On the Expressivity of Persistent Homology in Graph Learning (2302.09826v4)

Abstract: Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.

• The paper demonstrates that persistent homology matches and surpasses the WL algorithm by leveraging higher-order topological features.
• It employs various filtrations—such as degree, Laplacian, and Ollivier–Ricci curvature—to empirically validate graph differentiability.
• The research illustrates that topological insights like graph diameter and girth enhance the capability of graph learning models.

Expressivity of Persistent Homology in Graph Learning

The paper "On the Expressivity of Persistent Homology in Graph Learning" by Bastian Rieck addresses the integration of persistent homology, a computational topology tool, into the field of graph learning. The author provides a comprehensive exploration of the theoretical and empirical expressivity of persistent homology when leveraged in graph machine learning, primarily focusing on its compatibility with the Weisfeiler--Leman (WL) hierarchy of graph isomorphism tests.

Key Contributions and Findings

The core contribution of the paper is the formal characterization of the expressivity of persistent homology. The research demonstrates that persistent homology is at least as expressive as the Weisfeiler--Leman algorithm (and its generalizations) in distinguishing non-isomorphic graphs. The paper establishes that persistent homology can even surpass these algorithms by leveraging higher-order topological features, such as those provided by clique complexes, which are not captured within the standard WL framework.

• Theoretical Expressivity: The paper provides a rigorous examination of the Schnyder hierarchy, including \kWL variations, aligning its results with the expressivity achieved through topological analyses. It proves that persistent homology can distinguish graphs in instances where specific WL algorithms fall short.
• Empirical Validation: An experimental setup is introduced where different filtrations, including degree, Laplacian, and Ollivier–Ricci curvature filtrations, are employed to analyze various graph datasets. Persistent homology's ability to differentiate between certain classes of graphs, such as strongly-regular graphs, highlights its empirical advantages over purely algebraic approaches like WL.
• Descriptive Insights: The paper outlines how persistent homology captures more intricate graph properties like diameter and girth, adding a layer of geometric and topological insight beyond what is typically modeled in standard graph isomorphism tests.

Implications for AI and Future Directions

The implications of this research lie in its potential to enhance graph learning models with topological insights, enabling them to capture and utilize geometric and structural information inherently present in many graph-based datasets. As AI continually evolves, integrating such topology-driven methodologies could advance the capabilities of machine learning models, particularly in domains where underlying graph structures hold pivotal information, e.g., molecular graph representation in chemistry.

Several future directions arise from this work:

• Spectral and Curvature-Based Filtrations: The pronounced success of Laplacian-based filtrations suggests opportunities for further exploration of spectral graph theory methods in tandem with persistent homology, potentially leading to more computationally efficient and expressive graph descriptors.
• Hybrid Models: Developing models that combine the strengths of neural architectures with topological features is an exciting avenue, particularly beneficial for datasets where connectivity patterns and graph invariants play crucial roles in the predictive modeling process.
• Beyond Expressivity: The work opens avenues for further investigation into other graph properties that can be inferred through persistent homology, encouraging a deeper exploration of practical graph learning challenges where topology-driven insights could be transformative.

Conclusion

The paper successfully bridges the theoretical gap between computational topology and graph learning, establishing persistent homology as a powerful tool for enhancing expressivity in graph-based AI applications. It lays a foundation for both rigorous theoretical advancements and practical algorithmic innovations, positioning persistent homology as a crucial technique in the toolbox of graph learning researchers. As the intersection between machine learning and topology expands, this work will likely serve as a key reference point for exploring graph isomorphism and related computational challenges.