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

• URL: http://arxiv.org/abs/2509.13139v1
• Date: Tue, 16 Sep 2025 14:54:54 GMT
• Title: Learning from Heterophilic Graphs: A Spectral Theory Perspective on the Impact of Self-Loops and Parallel Edges
• Authors: Kushal Bose, Swagatam Das,
• Abstract summary: Graph heterophily poses a formidable challenge to the performance of Message-passing Graph Neural Networks (MP-GNNs)<n>The performance of the low-pass filters on heterophilic graphs still requires an in-depth analysis.<n>We conduct several studies regarding the performance of GCN on various benchmark heterophilic networks by adding either self-loops or parallel edges.
• Score: 21.49824162847618
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Task-driven Heterophilic Graph Structure Learning [30.767828037086844]
  Graph neural networks (GNNs) often struggle to learn discriminative node representations for heterophilic graphs.<n>We propose frequency-guided graph structure learning (FgGSL), an end-to-end graph inference framework.
  arXiv  Detail & Related papers  (2025-12-29T11:59:16Z)
• Predicting the Performance of Graph Convolutional Networks with Spectral Properties of the Graph Laplacian [3.7958475517455947]
  A common observation in the Graph Convolutional Network (GCN) literature is that stacking GCN layers may or may not result in better performance on tasks like node classification and edge prediction.<n>We have found empirically that a graph's algebraic connectivity, which is known as the Fiedler value, is a good predictor of GCN performance.
  arXiv  Detail & Related papers  (2025-08-18T15:13:13Z)
• A Spectral Interpretation of Redundancy in a Graph Reservoir [51.40366905583043]
  This work revisits the definition of the reservoir in the Multiresolution Reservoir Graph Neural Network (MRGNN)<n>It proposes a variant based on a Fairing algorithm originally introduced in the field of surface design in computer graphics.<n>The core contribution of the paper lies in the theoretical analysis of the algorithm from a random walks perspective.
  arXiv  Detail & Related papers  (2025-07-17T10:02:57Z)
• It Takes a Graph to Know a Graph: Rewiring for Homophily with a Reference Graph [19.222317334613162]
  Graph Neural Networks (GNNs) excel at analyzing graph-structured data but struggle on heterophilic graphs, where connected nodes often belong to different classes.<n>We provide theoretical foundations linking edge homophily, GNN embedding smoothness, and node classification performance.<n>We introduce a rewiring framework that increases graph homophily using a reference graph, with theoretical guarantees on the homophily of the rewired graph.
  arXiv  Detail & Related papers  (2025-05-18T13:28:56Z)
• Dual-Frequency Filtering Self-aware Graph Neural Networks for Homophilic and Heterophilic Graphs [60.82508765185161]
  We propose Dual-Frequency Filtering Self-aware Graph Neural Networks (DFGNN) DFGNN integrates low-pass and high-pass filters to extract smooth and detailed topological features. It dynamically adjusts filtering ratios to accommodate both homophilic and heterophilic graphs.
  arXiv  Detail & Related papers  (2024-11-18T04:57:05Z)
• Homophily-Related: Adaptive Hybrid Graph Filter for Multi-View Graph Clustering [29.17784041837907]
  We propose Adaptive Hybrid Graph Filter for Multi-View Graph Clustering (AHGFC) AHGFC learns the node embedding based on the graph joint aggregation matrix. Experimental results show that our proposed model performs well on six datasets containing homophilous and heterophilous graphs.
  arXiv  Detail & Related papers  (2024-01-05T07:27:29Z)
• Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs [122.06927400759021]
  We study the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold. We prove non-asymptotic error bounds showing that convolutional filters and neural networks on these graphs converge to convolutional filters and neural networks on the continuous manifold.
  arXiv  Detail & Related papers  (2023-05-29T08:27:17Z)
• Graph Contrastive Learning under Heterophily via Graph Filters [51.46061703680498]
  Graph contrastive learning (CL) methods learn node representations in a self-supervised manner by maximizing the similarity between the augmented node representations obtained via a GNN-based encoder. In this work, we propose an effective graph CL method, namely HLCL, for learning graph representations under heterophily. Our extensive experiments show that HLCL outperforms state-of-the-art graph CL methods on benchmark datasets with heterophily, as well as large-scale real-world graphs, by up to 7%, and outperforms graph supervised learning methods on datasets with heterophily by up to 10%.
  arXiv  Detail & Related papers  (2023-03-11T08:32:39Z)
• Beyond Low-pass Filtering: Graph Convolutional Networks with Automatic Filtering [61.315598419655224]
  We propose Automatic Graph Convolutional Networks (AutoGCN) to capture the full spectrum of graph signals. While it is based on graph spectral theory, our AutoGCN is also localized in space and has a spatial form.
  arXiv  Detail & Related papers  (2021-07-10T04:11:25Z)
• An Experimental Study of the Transferability of Spectral Graph Networks [5.736353542430439]
  Spectral graph convolutional networks are generalizations of standard convolutional networks for graph-structured data using the Laplacian operator. Recent works have proved the stability of spectral filters under graph benchmarks.
  arXiv  Detail & Related papers  (2020-12-18T14:15:07Z)
• Dirichlet Graph Variational Autoencoder [65.94744123832338]
  We present Dirichlet Graph Variational Autoencoder (DGVAE) with graph cluster memberships as latent factors. Motivated by the low pass characteristics in balanced graph cut, we propose a new variant of GNN named Heatts to encode the input graph into cluster memberships.
  arXiv  Detail & Related papers  (2020-10-09T07:35:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.