LaguerreNet: Advancing a Unified Solution for Heterophily and Over-smoothing with Adaptive Continuous Polynomials

Spectral Graph Neural Networks (GNNs) suffer from two critical limitations: poor performance on "heterophilic" graphs and performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters (e.g., ChebyNet). While adaptive polynomial filters, such as the discrete MeixnerNet, have emerged as a potential unified solution, their extension to the continuous domain and stability with unbounded coefficients remain open questions. In this work, we propose `LaguerreNet`, a novel GNN filter based on continuous Laguerre polynomials. `LaguerreNet` learns the filter's spectral shape by making its core alpha parameter trainable, thereby advancing the adaptive polynomial approach. We solve the severe O(k^2) numerical instability of these unbounded polynomials using a `LayerNorm`-based stabilization technique. We demonstrate experimentally that this approach is highly effective: 1) `LaguerreNet` achieves state-of-the-art results on challenging heterophilic benchmarks. 2) It is exceptionally robust to over-smoothing, with performance peaking at K=10, an order of magnitude beyond where ChebyNet collapses.