How does over-squashing affect the power of GNNs? (2306.03589v3)

Abstract: Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). Given their widespread use, understanding the expressive power of MPNNs is a key question. However, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of pairwise interactions between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that over-squashing hinders the expressive power of MPNNs. We validate our theoretical findings through extensive controlled experiments and ablation studies.

• The paper introduces a Hessian-based metric to quantitatively assess feature mixing, linking over-squashing to graph topology and model capacity.
• It establishes theoretical boundaries on GNN expressive power, showing that deeper and wider models are needed for effective long-range node interactions.
• Empirical validation confirms that increasing depth or weight mitigates over-squashing, guiding the development of optimized GNN architectures.

Analysis of Over-Squashing in Graph Neural Networks: Impact on Expressive Power

The paper "How does over-squashing affect the power of GNNs?" presents a rigorous examination of the limitations posed by over-squashing in Message Passing Neural Networks (MPNNs), a prevalent subclass of Graph Neural Networks (GNNs). This paper is pivotal for understanding the expressive capabilities of MPNNs when tasked with learning functions of node features in graph-structured data, a crucial aspect as these models are widely applied in both scientific and technological domains.

Summary and Key Contributions

At the heart of this research is the novel problem posed by over-squashing, which occurs when a substantial volume of messages from graph nodes is compressed into fixed-size vectors, potentially limiting the expressive power of MPNNs. The authors develop a formal framework to quantify this phenomenon and examine its impact on the ability of MPNNs to learn pairwise interactions across nodes.

The primary contributions of the paper include:

1. Quantitative Measure of Mixing: The authors introduce a robust metric based on the Hessian of the function estimated by an MPNN, which gauges the degree to which an MPNN can mix features between different nodes. This metric offers a new avenue to assess the expressive power of GNNs.
2. Theoretical Boundaries on Expressive Power: Through rigorous theoretical analysis, the paper derives upper limitations on the ability of MPNNs to model interactions between node features, determined by factors such as graph topology and model capacity. This includes comprehensive considerations of weights, depth, and commute times as pivotal elements influencing mixing capabilities.
3. Characterization of Over-Squashing: Over-squashing is characterized as the inverse of the mixing facilitated by an MPNN. The findings reveal that to achieve a desired level of communication (mixing) between nodes, especially those separated by large commute times, substantial model capacity is required.
4. Empirical Validation: The theoretical claims are substantiated through controlled experiments, demonstrating that increased depth or weights are necessary to mitigate over-squashing effects, particularly in tasks demanding high interaction among nodes with significant commute times.

Implications and Future Directions

The findings of this paper have several implications for both theoretical research and practical applications of GNNs:

• Design of More Effective GNN Architectures: Understanding the limitations imposed by over-squashing can drive innovations in GNN architectures, potentially leading to the design of models that avoid bottlenecks in information propagation across nodes.
• Guidance for Rewiring Graph Structures: The research suggests potential benefits of optimizing graph structure to reduce effective resistance, thereby enhancing model capacity to handle tasks necessitating long-range node interactions.
• Extension to More Complex Graph Models: While focused on MPNNs, insights from this framework can inform the design and improvement of more sophisticated models, including those leveraging attention mechanisms or graph transformers.

In concluding, this paper significantly contributes to our understanding of how topology and model constraints affect the learning abilities of MPNNs. It paves the way for future explorations into overcoming these challenges, refining both theoretical constructs and practical implementations in graph neural network research.