Graph Alignment for Benchmarking Graph Neural Networks and Learning Positional Encodings

TL;DR

The paper introduces Graph Alignment as a flexible, topology-centric benchmark for graph neural networks, enabling self-supervised dataset generation with controllable difficulty and providing an empirical framework to compare GNN architectures across diverse graph topologies. It shows that anisotropic GNNs outperform isotropic ones on the alignment task and demonstrates that node embeddings learned through graph alignment can serve as powerful positional encodings for graph transformers, achieving state-of-the-art results on PCQM4Mv2 with substantially fewer parameters. The authors also present GAPE, a method for generating graph-alignment-based positional encodings, and provide an open-source toolkit to reproduce and extend the benchmarks. Collectively, these contributions offer a scalable, reproducible pathway to assess structural understanding in GNNs and to enhance transformer performance on graph-structured data, particularly in molecular regression tasks.

Abstract

We propose a novel benchmarking methodology for graph neural networks (GNNs) based on the graph alignment problem, a combinatorial optimization task that generalizes graph isomorphism by aligning two unlabeled graphs to maximize overlapping edges. We frame this problem as a self-supervised learning task and present several methods to generate graph alignment datasets using synthetic random graphs and real-world graph datasets from multiple domains. For a given graph dataset, we generate a family of graph alignment datasets with increasing difficulty, allowing us to rank the performance of various architectures. Our experiments indicate that anisotropic graph neural networks outperform standard convolutional architectures. To further demonstrate the utility of the graph alignment task, we show its effectiveness for unsupervised GNN pre-training, where the learned node embeddings outperform other positional encodings on three molecular regression tasks and achieve state-of-the-art results on the PCQM4Mv2 dataset with significantly fewer parameters. To support reproducibility and further research, we provide an open-source Python package to generate graph alignment datasets and benchmark new GNN architectures.

Figures (4)

• Figure 1: Visualization of graph topologies: Graph topologies differ widely across application domains, and the performance of GNN architectures varies accordingly based on these topological differences. Furthermore, different tasks may share the same graph topologies.
• Figure 2: The siamese algorithm for predicting a solution to the Graph Alignment Problem. During training, we use the binary cross-entropy loss between the predicted similarity matrix $\Sigma$ and a solution $\pi^*$ of the Graph Alignment Problem. During evaluation, we use the Hungarian algorithm to extract a permutation from $\Sigma$ by solving the Linear Assignment Problem. In practice, using PyTorch Geometric, we represent the graph solely by its edge_index, with all node features set to a constant value (e.g., 1) to focus exclusively on the graph structure.
• Figure 3: Relative performance of well-known Message Passing Neural Networks for different graph alignment datasets based on 3 base datasets: Erdös-Rényi, CoraFull and AQSOL.
• Figure E.1: Comparison of the generalization error in function of the noise used to create the validation Graph Alignment dataset. The decrease in performance represents the accuracy gap to the best model

Theorems & Definitions (2)

• Theorem A.1
• proof