Graph Coarsening with Message-Passing Guarantees (2405.18127v1)

Abstract: Graph coarsening aims to reduce the size of a large graph while preserving some of its key properties, which has been used in many applications to reduce computational load and memory footprint. For instance, in graph machine learning, training Graph Neural Networks (GNNs) on coarsened graphs leads to drastic savings in time and memory. However, GNNs rely on the Message-Passing (MP) paradigm, and classical spectral preservation guarantees for graph coarsening do not directly lead to theoretical guarantees when performing naive message-passing on the coarsened graph. In this work, we propose a new message-passing operation specific to coarsened graphs, which exhibit theoretical guarantees on the preservation of the propagated signal. Interestingly, and in a sharp departure from previous proposals, this operation on coarsened graphs is oriented, even when the original graph is undirected. We conduct node classification tasks on synthetic and real data and observe improved results compared to performing naive message-passing on the coarsened graph.

• The paper introduces a novel message-passing coarsening method that approximates original graph propagation using an oriented paradigm.
• It demonstrates enhanced node classification and spectral preservation by maintaining low RSA error bounds on both synthetic and real-world datasets.
• The study offers practical techniques for scalable GNN training while motivating future research into advanced coarsening algorithms.

Overview of "Graph Coarsening with Message-Passing Guarantees"

This paper introduces a novel approach to graph coarsening with a specific focus on ensuring message-passing guarantees, which are pivotal for the efficient training and inference of Graph Neural Networks (GNNs). Graph coarsening refers to the process of reducing a graph's size while preserving its essential structural properties. This reduction aims to alleviate computational and memory limitations, especially relevant for large-scale graphs encountered in modern-day data science and machine learning applications.

Key Contributions

The authors propose a novel message-passing operation tailored for coarsened graphs, proposing a propagation matrix denoted as $S^$, defined as $S^ = Q S Q^+$, where $Q$ denotes the coarsening matrix and $S$ denotes the original propagation matrix on the complete graph. This proposal diverges from traditional symmetric propagation matrices, introducing an oriented message-passing paradigm that accommodates undirected graphs. The primary advantage of this approach lies in its ability to approximate the message-passing process on the original graph using the coarsened graph representation, promising computational efficiency without compromising on the model's performance.

Numerical Results and Claims

The paper provides empirical evidence of the advantages of the proposed method by conducting node classification experiments on both synthetic and real-world datasets. These experiments reveal that the newly defined $S^$ significantly improves classification performance over naive message-passing strategies on coarsened graphs. Theoretical analysis supports these results, establishing a link between spectral preservation, measured by the Restricted Spectral Approximation (RSA) constant, and message-passing guarantees.

The adoption of the proposed message-passing operation translates spectral preservation properties of graph Laplacians into message-passing guarantees for GNN training. The analysis hinges on maintaining low RSA errors, thereby extending spectral approximation guarantees to message-passing guarantees. Further, the bounds on training loss underscore the practical utility of the proposed operation by ensuring that the risk associated with training on the coarsened graph remains close to that of the original graph.

Implications and Future Directions

From a practical standpoint, this work proffers substantial improvements in the training efficiency of GNNs. By adopting the proposed coarsening technique, practitioners can handle larger graphs that would otherwise be computationally prohibitive, thereby facilitating more scalable machine learning applications. Furthermore, the paper lays the groundwork for a novel class of coarsening algorithms that prioritize message-passing fidelity, potentially inspiring future research to develop scalable algorithms that satisfy the requisite spectral guarantees.

Theoretically, the results pose intriguing questions regarding the broader applicability of oriented message-passing operations in other graph-based learning tasks. A promising area for future exploration would be the extension of these techniques to non-linear GNN architectures, for which the current analysis chiefly applies to linear models like the Simplified Graph Convolution (SGC). Additionally, investigating more sophisticated coarsening methods and other graph transformations that preserve high-frequency signals could further enhance the model's flexibility and performance.

Overall, this paper provides a rigorous examination of the interaction between graph coarsening and message-passing, offering a robust framework for researchers seeking to optimize GNN operations on coarsened graphs. Its implications stretch across both theoretical insights and practical methodologies, contributing to the enhanced efficiency of graph-based learning models.