A NON-ASYMPTOTIC ANALYSIS OF OVERSMOOTHING IN GRAPH NEURAL NETWORKS

Abstract

Oversmoothing is a central challenge of building more powerful Graph Neural Networks (GNNs). While previous works have only demonstrated that oversmoothing is inevitable when the number of graph convolutions tends to infinity, in this paper, we precisely characterize the mechanism behind the phenomenon via a non-asymptotic analysis. Specifically, we distinguish between two different effects when applying graph convolutions-an undesirable mixing effect that homogenizes node representations in different classes, and a desirable denoising effect that homogenizes node representations in the same class. By quantifying these two effects on random graphs sampled from the Contextual Stochastic Block Model (CSBM), we show that oversmoothing happens once the mixing effect starts to dominate the denoising effect, and the number of layers required for this transition is O(log N/ log(log N )) for sufficiently dense graphs with N nodes. We also extend our analysis to study the effects of Personalized PageRank (PPR), or equivalently, the effects of initial residual connections on oversmoothing. Our results suggest that while PPR mitigates oversmoothing at deeper layers, PPR-based architectures still achieve their best performance at a shallow depth and are outperformed by the graph convolution approach on certain graphs. Finally, we support our theoretical results with numerical experiments, which further suggest that the oversmoothing phenomenon observed in practice can be magnified by the difficulty of optimizing deep GNN models.

1. INTRODUCTION

Graph Neural Networks (GNNs) are a powerful framework for learning with graph-structured data (Gori et al., 2005; Scarselli et al., 2009; Bruna et al., 2014; Duvenaud et al., 2015; Defferrard et al., 2016; Battaglia et al., 2016; Li et al., 2016) . Most GNN models are built by stacking graph convolutions or message-passing layers (Gilmer et al., 2017) , where the representation of each node is computed by recursively aggregating and transforming the representations of its neighboring nodes. The most representative and popular example is the Graph Convolutional Network (GCN) (Kipf & Welling, 2017) , which has demonstrated success in node classification, a primary graph task which asks for node labels and identifies community structures in real graphs. Despite these achievements, the choice of depth for these GNN models remains an intriguing question. GNNs often achieve optimal classification performance when networks are shallow. Many widely used GNNs such as the GCN are no deeper than 4 layers (Kipf & Welling, 2017; Wu et al., 2019) , and it has been observed that for deeper GNNs, repeated message-passing makes node representations in different classes indistinguishable and leads to lower node classification accuracy-a phenomenon known as oversmoothing (Kipf & Welling, 2017; Li et al., 2018; Klicpera et al., 2019; Wu et al., 2019; Oono & Suzuki, 2020; Chen et al., 2020a; b; Keriven, 2022) . Through the insight that graph convolutions can be regarded as low-pass filters on graph signals, prior studies have established that oversmoothing is inevitable when the number of layers in a GNN increases to infinity (Li et al., 2018; Oono & Suzuki, 2020) . However, these asymptotic analyses do not fully explain the rapid occurrence of oversmoothing when we increase the network depth, let alone the fact that for some datasets, having no graph convolution is even optimal (Liu et al., 2021) . These observations motivate the following key questions about oversmoothing in GNNs:

