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arxiv_ml 95% Match Short Note / Survey Researchers in Graph Neural Networks,Theoretical machine learning researchers,Data scientists working with graph data 1 week ago

A Short Note on Upper Bounds for Graph Neural Operator Convergence Rate

graph-neural-networks › graph-learning
📄 Abstract

Abstract: Graphons, as limits of graph sequences, provide a framework for analyzing the asymptotic behavior of graph neural operators. Spectral convergence of sampled graphs to graphons yields operator-level convergence rates, enabling transferability analyses of GNNs. This note summarizes known bounds under no assumptions, global Lipschitz continuity, and piecewise-Lipschitz continuity, highlighting tradeoffs between assumptions and rates, and illustrating their empirical tightness on synthetic and real data.
Authors (2)
Roxanne Holden
Luana Ruiz
Submitted
October 23, 2025
arXiv Category
stat.ML
arXiv PDF

Key Contributions

This note summarizes and analyzes known upper bounds for the convergence rate of graph neural operators based on spectral convergence to graphons. It highlights the trade-offs between different assumptions (e.g., Lipschitz continuity) and the resulting convergence rates, providing a theoretical foundation for understanding GNN transferability.

Business Value

Enhances the theoretical understanding and reliability of graph-based machine learning models, leading to more robust and predictable performance in applications involving network data.

Paper Metadata

Innovation Type

Theoretical Analysis

Deployment Feasibility

High, as it provides theoretical foundations that guide the development and application of GNNs, rather than proposing a new deployable system.

Limitations Addressed

Lack of clear theoretical understanding of the convergence rates and transferability of Graph Neural Networks (GNNs) and Graph Neural Operators (GNOs) in the limit of large graphs.

Performance Gains

Provides theoretical guarantees and insights into the performance and transferability of GNNs/GNOs.

Technical Tags

graph neural operatorsgraphonsspectral convergenceoperator convergencegraph neural networks (GNNs)asymptotic behaviortransferabilityLipschitz continuitytheoretical bounds

Research Topics

Graph Neural NetworksGraph TheoryTheoretical Machine LearningOperator TheoryNetwork Analysis

Methods & Architectures

Analysis of spectral convergence of sampled graphs to graphonsDerivation of operator-level convergence ratesSummarizing known bounds under different assumptions (no assumptions, global Lipschitz, piecewise-Lipschitz) Graph Neural OperatorsGraph Neural Networks (GNNs)

Applications & Tasks

Network Analysis Machine Learning on Graphs Data Science Analyzing asymptotic behavior of graph operatorsUnderstanding convergence rates of GNNsQuantifying transferability of GNNs Theoretical analysis of graph neural operatorsEstablishing convergence bounds for GNNs

Datasets & Benchmarks

Datasets
Synthetic data, Real data
Convergence rate bounds

Related Fields

Graph TheoryMachine LearningNetwork ScienceFunctional AnalysisTheoretical Computer Science

Keywords

Graph Neural NetworksGraph Neural OperatorsGraphonsConvergence RateSpectral ConvergenceLipschitz ContinuityTransferabilityTheoretical AnalysisAsymptotic BehaviorMachine Learning on Graphs

Academic Context

#Graph Neural Networks#Graph Theory#Theoretical Machine Learning#Operator Theory#Network Analysis

Commercial Potential

Target Industries
Social MediaRecommender SystemsDrug DiscoveryCybersecurity (network analysis)
Use Case Examples
Predicting properties of large-scale networksAnalyzing the behavior of GNNs on evolving graphsEnsuring reliable performance of GNNs across different graph sizes
Competitive Edge
Provides a foundational theoretical framework for understanding the behavior and limitations of graph neural operators, complementing empirical studies.
Market Opportunity
N/A
Revenue Models
N/A

Resource Requirements

Compute Needs
Not applicable (theoretical paper).
Data Requirements
Not applicable (theoretical paper).
Deployment Constraints
Not applicable (theoretical paper).
Scalability
Focuses on the theoretical scalability of GNNs/GNOs as graph size increases.

Production Readiness

Maturity Level

Theoretical Foundation

Time to Market

N/A

Patent Potential

Very low (theoretical analysis).

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