# Path Integral Based Convolution and Pooling for Graph Neural Networks

arXiv: https://arxiv.org/abs/2006.16811 · HTML: https://mlsanity.com/paper/2006.16811/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Jun 2020.

## Built on by (most cited first, 30 of 35)

- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [A Survey on Graph Representation Learning Methods](https://mlsanity.com/paper/2204.01855/) (2022)
- [Locality-Aware Graph-Rewiring in GNNs](https://mlsanity.com/paper/2310.01668/) (2023)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [On Positional and Structural Node Features for Graph Neural Networks on Non-attributed Graphs](https://mlsanity.com/paper/2107.01495/) (2021)
- [Walking Out of the Weisfeiler Leman Hierarchy: Graph Learning Beyond Message Passing](https://mlsanity.com/paper/2102.08786/) (2021)
- [How Framelets Enhance Graph Neural Networks](https://mlsanity.com/paper/2102.06986/) (2021)
- [Shortest Path Networks for Graph Property Prediction](https://mlsanity.com/paper/2206.01003/) (2022)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Challenges and Opportunities in Deep Reinforcement Learning with Graph Neural Networks: A Comprehensive review of Algorithms and Applications](https://mlsanity.com/paper/2206.07922/) (2022)
- [High-Level Synthesis Performance Prediction using GNNs: Benchmarking, Modeling, and Advancing](https://mlsanity.com/paper/2201.06848/) (2022)
- [OpenFGL: A Comprehensive Benchmark for Federated Graph Learning](https://mlsanity.com/paper/2408.16288/) (2024)
- [pathGCN: Learning General Graph Spatial Operators from Paths](https://mlsanity.com/paper/2207.07408/) (2022)
- [Subgraph-level Universal Prompt Tuning](https://mlsanity.com/paper/2402.10380/) (2024)
- [A Comprehensive Graph Pooling Benchmark: Effectiveness, Robustness and Generalizability](https://mlsanity.com/paper/2406.09031/) (2024)
- [Generalizing Downsampling from Regular Data to Graphs](https://mlsanity.com/paper/2208.03523/) (2022)
- [MaxCutPool: differentiable feature-aware Maxcut for pooling in graph neural networks](https://mlsanity.com/paper/2409.05100/) (2024)
- [On Exploring Node-feature and Graph-structure Diversities for Node Drop Graph Pooling](https://mlsanity.com/paper/2306.12726/) (2023)
- [From Anchors to Answers: A Novel Node Tokenizer for Integrating Graph Structure into Large Language Models](https://mlsanity.com/paper/2410.10743/) (2024)
- [Sort & Slice: A Simple and Superior Alternative to Hash-Based Folding for Extended-Connectivity Fingerprints](https://mlsanity.com/paper/2403.17954/) (2024)
- [Embedding Graphs on Grassmann Manifold](https://mlsanity.com/paper/2205.15068/) (2022)
- [Deep Surrogate Docking: Accelerating Automated Drug Discovery with Graph Neural Networks](https://mlsanity.com/paper/2211.02720/) (2022)
- [Scalable Graph Compressed Convolutions](https://mlsanity.com/paper/2407.18480/) (2024)
- [DELTA: Dual Consistency Delving with Topological Uncertainty for Active Graph Domain Adaptation](https://mlsanity.com/paper/2409.08946/) (2024)
- [BN-Pool: Bayesian Nonparametric Pooling for Graphs](https://mlsanity.com/paper/2501.09821/) (2025)
- [Torch Geometric Pool: the PyTorch library for pooling in Graph Neural Networks](https://mlsanity.com/paper/2512.12642/) (2025)
- [Program-to-Circuit: Exploiting GNNs for Program Representation and Circuit Translation](https://mlsanity.com/paper/2109.06265/) (2021)
- [Every Node Counts: Improving the Training of Graph Neural Networks on Node Classification](https://mlsanity.com/paper/2211.16631/) (2022)
- [MQENet: A Mesh Quality Evaluation Neural Network Based on Dynamic Graph Attention](https://mlsanity.com/paper/2309.01067/) (2023)
- [Careful Selection and Thoughtful Discarding: Graph Explicit Pooling Utilizing Discarded Nodes](https://mlsanity.com/paper/2311.12644/) (2023)

## Abstract (arXiv)

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.