Flexible Diffusion Scopes with Parameterized Laplacian for Heterophilic Graph Learning

TL;DR

The paper tackles limited diffusion reach in GNNs on heterophilic graphs by introducing a parameterized normalized Laplacian $L^{(\alpha,\gamma)}$ that flexibly controls diffusion distance. It proves that spectral distance can serve as a surrogate for diffusion distance and leverages this insight to build two diffusion-flexible GNNs, PD-GCN and PD-GAT, along with a topology-guided rewiring mechanism. The approach unifies and extends existing Laplacian design, demonstrates stronger performance across synthetic and real-world heterophily benchmarks, and provides interpretable parameters $\alpha$ and $\gamma$ to adapt diffusion to graph homophily. This yields a practical framework for capturing global information in non-Euclidean graphs while remaining compatible with standard GNN architectures, with notable improvements on heterophilic datasets. The work thus offers a principled, scalable path to robust long-range information propagation in graphs.

Abstract

The ability of Graph Neural Networks (GNNs) to capture long-range and global topology information is limited by the scope of conventional graph Laplacian, leading to unsatisfactory performance on some datasets, particularly on heterophilic graphs. To address this limitation, we propose a new class of parameterized Laplacian matrices, which provably offers more flexibility in controlling the diffusion distance between nodes than the conventional graph Laplacian, allowing long-range information to be adaptively captured through diffusion on graph. Specifically, we first prove that the diffusion distance and spectral distance on graph have an order-preserving relationship. With this result, we demonstrate that the parameterized Laplacian can accelerate the diffusion of long-range information, and the parameters in the Laplacian enable flexibility of the diffusion scopes. Based on the theoretical results, we propose topology-guided rewiring mechanism to capture helpful long-range neighborhood information for heterophilic graphs. With this mechanism and the new Laplacian, we propose two GNNs with flexible diffusion scopes: namely the Parameterized Diffusion based Graph Convolutional Networks (PD-GCN) and Graph Attention Networks (PD-GAT). Synthetic experiments reveal the high correlations between the parameters of the new Laplacian and the performance of parameterized GNNs under various graph homophily levels, which verifies that our new proposed GNNs indeed have the ability to adjust the parameters to adaptively capture the global information for different levels of heterophilic graphs. They also outperform the state-of-the-art (SOTA) models on 6 out of 7 real-world benchmark datasets, which further confirms their superiority.

Figures (1)

• Figure 1: Experiments on synthetic graphs with varying levels of homophily. The y-axis represents averaged test accuracy. (a) Comparison of PD-GCN and PD-GAT with baseline models. The x-axis denotes the graph homophily level, where a larger value indicates a more homophilic graph. The solid lines for the proposed models represent performance with the optimal $\gamma$, while the dashed lines show performance with the worst $\gamma$. (b) Each line corresponds to synthetic graphs with a specific homophily level, illustrating the relationship between the performance of a one-layered PD-GCN and $\gamma$. The line color indicates the homophily coefficient, with blue representing low homophily and yellow indicating high homophily. The dot markers denote the optimal $\gamma$ for each homophily level.

Theorems & Definitions (7)

• Definition 3.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• proof
• proof
• proof