A Spectral Theory of Normalized Corrected GNN Propagation

2026-06-22

AI Summary

This work addresses the over-smoothing problem in deep graph neural networks (GNNs), which leads to the loss of class-discriminative signals during multi-layer propagation. Drawing on spectral graph theory, the authors propose a normalized correction mechanism for GNN message passing that removes the degree-stationary component from the symmetrically normalized adjacency matrix, thereby enhancing the preservation of class information across layers. Under the dense multi-logarithmic stochastic block model, they establish—for the first time—that exact recovery of binary community structure is achievable with high probability after $O(\log n)$ propagation layers, and provide partial recovery guarantees for the multi-class setting. The theoretical analysis, grounded in the contextual stochastic block model and signal-to-noise ratio conditions, elucidates how graph signals and node feature noise jointly influence the efficacy of deep propagation. Empirical results on both synthetic and real-world datasets validate the proposed approach.

Abstract

We develop a spectral theory for \emph{normalized corrected GNN propagation}. The object of study is the symmetric normalized adjacency with its degree-stationary component removed, matching the normalization used by standard GCN-style models while isolating the stationary direction most directly tied to oversmoothing. The central theoretical question is whether this corrected normalized operator preserves class-discriminative signal after many propagation layers. Our main result is a high-probability exact-recovery theorem for the binary Contextual Stochastic Block Model after \(k=O(\log n)\) propagation steps in the dense polylogarithmic regime \(p\ge C\log^B n/n\), for any fixed \(B>4\), under explicit graph-signal and feature-SNR conditions. We also establish a multi-class partial recovery theorem showing contraction toward class centers for most nodes. Synthetic and real node-classification experiments are included as empirical checks of the theory's predicted dependence on depth, graph signal, and feature noise.

Problem

• oversmoothing
• graph neural networks
• spectral theory
• normalized propagation
• class-discriminative signal

Innovation

• normalized corrected GNN propagation
• spectral theory
• oversmoothing
• Contextual Stochastic Block Model
• exact recovery