KrawtchoukNet: A Unified GNN Solution for Heterophily and Over-smoothing with Adaptive Bounded Polynomials

TL;DR

KrawtchoukNet introduces a spectral GNN filter using bounded discrete Krawtchouk polynomials with learnable shape, effectively addressing heterophily and over-smoothing issues, and achieving state-of-the-art results on challenging benchmarks.

Contribution

The paper proposes KrawtchoukNet, a novel GNN filter with inherently bounded recurrence coefficients and a learnable shape parameter, unifying solutions for heterophily and over-smoothing.

Findings

Achieves state-of-the-art results on heterophilic benchmarks.

Robust to over-smoothing at high polynomial degrees.

Outperforms standard GNNs like GAT and APPNP.

Abstract

Spectral Graph Neural Networks (GNNs) based on polynomial filters, such as ChebyNet, suffer from two critical limitations: 1) performance collapse on "heterophilic" graphs and 2) performance collapse at high polynomial degrees (K), known as over-smoothing. Both issues stem from the static, low-pass nature of standard filters. In this work, we propose `KrawtchoukNet`, a GNN filter based on the discrete Krawtchouk polynomials. We demonstrate that `KrawtchoukNet` provides a unified solution to both problems through two key design choices. First, by fixing the polynomial's domain N to a small constant (e.g., N=20), we create the first GNN filter whose recurrence coefficients are \textit{inherently bounded}, making it exceptionally robust to over-smoothing (achieving SOTA results at K=10). Second, by making the filter's shape parameter p learnable, the filter adapts its spectral response to the graph data. We show this adaptive nature allows `KrawtchoukNet` to achieve SOTA performance on challenging heterophilic benchmarks (Texas, Cornell), decisively outperforming standard GNNs like GAT and APPNP.