Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells

Topological Learning with Contraction and Hourglass Persistence

Introduction and Motivation

Persisting limitations in GNN expressivity, particularly their inability to capture higher-order topological features, motivate the search for new topological descriptors beyond conventional persistent homology (PH). The prevailing paradigm in graph representation learning leverages inclusion-based PH via filtrations, but this approach is fundamentally limited: certain topological events—like the explicit disappearance of cycles or bounded lifetimes for persistent features—cannot be faithfully registered. The paper "Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells" (2604.17548) systematically analyzes and addresses these shortcomings by embedding sequences of contractions within topological pipelines, introducing the notions of contraction homology (CH), backward persistence, forward-backward (FB) persistence, and ultimately hourglass persistence. These constructions aim to close the expressivity and metrizability gaps that hamper current PH-augmented GNN models.

Theoretical Framework: Contraction Homology and Hourglass Persistence

The authors formalize contraction homology (CH) as the persistent homology of contraction sequences, systematically contrasting it with the traditional, inclusion-based persistent homology (forward PH). Notably, they construct backward PH by contracting subgraphs in a schedule dictated by a given filtration and rigorously demonstrate the incomparability in expressivity between forward and backward PH: there exist graphs with identical forward PH but distinct backward PH and vice versa.

To fuse the complementary expressivity of forward and backward PH, forward-backward (FB) persistence is introduced. FB persistence concatenates an inclusion filtration phase with a contraction phase, thereby capturing both the emergence and disappearance of topological features and assigning finite lifetimes to those which would persist to infinity under ordinary PH. Theoretical results certify that FB persistence subsumes the descriptive power of forward and backward PH individually, yet it can distinguish graphs that neither forward nor backward PH can separate.

Generalizing further, the hourglass persistence paradigm is defined as any sequence of arbitrary interleavings of inclusions and contractions (with the constraint that a substructure must be included before it is contracted). The key expressivity result is that hourglass persistence strictly dominates FB-persistence—there exist pairs of graphs with identical FB-persistence but differing hourglass persistence diagrams, a property established directly via construction and minimal witness graphs.

The paper also situates these constructions within the larger landscape of generalized persistence modules, establishing connections and separation to extended persistence, zigzag persistence, and bipath persistence. The (f, g)-FB persistence generalizes both FB and extended persistence; crucially, the authors show that extended persistence is just a special case, and that hourglass persistence can strictly distinguish graphs that extended persistence cannot.

Algorithmic and Practical Contributions

Algorithms are developed for efficient persistence computation in all settings—forward, backward, FB, and hourglass—which are crucial for their integration into GNN pipelines. Notably, hourglass persistence enables practitioners to interleave contractions and inclusions, potentially truncating the size of intermediate complexes and mitigating the cubic scaling problems that afflict traditional PH computations on large graphs and complexes.

Empirical validation is performed on standard graph prediction benchmarks (NCI109, PROTEINS, IMDB-BINARY, NCI1, ZINC, and OGBG-MOLHIV) with both GIN and GCN backbones. The forward-backward framework achieves the best or second-best performance in the majority of classification, regression, and molecular prediction settings, with consistent gains over standard PH and even PH with learned filtrations (RePHINE). Ablations clarify the complementary nature: learning both inclusion and contraction orders is essential, as both forward-only and backward-only models are consistently outperformed by the full FB or hourglass persistence models.

The framework naturally extends to simplicial and cellular complexes. For simplicial complexes, the authors supply methods for constructing "simplicial quotients" via coning, ensuring the compatibility of their persistence computations in higher-dimensional combinatorial settings. For cellular complexes, the closure under quotients ensures that all constructions and algorithms generalize, allowing extension to architectures such as simplicial networks and CW-complex-based neural architectures.

Stability, Metrizability, and Theoretical Guarantees

Addressing practical aspects beyond expressivity, the paper provides stability guarantees for the (f, g)-FB persistence diagrams. The derived bottleneck stability bounds show that perturbations in filtrations lead to controlled perturbations in the diagrams, up to explicit constants depending on the filtration functions. This stability is critical for learnability and robustness in end-to-end differentiable GNNs. The authors also analyze the metrizability of their constructions, showing that the combination of inclusions and contractions resolves the issues of infinite bottleneck distances that can plague standard PH comparisons.

For hourglass persistence, combinatorial-time stability is proven, and existence of well-behaved metrics is established, although the authors highlight that establishing a notion of function-time stability in the arbitrary interleaving case remains an open problem.

Implications and Future Directions

From a practical perspective, these developments enrich the set of topological representations for structured data in GNNs, enabling both finer discrimination between graph structures and scalable computation on large graphs and complexes by permitting locality and truncation in the filtration-contraction schedule. The extension to higher-order complexes aligns with recent trends in topological deep learning, supporting architectures that operate on arbitrary cell complexes (e.g., simplicial, regular cell, or CW complexes).

The theoretical separation between FB/hourglass persistence and extended persistence, zigzag persistence, and bipath persistence points to a growing landscape of non-equivalent topological invariants with differing algorithmic and practical profiles, providing new research directions for both the topological data analysis and graph representation learning communities.

Potential developments include:

• Functional stability theory for arbitrary hourglass schedules,
• Further integration with geometric and hierarchical pooling algorithms,
• Application to non-graph complexes in computational biology, chemistry, and beyond,
• Joint optimization of GNN architectures and their induced topological descriptors in an end-to-end learnable fashion,
• Efficient specialization of the hourglass persistence computations for specific hardware and large-scale, streaming graph settings.

Conclusion

The development of contraction-based and hourglass persistent homology operators represents a rigorous advance in the expressive, stable, and practical augmentation of GNNs and topological neural architectures. By systematically integrating both inclusion and contraction sequences—and providing both the theoretical and algorithmic framework for hourglass persistence—the paper demonstrates a principled method for extracting richer topological features, with empirical gains across datasets and architectures and foundational implications for the theory of topological machine learning (2604.17548).