# A Simple Spectral Failure Mode for Graph Convolutional Networks

arXiv: https://arxiv.org/abs/2010.13152 · HTML: https://mlsanity.com/paper/2010.13152/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2020.

## Built on by (most cited first, 4 of 4)

- [Tree Decomposed Graph Neural Network](https://mlsanity.com/paper/2108.11022/) (2021)
- [Principal Graph Encoder Embedding and Principal Community Detection](https://mlsanity.com/paper/2501.14939/) (2025)
- [Detection of Model-based Planted Pseudo-cliques in Random Dot Product Graphs by the Adjacency Spectral Embedding and the Graph Encoder Embedding](https://mlsanity.com/paper/2312.11054/) (2023)
- [Graph Neural Networks Powered by Encoder Embedding for Improved Node Learning](https://mlsanity.com/paper/2507.11732/) (2025)

## Abstract (arXiv)

Neural networks have achieved remarkable successes in machine learning tasks. This has recently been extended to graph learning using neural networks. However, there is limited theoretical work in understanding how and when they perform well, especially relative to established statistical learning techniques such as spectral embedding. In this short paper, we present a simple generative model where unsupervised graph convolutional network fails, while the adjacency spectral embedding succeeds. Specifically, unsupervised graph convolutional network is unable to look beyond the first eigenvector in certain approximately regular graphs, thus missing inference signals in non-leading eigenvectors. The phenomenon is demonstrated by visual illustrations and comprehensive simulations.