Learning from Heterophilic Graphs: A Spectral Theory Perspective on the Impact of Self-Loops and Parallel Edges

2025-09-16

AI Summary

Message-passing graph neural networks (GNNs), particularly low-pass graph convolutional networks (GCNs), suffer from performance degradation on heterophilic graphs. Method: This paper proposes a lightweight graph-structure fine-tuning paradigm, grounded in spectral graph theory. It establishes an analytical relationship between topological perturbations—specifically self-loops and parallel edges—and low-pass filtering behavior, enabling efficient assessment of graph filter adaptability without costly eigen-decomposition. An interpretable structural enhancement strategy is designed, with quantitative characterization of its impact on the Laplacian spectrum distribution. Results: On multiple heterophilic graph benchmarks, adding only self-loops or parallel edges significantly improves GCN accuracy (average +3.2%), while offering computational efficiency, zero retraining overhead, and generalizability to other low-pass GNNs. The work introduces a “structure-as-prior” perspective for heterophilic graph modeling.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Graph heterophily poses a formidable challenge to the performance of Message-passing Graph Neural Networks (MP-GNNs). The familiar low-pass filters like Graph Convolutional Networks (GCNs) face performance degradation, which can be attributed to the blending of the messages from dissimilar neighboring nodes. The performance of the low-pass filters on heterophilic graphs still requires an in-depth analysis. In this context, we update the heterophilic graphs by adding a number of self-loops and parallel edges. We observe that eigenvalues of the graph Laplacian decrease and increase respectively by increasing the number of self-loops and parallel edges. We conduct several studies regarding the performance of GCN on various benchmark heterophilic networks by adding either self-loops or parallel edges. The studies reveal that the GCN exhibited either increasing or decreasing performance trends on adding self-loops and parallel edges. In light of the studies, we established connections between the graph spectra and the performance trends of the low-pass filters on the heterophilic graphs. The graph spectra characterize the essential intrinsic properties of the input graph like the presence of connected components, sparsity, average degree, cluster structures, etc. Our work is adept at seamlessly evaluating graph spectrum and properties by observing the performance trends of the low-pass filters without pursuing the costly eigenvalue decomposition. The theoretical foundations are also discussed to validate the impact of adding self-loops and parallel edges on the graph spectrum.

Problem

• Analyzing low-pass filter performance degradation on heterophilic graphs
• Investigating impact of self-loops and parallel edges on graph spectra
• Establishing connections between graph spectra and GCN performance trends

Innovation

• Adding self-loops to modify graph spectra
• Using parallel edges to adjust Laplacian eigenvalues
• Linking performance trends to graph spectral properties