Metric Graph Kernels via the Tropical Torelli Map

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.