Structural Node Features for Graph Neural Networks on Non-attributed Graphs
• How Framelets Enhance Graph Neural Networks
• MathNet: Haar-Like Wavelet Multiresolution-Analysis for Graph Representation and Learning
• Generalizing Downsampling from Regular Data to Graphs
• Embedding Graphs on Grassmann Manifold