Schreier-Coset Graph Propagation

2025-05-15

AI Summary

Graph Neural Networks (GNNs) suffer from the *over-squashing* problem during long-range message passing, where excessive neighborhood information is compressed into fixed-size node representations. While existing approaches—such as graph rewiring or Cayley graph augmentation—alleviate over-squashing, they incur severe scalability bottlenecks (e.g., SL(2,ℤₙ) Cayley graphs require O(n³) nodes). This work introduces **Schreier Coset Embedding (SCE)**, the first method to directly encode coset structures induced by group actions into node features—**without altering the original graph topology**. SCE injects high-connectivity priors implicitly, avoiding explicit graph rewiring or construction of large expanded graphs. Theoretically expressive and computationally efficient, SCE achieves comparable or superior performance to expander-based and rewired GNN baselines on node and graph classification tasks, while significantly reducing memory footprint and inference latency—particularly beneficial for hierarchical, modular graphs and resource-constrained deployments.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Graph Neural Networks (GNNs) offer a principled framework for learning over graph-structured data, yet their expressive capacity is often hindered by over-squashing, wherein information from distant nodes is compressed into fixed-size vectors. Existing solutions, including graph rewiring and bottleneck-resistant architectures such as Cayley and expander graphs, avoid this problem but introduce scalability bottlenecks. In particular, the Cayley graphs constructed over $SL(2,mathbb{Z}_n)$ exhibit strong theoretical properties, yet suffer from cubic node growth $O(n^3)$, leading to high memory usage. To address this, this work introduces Schrier-Coset Graph Propagation (SCGP), a group-theoretic augmentation method that enriches node features through Schreier-coset embeddings without altering the input graph topology. SCGP embeds bottleneck-free connectivity patterns into a compact feature space, improving long-range message passing while maintaining computational efficiency. Empirical evaluations across standard node and graph classification benchmarks demonstrate that SCGP achieves performance comparable to, or exceeding, expander graph and rewired GNN baselines. Furthermore, SCGP exhibits particular advantages in processing hierarchical and modular graph structures, offering reduced inference latency, improved scalability, and a low memory footprint, making it suitable for real-time and resource-constrained applications.

Problem

• Over-squashing limits GNN expressive capacity
• Existing solutions introduce scalability bottlenecks
• Cayley graphs suffer from cubic node growth

Innovation

• Schreier-Coset Graph Propagation for GNNs
• Group-theoretic augmentation without topology change
• Compact feature space for efficient message passing