Geometry-Induced Diffusion on Graphs: Learning Task-Adaptive Propagation for GNNs

Long-range information propagation remains challenging in Graph Neural Networks (GNNs), where repeated local aggregation can lead to oversmoothing and oversquashing. Existing approaches often rely on global attention mechanisms or modify the graph topology through rewiring, potentially increasing computational or architectural complexity. We introduce a learnable node-wise function $\mu$ that induces a weighted graph Laplacian $L_{\mu }$, allowing the propagation geometry to adapt to the task while preserving the original graph topology. This construction is not specific to a particular GNN architecture; here, we instantiate it within Chebyshev spectral convolutions, yielding $\mu$-ChebNet. We show that learning $\mu$ reshapes the spectrum of the diffusion operator, modifying how long different graph signal components persist during propagation and thus the timescale over which oversmoothing occurs. On a controlled bottleneck example, we further show that reweighting the graph through $\mu$ can reduce effective resistance, a quantity previously connected to oversquashing. Experiments on synthetic long-range tasks and real-world graph benchmarks show improved predictive performance while retaining the computational structure of sparse polynomial graph filtering.