Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View
Introduction
This paper addresses a critical issue faced by Graph Neural Networks (GNNs): the over-smoothing problem, where node representations across different classes become indistinguishable when multiple layers are stacked, degrading the model's performance in tasks such as node classification. The paper introduces two novel metrics—Mean Average Distance (MAD) and MADGap—to quantify smoothness and over-smoothness in GNNs, respectively. Additionally, the authors propose two methods to mitigate over-smoothing: MADReg and Adaptive Edge Optimization (AdaEdge).
Over-smoothing Analysis with MAD and MADGap
The introduction of the MAD metric provides a quantitative measure of the smoothness of graph node representations. Calculating the cosine distance between node pairs, MAD reflects how similar these representations are within the same graph. The empirical analysis using MAD demonstrates that the smoothness of graph representations increases as the number of GNN layers increases. This finding supports the hypothesis that smoothing is an intrinsic property of GNNs but poses a risk for over-smoothing, characterized by too much similarity among node representations from different classes.
To identify the root cause of over-smoothing, the authors extend MAD to MADGap, which differentiates between the smoothness of node representations among nearby and remote nodes. The definition involves calculating the MAD for neighbor nodes (MADneb) and for remote nodes (MADrmt) and then measuring the gap (MADGap). A high MADGap indicates that useful intra-class information outweighs inter-class noise, leading to better model performance and reduced over-smoothing.
Relationship Between Topology and Over-smoothing
The authors argue that the graph topology significantly influences the information-to-noise ratio, which in turn affects the smoothness of node representations. Specifically, graphs with many inter-class edges tend to propagate more noise, leading to over-smoothing. Experimental manipulation of graph topology (i.e., removing inter-class edges and adding intra-class edges) affirms that optimizing the topology can effectively mitigate over-smoothing and enhance model performance.
Mitigating Over-smoothing: MADReg and AdaEdge
To address over-smoothing, the paper proposes two methods: MADReg and AdaEdge. MADReg incorporates MADGap as a regularizer in the training objective, explicitly encouraging the model to maintain distinct representations for different classes. Meanwhile, AdaEdge iteratively refines the graph topology based on the model’s predictions, optimizing it dynamically for better performance in downstream tasks.
Experimental Validation
Extensive experiments are conducted on seven widely-used graph datasets and ten typical GNN models. The paper shows that both MADReg and AdaEdge are effective in reducing over-smoothing and improving the performance of GNNs. For instance, in experiments with 4-layer GNNs—a scenario prone to severe over-smoothing—both methods significantly increase MADGap and the model’s prediction accuracy across different datasets.
Implications and Future Directions
The findings underscore the importance of considering graph topology when designing and training GNNs. While current approaches mostly focus on novel architectures for information propagation, this paper highlights that aligning the graph topology with the task objective is equally crucial. Optimizing topology not only alleviates over-smoothing but also leads to better utilization of GNN capabilities.
Moving forward, an intriguing research direction involves further refining the AdaEdge methodology to reduce incorrect edge adjustment operations, which could potentially introduce noise rather than useful information. Additionally, exploring the application of these methods to dynamic graphs where the topology changes over time could open new avenues for enhancing GNN performance in real-world scenarios.
Conclusion
This paper provides a comprehensive and quantitative investigation into the over-smoothing problem in GNNs, introducing effective metrics and methods for mitigation. By demonstrating the critical role of topology in GNN performance, the paper sets a foundation for future research focused on graph topology optimization. Both MADReg and AdaEdge offer valuable strategies for enhancing GNN robustness and efficacy, particularly in challenging tasks involving complex graph structures.