Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Huseyin Goksu

TL;DR

This work proposes `KrawtchoukNet', a GNN filter based on the discrete Krawtchouk polynomials, which is the first GNN filter whose recurrence coefficients areherently bounded, making it exceptionally robust to over-smoothing.

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.

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

TL;DR

This work proposes `KrawtchoukNet', a GNN filter based on the discrete Krawtchouk polynomials, which is the first GNN filter whose recurrence coefficients areherently bounded, making it exceptionally robust to over-smoothing.

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.

Paper Structure

This paper contains 21 sections, 4 equations, 3 figures, 5 tables.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. Spectral Filter Design in GNNs
  4. Solutions for Heterophily
  5. Solutions for Over-smoothing
  6. PROPOSED METHOD: KRAWTCHOUKNET
  7. Krawtchouk Polynomials
  8. Parameterization for a Unified Solution
  9. Spectral Analysis of the Krawtchouk Filter
  10. The KrawtchoukConv Layer
  11. The KrawtchoukConv Layer
  12. EXPERIMENTS
  13. Experimental Setup
  14. Performance on Homophilic Graphs (K=3)
  15. Hypothesis 1: Performance on Heterophilic Graphs
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Training dynamics comparison (K=3, H=16). Top 3 rows (homophilic): All models are stable. Bottom 2 rows (heterophilic): 'GAT' and 'APPNP' fail to converge or are highly unstable, while the adaptive polynomial filters ('MeixnerNet', 'KrawtchoukNet') converge quickly to a high, stable accuracy. This is the visual proof for Hypothesis 1.
  • Figure 2: $K$ (Polynomial Degree) vs. Test Accuracy (PubMed). 'ChebyNet' (blue) collapses at $K=3$. 'MeixnerNet' (orange), with $O(k^2)$ coefficients, collapses at $K=19$. 'KrawtchoukNet' (green) is stable by design and performance increases.
  • Figure 3: Hidden Dimension (Capacity) vs. Test Accuracy (PubMed, K=10). 'KrawtchoukNet''s (green) superior performance is robust across all tested capacities.