Path Integral Based Convolution and Pooling for Graph Neural Networks

TL;DR

This paper introduces a novel graph neural network framework based on path integrals, which generalizes existing methods and achieves state-of-the-art results in graph classification and regression tasks, including applications in physical sciences.

Contribution

The paper proposes a new path integral based convolution and pooling method for GNNs, unifying and extending existing architectures with a physics-inspired approach.

Findings

Achieves state-of-the-art performance on multiple graph tasks.

Introduces a new benchmark dataset from statistical mechanics.

Provides a versatile framework adaptable to various graph structures.

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.