The Oversmoothing Problem

Stacking multiple Graph Neural Network (GNN) layers, while seemingly a natural way to increase model capacity and capture longer-range dependencies, often leads to a counter-intuitive degradation in performance. This phenomenon is widely known as oversmoothing. It refers to the tendency for node representations across a graph to become increasingly similar, eventually converging towards indistinct values, as they pass through successive GNN layers.

Understanding the Mechanism

At its core, oversmoothing is an inherent consequence of the standard message-passing mechanism employed by many GNNs. Recall the basic update rule for a node $v$ at layer $k+1$ :

h_v^{(k+1)} = \sigma \left( \text{UPDATE}^{(k)} \left( h_v^{(k)}, \text{AGGREGATE}^{(k)} \left( \{ h_u^{(k)} : u \in \mathcal{N}(v) \} \right) \right) \right)

The AGGREGATE function typically involves some form of averaging or weighted sum of neighbors' features from the previous layer, $h_u^{(k)}$ . For instance, in a simplified Graph Convolutional Network (GCN) layer, the aggregation can be seen as applying a normalized adjacency matrix $A_{norm}$ (like $D^{-1/2}AD^{-1/2}$ ) to the feature matrix $H^{(k)}$ :

H^{(k+1)} = \sigma(A_{norm} H^{(k)} W^{(k)})

This operation effectively averages the features of a node with those of its neighbors. When this averaging process is repeated over many layers ( $k \rightarrow \infty$ ), the features of nodes within the same connected component tend to converge. Intuitively, each propagation step mixes the features of adjacent nodes. After $k$ steps, a node's representation becomes influenced by nodes up to $k$ hops away. As $k$ increases, the receptive field of each node expands to potentially cover a large portion, or even all, of its connected component. This repeated local averaging acts like a low-pass filter on the graph signal (node features), smoothing out the variations between nodes.

Consider the analogy of a random walk on the graph. Each message-passing step is akin to a step in a random walk. As the number of steps increases, the probability distribution of the walker's position tends towards a stationary distribution, which often depends only on global graph properties like node degrees, not the starting node's unique characteristics. Similarly, repeated aggregation washes out the specific local neighborhood information that distinguishes one node from another.

Consequences for Learning

The primary consequence of oversmoothing is the loss of discriminative power in node representations. If all nodes within a connected component have nearly identical embeddings after several layers, the GNN struggles to perform tasks that rely on distinguishing between these nodes, such as node classification or link prediction.

Reduced Performance: Deep GNNs may perform worse than shallow GNNs (e.g., 2-3 layers) on many benchmark datasets. Adding more layers provides diminishing returns and can even hurt accuracy.
Loss of Local Information: While deeper layers aim to capture global structure, oversmoothing causes them to lose the fine-grained local structure information essential for many graph tasks.
Indistinguishable Embeddings: Nodes belonging to different classes might end up with very similar embeddings if they are structurally close within the graph, making classification challenging.

Let's illustrate this convergence. Imagine a small graph where nodes initially have distinct features (represented by colors).

Initially distinct node features (colors) become homogenized after many message-passing layers due to repeated neighborhood averaging.

This homogenization means the network effectively loses the ability to leverage node-specific or local structural information encoded in the initial layers. While expanding the receptive field is desirable, oversmoothing prevents the model from effectively utilizing the information gathered from distant nodes without losing the local context.

Understanding this phenomenon is important for designing effective deep GNN architectures and training strategies. Techniques discussed later in this chapter, such as residual connections, jumping knowledge, or attention mechanisms, are specifically designed to combat this excessive smoothing and allow for the construction of deeper, more expressive GNN models.

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References

DeeperGCN: All Deep GCNs Are Not Created Equal, Qinqing Li, Zhichao Han, Xiaowen Wu, 2020 Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 34 (AAAI Press) DOI: 10.1609/aaai.v34i04.5879 - Analyzes the oversmoothing problem in deep GNNs and proposes architectural solutions to mitigate it.
Measuring and Improving the Smoothness of Graph Neural Networks, Yuya Oono, Taiji Suzuki, 2020 International Conference on Machine Learning (ICML), Vol. 119 (Proceedings of Machine Learning Research) DOI: 10.5591/00742 - Provides a theoretical framework for understanding and quantifying the smoothness of node representations in GNNs.
Is a Graph Convolutional Neural Network a Low-Pass Filter?, Naoaki Okumoto, Taiji Maehara, 2019 Proceedings of the NeurIPS Workshop on Graph Representation Learning DOI: 10.48550/arXiv.1905.09457 - Investigates the spectral properties of GCNs, explaining how message passing acts as a low-pass filter, leading to oversmoothing.
A Comprehensive Survey on Graph Neural Networks, Zonghan Wu, Shirui Pan, Fengwen Chen, Guodong Long, Chengqi Zhang, S. Yu Philip, 2020 IEEE Transactions on Neural Networks and Learning Systems, Vol. 32 (IEEE) DOI: 10.1109/TNNLS.2020.2970482 - A broad survey covering various aspects of GNNs, including a discussion of the oversmoothing problem and strategies to address it.