On the Interplay between Graph Structure and Learning Algorithms in Graph Neural Networks

2025-08-19

AI Summary

This work investigates the coupling mechanism between graph structure and learning algorithms (e.g., SGD, ridge regression) in graph neural networks (GNNs) under noisy generalization settings—a scenario largely overlooked by existing theory, which predominantly assumes noiseless interpolation and only loosely links graph properties (e.g., maximum degree) to performance. Leveraging spectral graph theory, we establish a unified analytical framework that quantitatively characterizes how regular and power-law graphs affect excess risk and explains oversmoothing from a spectral perspective. Our analysis extends to multi-layer linear GNNs, integrating excess risk bounds with empirical validation. Results demonstrate that spectral properties—particularly eigenvalue distribution—critically govern both optimization convergence and generalization behavior. This provides interpretable, theoretically grounded principles for GNN architecture design and optimization. (149 words)

Technology Category

• Machine Learning: Graph-based Machine Learning
• Natural Language Processing: Learning & Optimization for NLP
• 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
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

This paper studies the interplay between learning algorithms and graph structure for graph neural networks (GNNs). Existing theoretical studies on the learning dynamics of GNNs primarily focus on the convergence rates of learning algorithms under the interpolation regime (noise-free) and offer only a crude connection between these dynamics and the actual graph structure (e.g., maximum degree). This paper aims to bridge this gap by investigating the excessive risk (generalization performance) of learning algorithms in GNNs within the generalization regime (with noise). Specifically, we extend the conventional settings from the learning theory literature to the context of GNNs and examine how graph structure influences the performance of learning algorithms such as stochastic gradient descent (SGD) and Ridge regression. Our study makes several key contributions toward understanding the interplay between graph structure and learning in GNNs. First, we derive the excess risk profiles of SGD and Ridge regression in GNNs and connect these profiles to the graph structure through spectral graph theory. With this established framework, we further explore how different graph structures (regular vs. power-law) impact the performance of these algorithms through comparative analysis. Additionally, we extend our analysis to multi-layer linear GNNs, revealing an increasing non-isotropic effect on the excess risk profile, thereby offering new insights into the over-smoothing issue in GNNs from the perspective of learning algorithms. Our empirical results align with our theoretical predictions, emph{collectively showcasing a coupling relation among graph structure, GNNs and learning algorithms, and providing insights on GNN algorithm design and selection in practice.}

Problem

• Analyzing excessive risk of GNN learning algorithms with noise
• Investigating graph structure impact on SGD and Ridge regression
• Exploring coupling between graph topology and learning algorithm performance

Innovation

• Extends learning theory to GNN generalization regime
• Connects excess risk to graph structure spectrally
• Reveals non-isotropic effects in multi-layer GNNs