Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks (1810.02244v5)

Abstract: In recent years, graph neural networks (GNNs) have emerged as a powerful neural architecture to learn vector representations of nodes and graphs in a supervised, end-to-end fashion. Up to now, GNNs have only been evaluated empirically -- showing promising results. The following work investigates GNNs from a theoretical point of view and relates them to the $1$-dimensional Weisfeiler-Leman graph isomorphism heuristic ($1$-WL). We show that GNNs have the same expressiveness as the $1$-WL in terms of distinguishing non-isomorphic (sub-)graphs. Hence, both algorithms also have the same shortcomings. Based on this, we propose a generalization of GNNs, so-called $k$-dimensional GNNs ($k$-GNNs), which can take higher-order graph structures at multiple scales into account. These higher-order structures play an essential role in the characterization of social networks and molecule graphs. Our experimental evaluation confirms our theoretical findings as well as confirms that higher-order information is useful in the task of graph classification and regression.

• The paper establishes that conventional GNNs match 1-WL expressiveness, setting clear limits in graph isomorphism detection.
• The paper introduces k-GNNs that operate on k-node subsets to capture intricate, higher-order graph structures beyond traditional GNNs.
• The paper demonstrates that hierarchical k-GNN architectures yield significant empirical gains, reducing QM9 mean absolute error by over 54%.

Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks

This paper addresses the burgeoning domain of Graph Neural Networks (GNNs) from a rigorous theoretical perspective while proposing a compelling extension termed $k$-dimensional GNNs ($k$-GNNs). The authors aim to ground the practical efficacy of GNNs with the theoretical underpinnings provided by the Weisfeiler-Leman (WL) test, a prominent algorithm used in graph isomorphism problems.

Summary and Contributions

The paper tackles two central themes: the theoretical expressiveness of GNNs in distinguishing graph structures, and the introduction of a more powerful class of GNNs capable of leveraging higher-dimensional graph features.

1. Expressiveness of GNNs: The paper establishes an equivalence between the expressiveness of traditional GNNs and the 1-dimensional WL heuristic (1-WL). It shows conclusively that GNNs cannot outperform 1-WL in their capacity to differentiate between non-isomorphic graphs. Specifically, for all choices of the parameters and architectures, the generalized multi-layer GNNs are shown to neither surpass nor fall short of 1-WL in distinguishing graph structures.
2. k-dimensional GNNs: Extending beyond 1-WL, which primarily aggregates node-level information, the paper introduces $k$-GNNs. These novel architectures operate on $k$-subsets of graph nodes rather than individual nodes, capturing more complex and higher-order graph structures. This is akin to generalizing WL-test to $k$-WL, thereby enabling these networks to capture nuances in graph structures that traditional GNNs, restricted to 1-WL, are unable to discern.
3. Hierarchical Learning: Additionally, the paper proposes hierarchical $k$-GNNs, where higher-dimensional features are recursively extracted and integrated from lower-dimensional GNNs. This multi-scale, hierarchical architecture enhances the models' ability to learn both fine-grained and coarse representations from graph data, making these models particularly adept for tasks inherent to social networks and molecular graphs.

Empirical Validation

The paper substantiates its theoretical claims with extensive empirical validation across well-known graph classification benchmarks and the QM9 quantum chemistry dataset. Through empirical results, it is evident that:

• $k$-GNNs outperform traditional 1-GNNs and existing state-of-the-art graph kernels on several tasks.
• Hierarchical $k$-GNNs consistently exhibit improved performance, highlighting the importance of capturing hierarchical graph features for graph classification and regression tasks.

For instance, on the QM9 dataset, hierarchical models demonstrated significant reductions in mean absolute error—averaging a 54.45% reduction across twelve graph regression tasks. This attests to the practical advantages of incorporating multi-dimensional node information.

Implications and Future Directions

The theoretical insights and empirical validations presented in the paper hold substantial implications for future advancements in graph-based machine learning models. The insights that traditional GNNs are fundamentally bound by the limitations of 1-WL guide future research towards exploring higher-dimensional and more expressive graph architectures.

1. Practical Implications: Practitioners developing applications in cheminformatics, bioinformatics, and social network analysis can leverage $k$-GNNs to capture more intricate relationships, potentially leading to more accurate models in tasks like molecular property prediction and social network dynamics analysis.
2. Algorithmic Advances: Future research can explore optimizing $k$-GNN architectures, exploring efficient training paradigms, and integrating domain-specific knowledge to enhance interpretability and performance. Implementations could also focus on balancing computational costs with the expressive power afforded by higher-dimensional structures.
3. Theoretical Exploration: From a theoretical standpoint, understanding the limitations and potential extensions of $k$-WL and associated GNNs could provide deeper insights into the boundaries of graph representational learning. Investigating the trade-offs between computational feasibility and representational power remains an open and fertile ground for exploration.

Overall, this paper significantly contributes by exacting connections between GNNs and graph-theoretic algorithms, elucidating their expressiveness and charting a course towards more potent graph-based models. The introduction of $k$-GNNs marks a pivotal step forward, facilitating a nuanced capture of graph structures that aligns with both theoretical rigor and empirical necessity.