# A Spectral Theory of Normalized Corrected GNN Propagation

arXiv: https://arxiv.org/abs/2606.23572 · HTML: https://mlsanity.com/paper/2606.23572/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 40 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 30 of 34)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [PairNorm: Tackling Oversmoothing in GNNs](https://mlsanity.com/paper/1909.12223/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Contextual Stochastic Block Models](https://mlsanity.com/paper/1807.09596/) (2018)
- [Not too little, not too much: a theoretical analysis of graph (over)smoothing](https://mlsanity.com/paper/2205.12156/) (2022)
- [Graph Convolution for Semi-Supervised Classification: Improved Linear Separability and Out-of-Distribution Generalization](https://mlsanity.com/paper/2102.06966/) (2021)
- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [Stochastic Blockmodels meet Graph Neural Networks](https://mlsanity.com/paper/1905.05738/) (2019)
- [Residual Connections and Normalization Can Provably Prevent Oversmoothing in GNNs](https://mlsanity.com/paper/2406.02997/) (2024)
- [Effects of Graph Convolutions in Multi-layer Networks](https://mlsanity.com/paper/2204.09297/) (2022)
- [Contextual Stochastic Block Model: Sharp Thresholds and Contiguity](https://mlsanity.com/paper/2011.09841/) (2020)
- [Generalization Bounds for Message Passing Networks on Mixture of Graphons](https://mlsanity.com/paper/2404.03473/) (2024)
- [Optimality of Message-Passing Architectures for Sparse Graphs](https://mlsanity.com/paper/2305.10391/) (2023)
- [Mitigating Oversmoothing Through Reverse Process of GNNs for Heterophilic Graphs](https://mlsanity.com/paper/2403.10543/) (2024)
- [Almost Surely Asymptotically Constant Graph Neural Networks](https://mlsanity.com/paper/2403.03880/) (2024)
- [Oversmoothing Alleviation in Graph Neural Networks: A Survey and Unified View](https://mlsanity.com/paper/2405.01663/) (2024)
- [Mamba-Based Graph Convolutional Networks: Tackling Over-smoothing with Selective State Space](https://mlsanity.com/paper/2501.15461/) (2025)
- [KrawtchoukNet: A Unified GNN Solution for Heterophily and Over-smoothing with Adaptive Bounded Polynomials](https://mlsanity.com/paper/2511.15327/) (2025)

## Abstract (arXiv)

We develop a spectral theory for 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≥ ClogB 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.