On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learning

TL;DR

This paper analyzes the issues of over-smoothing and over-squashing in GNNs through the lens of vanishing gradients, proposing a state-space formulation that mitigates these problems without extra parameters.

Contribution

It offers a unified view of GNN problems via control theory, interprets GNNs as recurrent models, and introduces a simple formulation that alleviates over-smoothing and over-squashing.

Findings

GNNs are prone to extreme gradient vanishing after few layers.

Over-smoothing is linked to vanishing gradients.

Graph rewiring combined with gradient mitigation alleviates over-squashing.

Abstract

Graph Neural Networks (GNNs) are models that leverage the graph structure to transmit information between nodes, typically through the message-passing operation. While widely successful, this approach is well known to suffer from the over-smoothing and over-squashing phenomena, which result in representational collapse as the number of layers increases and insensitivity to the information contained at distant and poorly connected nodes, respectively. In this paper, we present a unified view of these problems through the lens of vanishing gradients, using ideas from linear control theory for our analysis. We propose an interpretation of GNNs as recurrent models and empirically demonstrate that a simple state-space formulation of a GNN effectively alleviates over-smoothing and over-squashing at no extra trainable parameter cost. Further, we show theoretically and empirically that (i) GNNs are by design prone to extreme gradient vanishing even after a few layers; (ii) Over-smoothing is directly related to the mechanism causing vanishing gradients; (iii) Over-squashing is most easily alleviated by a combination of graph rewiring and vanishing gradient mitigation. We believe our work will help bridge the gap between the recurrent and graph neural network literature and will unlock the design of new deep and performant GNNs.