Metric Graph Kernels via the Tropical Torelli Map

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.

💡 Research Summary

This paper introduces the first family of graph kernels specifically designed for metric graphs—graphs equipped with a positive length function on each edge, thus representing a one‑dimensional metric space. Traditional graph kernels (e.g., Weisfeiler–Lehman, graphlet, shortest‑path) rely on combinatorial features such as node labels, edge labels, or subgraph counts and are not invariant under edge subdivision, a fundamental operation that refines a graph without changing its underlying metric space. The authors formalize a “metric graph kernel” as any kernel that yields identical similarity values for any pair of refinements of the original graphs (Definition 2.1).

The core technical contribution is the use of the tropical Torelli map from tropical algebraic geometry. For a weighted graph with a generic length function (distinct edge lengths and a unique minimum spanning tree), the tropical Torelli map produces a unique symmetric positive‑definite (SPD) matrix Q(G) of size g × g, where g is the graph’s genus (the first Betti number, i.e., the number of independent cycles). The construction proceeds as follows: a minimal spanning tree T is computed; each non‑tree edge defines a fundamental cycle; the cycle‑edge incidence matrix M (size g × m) is built; a diagonal matrix L of edge lengths is formed; and finally Q = M L Mᵀ. Algorithm 1 details this process and proves that the resulting matrix is independent of the chosen orientation and invariant under any edge subdivision, provided the length function is generic—a condition satisfied almost surely for real‑world data with measurement noise.

To compare graphs of different genus, the authors embed each SPD matrix into a common ambient space of positive semi‑definite (PSD) matrices of fixed dimension g₀ (the maximal genus among the dataset) by padding with zeros. Within this unified PSD space they define two distance‑based kernels:

1. Tropical Torelli–Wasserstein (TTW) kernel – uses the Bures–Wasserstein distance, a Riemannian metric on PSD matrices that is invariant under orthogonal transformations. The kernel is the standard radial basis function k(x,y)=exp(−γ d_BW²).

2. Tropical Torelli–Euclidean (TTE) kernel – employs the ordinary Euclidean (Frobenius) norm on PSD matrices, yielding a computationally cheaper but geometrically less expressive similarity measure.

Complexity analysis shows that constructing the cycle‑edge incidence matrix costs O(g n log n) and forming Q costs O(g² n), leading to an overall runtime of O(g n(g+log n)). Because the genus g captures topological complexity rather than raw size, the algorithm is near‑linear for sparse graphs (g = O(1)), cubic for semi‑sparse graphs (g = O(n)), and up to O(n³⁺²c) for dense graphs (where m ≈ n¹⁺c). This scaling is competitive with, and often superior to, traditional combinatorial kernels, especially on datasets where the underlying geometry is the primary source of information.

Empirical evaluation comprises two parts. Synthetic experiments generate random metric graphs with varying genus and edge‑length distributions to validate the theoretical runtime and to illustrate that the kernels remain stable under random refinements. Real‑world experiments focus on urban road networks (URNs), a canonical example of metric graphs where intersections can be added arbitrarily without altering the physical layout. In a label‑free classification task (e.g., distinguishing residential versus commercial road patterns), both TTW and TTE outperform state‑of‑the‑art combinatorial kernels, achieving higher accuracy and F1 scores. Moreover, when artificial landmarks are inserted (simulating edge subdivision), the proposed kernels’ performance remains unchanged, confirming the desired invariance property.

The paper’s contributions are threefold: (i) a rigorous definition and construction of metric‑graph‑specific kernels grounded in tropical geometry; (ii) efficient algorithms whose complexity depends on topological genus rather than graph size; (iii) demonstration of practical utility on both synthetic and real datasets, highlighting complementary information captured by geometry and topology that combinatorial kernels miss.

In summary, by bridging tropical algebraic geometry, information geometry, and machine learning, the authors open a new avenue for analyzing metric‑valued network data such as road maps, vascular systems, or power grids, where traditional label‑centric graph kernels are inadequate. Future work may explore low‑rank approximations for high‑genus graphs, learning optimal kernel parameters, and extending the framework to dynamic or probabilistic metric graphs.