Homomorphism Counts for Graph Neural Networks: All About That Basis (2402.08595v5)

Abstract: A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at the heart of such limitations, since many functions to be learned rely on the ability of counting such patterns. Two prominent paradigms aim to address this limitation by enriching the graph features with subgraph or homomorphism pattern counts. In this work, we show that both of these approaches are sub-optimal in a certain sense and argue for a more fine-grained approach, which incorporates the homomorphism counts of all structures in the ``basis'' of the target pattern. This yields strictly more expressive architectures without incurring any additional overhead in terms of computational complexity compared to existing approaches. We prove a series of theoretical results on node-level and graph-level motif parameters and empirically validate them on standard benchmark datasets.

• The paper introduces a homomorphism basis framework that enhances GNN expressiveness beyond traditional subgraph counting methods.
• It demonstrates that computing homomorphism counts is computationally efficient and avoids redundancy in graph representations.
• Empirical benchmarks on datasets like ZINC and QM9 confirm improved performance in graph regression and quantum property prediction.

Homomorphism Counts for Graph Neural Networks: All About That Basis

This paper investigates the expressive power and computational efficiency of Graph Neural Networks (GNNs) when augmented with homomorphism counts for graph structures, emphasizing that homomorphism bases offer significant advantages over existing methods for subgraph and homomorphism counting. At the core is the proposal that the utilization of a homomorphism basis—consisting of all quotient graphs of a target graph motif—enhances the expressive power of GNNs without increasing computational complexity.

Key Contributions and Framework

The central claim is that the expressiveness of GNNs is significantly enhanced by the incorporation of homomorphism counts of the full basis of target graph motifs. The authors establish through theoretical analysis that using the homomorphism basis is strictly more expressive compared to employing direct subgraph or homomorphism counts. Specifically, they show that for any connected graph motif parameter, the Weisfeiler-Leman (WL) test augmented with homomorphism basis counts is strictly more expressive if certain conditions are met.

Several consequences of adopting homomorphism bases are explored:

1. Expressive Power: The use of a homomorphism basis is more expressive than subgraph counts for GNNs that cannot account for all graphs within a pattern's homomorphism basis. Theoretical results suggest that this expressiveness is realized without redundancy or excessive computational burden.
2. Computational Efficiency: The computation of homomorphism counts is tractable and efficient, often leveraging parameterized complexity to operate efficiently even within challenging computational domains such as networks containing many nodes or edges.
3. Avoiding Redundancy: The paper proposes methods for selecting minimal sets of homomorphism counts to avoid redundancy, optimizing the information provided to the GNN. This includes avoiding disconnected graphs in homomorphism bases where the additional counts do not contribute to expressiveness.
4. Practical Implications: The paper examines several standard benchmarks, demonstrating that models enhanced with homomorphism bases achieve a performance increase across tasks such as graph regression, link prediction, and expressiveness evaluation.

Empirical Validation

Extensive experiments support the theoretical claims. On the ZINC dataset for graph regression, GNN models enriched with homomorphism basis counts outperform those with direct subgraph counts or standard architectures by a significant margin. Similarly, for the QM9 dataset, the advantageous integration of homomorphism basis features aids in achieving state-of-the-art performance across various quantum property prediction tasks. Moreover, experimental evaluations on the COLLAB dataset and BREC benchmark underscore the versatility and robustness of the approach in improving model performance in a variety of contexts.

Implications and Future Directions

The work presents a compelling case for a paradigm shift in how graph motif information is incorporated into GNNs. It suggests broad implications for graph machine learning, where more expressive yet computationally feasible methods can now be harnessed effectively. One potential avenue for further research could be exploring the integration of logical information using similar motifs and homomorphism bases, extending the framework beyond simple substructure counting.

Overall, this work advances the field by offering a flexible, efficient, and more informative means of enhancing GNN expressiveness through graph homomorphism counts, laying robust foundations for further investigation and application development.