Metric Graph Kernels via the Tropical Torelli Map

2025-05-17

AI Summary

Existing graph kernels overly rely on combinatorial structures (e.g., nodes, edges) and struggle with graphs exhibiting heterogeneous discretization levels. Method: We propose the first metric graph kernel grounded in tropical algebraic geometry. Graphs are modeled as metric spaces, and we introduce the tropical Torelli map—leveraging its geometric and topological invariants (e.g., genus, cycle structure)—to define graph similarity. This ensures subdivision-invariance, enabling cross-resolution comparison. Kernel computation scales solely with graph genus, enabling efficient kernel matrix assembly. Contribution/Results: Our kernel achieves statistically significant improvements over state-of-the-art methods on label-free graph classification benchmarks. It further demonstrates robustness and practical utility in real-world applications, notably achieving high accuracy in classifying urban road networks. To our knowledge, this is the first kernel that unifies metric geometry, tropical algebraic geometry, and topological invariance for scalable, resolution-agnostic graph 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: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We propose new graph kernels grounded in the study of metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels that are based on graph combinatorics such as nodes, edges, and subgraphs, our graph kernels are purely based on the geometry and topology of the underlying metric space. A key characterizing property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs that represent different underlying spaces. We develop efficient algorithms for computing these kernels and analyze their complexity, showing that it depends primarily on the genus of the input graphs. Empirically, our kernels outperform existing methods in label-free settings, as demonstrated on both synthetic and real-world benchmark datasets. We further highlight their practical utility through an urban road network classification task.

Problem

• Proposing metric graph kernels using tropical geometry
• Developing invariant kernels for comparing diverse graphs
• Outperforming existing methods in label-free settings

Innovation

• Metric graph kernels via tropical algebraic geometry
• Invariance under edge subdivision for comparison
• Efficient algorithms based on graph genus