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

2026-04-19

AI Summary

This work addresses the limited expressivity and instability of conventional persistence-based methods in graph neural networks (GNNs) that rely solely on inclusion sequences. It presents the first systematic analysis of the topological properties induced by graph contractions and introduces the “hourglass persistence” framework, which constructs novel topological descriptors by alternately combining inclusion and contraction operations. The proposed method features an efficient, differentiable algorithm that seamlessly integrates into GNN architectures and naturally extends to simplicial and cell complexes. Empirical evaluations demonstrate that this framework significantly outperforms existing persistence-based approaches across multiple standard graph benchmarks, thereby enhancing the expressivity, learnability, and robustness of graph representation learning.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.

Problem

• Persistent Homology
• Graph Neural Networks
• Topological Representation
• Contraction
• Expressivity

Innovation

• Contraction Hom日消息
• Hourglass Persistence
• Graph Neural Networks
• Topological Data Analysis
• Persistent Homology