Metric Graph Kernels via the Tropical Torelli Map

TL;DR

This paper introduces novel metric graph kernels derived from tropical algebraic geometry, enabling geometry-based comparison of metric graphs that are invariant under edge subdivision, with demonstrated effectiveness on synthetic and real-world data.

Contribution

It presents the first graph kernels based on tropical geometry for metric graphs, offering invariance under edge subdivision and efficient algorithms for computation.

Findings

Kernels capture geometric and topological features overlooked by combinatorial methods.

Algorithms' complexity depends mainly on graph genus, not size.

Effective in real-world urban network classification.

Abstract

We introduce the first graph kernels for metric graphs via tropical algebraic geometry. In contrast to conventional graph kernels based on graph combinatorics such as nodes, edges, and subgraphs, our metric 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 representing different underlying metric spaces. We develop efficient algorithms to compute our kernels and analyze their complexity, which depends primarily on the genus of the input graphs rather than their size. Through experiments on synthetic data and selected real-world datasets, we demonstrate that our kernels capture complementary geometric and topological information overseen by standard combinatorial approaches, particularly in label-free settings. We further showcase their practical utility with an urban road network classification task.