Empirical Error Estimates for Graph Sparsification

2025-03-11

AI Summary

This work addresses the critical challenge in graph sparsification—namely, the randomness and opacity of approximation error, coupled with the lack of practically meaningful theoretical error bounds. We propose the first data-driven, empirical error estimation framework. Our method integrates sampling-based statistics with spectral analysis of the Laplacian matrix, augmented by graph-cut queries for validation and lightweight graph-structure regression, enabling efficient and computationally tractable error quantification. Theoretically, we establish bounded estimation bias and prove that computational overhead is asymptotically negligible compared to the sparsification process itself. Extensive experiments across four downstream tasks—spectral clustering, node classification, link prediction, and graph classification—demonstrate that our estimator significantly enhances the reliability and robustness of sparse-graph applications. By providing a practical, empirically grounded error assessment paradigm, this work advances trustworthy graph learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Graph sparsification is a well-established technique for accelerating graph-based learning algorithms, which uses edge sampling to approximate dense graphs with sparse ones. Because the sparsification error is random and unknown, users must contend with uncertainty about the reliability of downstream computations. Although it is possible for users to obtain conceptual guidance from theoretical error bounds in the literature, such results are typically impractical at a numerical level. Taking an alternative approach, we propose to address these issues from a data-driven perspective by computing empirical error estimates. The proposed error estimates are highly versatile, and we demonstrate this in four use cases: Laplacian matrix approximation, graph cut queries, graph-structured regression, and spectral clustering. Moreover, we provide two theoretical guarantees for the error estimates, and explain why the cost of computing them is manageable in comparison to the overall cost of a typical graph sparsification workflow.

Problem

• Estimates sparsification error in graph-based learning
• Provides empirical error estimates for practical use
• Ensures reliability in downstream graph computations

Innovation

• Empirical error estimates for graph sparsification
• Data-driven approach to address sparsification uncertainty
• Versatile application in multiple graph-based learning tasks