# Graph Neural Networks Exponentially Lose Expressive Power for Node Classification

arXiv: https://arxiv.org/abs/1905.10947 · HTML: https://mlsanity.com/paper/1905.10947/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2019.

## Built on by (most cited first, 16 of 16)

- [DropMessage: Unifying Random Dropping for Graph Neural Networks](https://mlsanity.com/paper/2204.10037/) (2022)
- [Training Robust Graph Neural Networks with Topology Adaptive Edge Dropping](https://mlsanity.com/paper/2106.02892/) (2021)
- [SmoothGNN: Smoothing-aware GNN for Unsupervised Node Anomaly Detection](https://mlsanity.com/paper/2405.17525/) (2024)
- [Asymptotics of Network Embeddings Learned via Subsampling](https://mlsanity.com/paper/2107.02363/) (2021)
- [Stochastic Aggregation in Graph Neural Networks](https://mlsanity.com/paper/2102.12648/) (2021)
- [Training Graph Neural Networks by Graphon Estimation](https://mlsanity.com/paper/2109.01918/) (2021)
- [Graph Relation Aware Continual Learning](https://mlsanity.com/paper/2308.08259/) (2023)
- [Towards Agentic Intelligence for Materials Science](https://mlsanity.com/paper/2602.00169/) (2026)
- [Mario: Multimodal Graph Reasoning with Large Language Models](https://mlsanity.com/paper/2603.05181/) (2026)
- [Local-to-global Perspectives on Graph Neural Networks](https://mlsanity.com/paper/2306.06547/) (2023)
- [Implicit Graph Neural Diffusion Networks: Convergence, Generalization, and Over-Smoothing](https://mlsanity.com/paper/2308.03306/) (2023)
- [Towards Efficient Few-shot Graph Neural Architecture Search via Partitioning Gradient Contribution](https://mlsanity.com/paper/2506.01231/) (2025)
- [Less is More: on the Over-Globalizing Problem in Graph Transformers](https://mlsanity.com/paper/2405.01102/) (2024)
- [Information propagation dynamics in Deep Graph Networks](https://mlsanity.com/paper/2410.10464/) (2024)
- [Bayesian Neighborhood Adaptation for Graph Neural Networks](https://mlsanity.com/paper/2602.05358/) (2026)
- [NEMORA: Neural Equivariant Multipole Operators for Long-Range Atomistic Learning](https://mlsanity.com/paper/2610.10776/) (2026)

## Abstract (arXiv)

Graph Neural Networks (graph NNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up many layers and add non-lineality. To tackle this problem, we investigate the expressive power of graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is a popular graph NN variant, as a specific dynamical system. In the case of a GCN, we show that when its weights satisfy the conditions determined by the spectra of the (augmented) normalized Laplacian, its output exponentially approaches the set of signals that carry information of the connected components and node degrees only for distinguishing nodes. Our theory enables us to relate the expressive power of GCNs with the topological information of the underlying graphs inherent in the graph spectra. To demonstrate this, we characterize the asymptotic behavior of GCNs on the ErdHos -- R'enyi graph. We show that when the ErdHos -- R'enyi graph is sufficiently dense and large, a broad range of GCNs on it suffers from the "information loss" in the limit of infinite layers with high probability. Based on the theory, we provide a principled guideline for weight normalization of graph NNs. We experimentally confirm that the proposed weight scaling enhances the predictive performance of GCNs in real data. Code is available at https://github.com/delta2323/gnn-asymptotics.