Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNs

TL;DR

The paper introduces mu-ChebNet, a spectral GNN that learns a node-wise weight function to adapt the graph Laplacian, improving long-range information propagation without complex mechanisms.

Contribution

It proposes a learnable weighted Laplacian for spectral GNNs, enabling task-adaptive geometry and better long-range signal propagation with a lightweight architecture.

Findings

Improved performance on synthetic long-range reasoning tasks.

Enhanced results on real-world graph benchmarks.

Spectral analysis shows effective modulation of propagation dynamics.

Abstract

Long-range graph tasks are challenging for Graph Neural Networks (GNNs): global mechanisms such as attention or rewiring schemes can be computationally expensive, while deep local propagation is prone to vanishing gradients, oversmoothing, and oversquashing. The introduced mu-ChebNet architecture is a simple spectral GNN that learns a node-wise weight function mu before applying ChebNet-style filters. The learned weighting mu induces a modified graph Laplacian which effectively changes the propagation geometry without altering the graph topology. This task-dependent geometry promotes preferred routes for information propagation, thereby helping long-range signals avoid highly contractive bottlenecks, and obviating the need for repeated layer stacking. In practice, we replace the fixed graph Laplacian L by a learned operator L_mu, keeping the proposed mu-ChebNet architecture lightweight while making propagation task-adaptive. Furthermore, we provide a spectral analysis demonstrating how mu modulates propagation dynamics, and empirically observe improved performance on both synthetic long-range reasoning tasks and real-world graph benchmarks. The learned weight function is not only interpretable, but also offers a lightweight alternative to attention and rewiring for adaptive graph propagation.