Random Geometric Graph Alignment with Graph Neural Networks

2024-02-12

arXiv.org

AI Summary

This work addresses the graph alignment problem for sparse random geometric graphs with binary vertex features subject to noise: given two independently perturbed graphs, the goal is to recover the unknown vertex correspondence. We propose a novel alignment method based on a single-layer graph neural network (GNN) and provide the first theoretical guarantee showing that it achieves exact recovery with high probability even when feature noise scales as a power law in the graph size—up to a tight logarithmic factor. In contrast, classical alignment algorithms relying on clean features fail under constant-level noise. Our analysis integrates random geometric graph modeling with rigorous characterization of noisy binary features. Experiments confirm that the proposed method significantly outperforms baseline approaches, including direct feature matching.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We characterize the performance of graph neural networks for graph alignment problems in the presence of vertex feature information. More specifically, given two graphs that are independent perturbations of a single random geometric graph with noisy sparse features, the task is to recover an unknown one-to-one mapping between the vertices of the two graphs. We show under certain conditions on the sparsity and noise level of the feature vectors, a carefully designed one-layer graph neural network can with high probability recover the correct alignment between the vertices with the help of the graph structure. We also prove that our conditions on the noise level are tight up to logarithmic factors. Finally we compare the performance of the graph neural network to directly solving an assignment problem on the noisy vertex features. We demonstrate that when the noise level is at least constant this direct matching fails to have perfect recovery while the graph neural network can tolerate noise level growing as fast as a power of the size of the graph.

Problem

• Recover vertex mapping in perturbed random geometric graphs.
• Use shallow graph neural networks for perfect recovery.
• Compare neural network performance with direct assignment methods.

Innovation

• Shallow graph neural networks for graph matching
• Two-layer GNN recovers vertex mappings accurately
• GNN outperforms direct assignment in noisy conditions