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Metric Graph Kernels via the Tropical Torelli Map

Yueqi Cao, Anthea Monod

TL;DR

This work addresses the limitation of conventional graph kernels by introducing geometry-based similarity measures for metric graphs using the tropical Torelli map, which embeds a graph's metric structure into a symmetric positive definite matrix $Q(G)$. It proposes two refinement-invariant kernels, the Tropical Torelli--Euclidean (TTE) and Tropical Torelli--Wasserstein (TTW) kernels, defined on the PSD space and using either the Frobenius norm or the Bures-Wasserstein distance with a closed-form expression for $W^2(Q_1,Q_2)$. A practical algorithm to compute $Q(G)$ via a minimal spanning tree and cycle basis yields a unique representation under generic edge lengths, with complexity scaling with the graph genus $g$ as $O(g n (g + \, ext{log} ))$. Empirical results on synthetic data, 23 benchmark datasets, and urban road networks show the proposed kernels outperform competing label-free kernels and demonstrate robustness on geometric graph data. This framework blends tropical geometry and information geometry to produce geometry-aware graph similarities applicable to real-world networks such as urban road systems, where edge refinements should not alter the underlying space.

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.

Metric Graph Kernels via the Tropical Torelli Map

TL;DR

This work addresses the limitation of conventional graph kernels by introducing geometry-based similarity measures for metric graphs using the tropical Torelli map, which embeds a graph's metric structure into a symmetric positive definite matrix . It proposes two refinement-invariant kernels, the Tropical Torelli--Euclidean (TTE) and Tropical Torelli--Wasserstein (TTW) kernels, defined on the PSD space and using either the Frobenius norm or the Bures-Wasserstein distance with a closed-form expression for . A practical algorithm to compute via a minimal spanning tree and cycle basis yields a unique representation under generic edge lengths, with complexity scaling with the graph genus as . Empirical results on synthetic data, 23 benchmark datasets, and urban road networks show the proposed kernels outperform competing label-free kernels and demonstrate robustness on geometric graph data. This framework blends tropical geometry and information geometry to produce geometry-aware graph similarities applicable to real-world networks such as urban road systems, where edge refinements should not alter the underlying space.

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.
Paper Structure (29 sections, 1 theorem, 17 equations, 7 figures, 8 tables, 2 algorithms)

This paper contains 29 sections, 1 theorem, 17 equations, 7 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. From Graphs to Metric Graphs
  3. The Tropical Torelli Map for Weighted Graphs
  4. Graph Homology
  5. Main Algorithm
  6. Generic Length Functions
  7. Computational Complexity
  8. Tropical Metric Graph Kernels
  9. The Space of Positive Semi-Definite Matrices
  10. The Tropical Torelli--Wasserstein Kernel
  11. Computational Complexity
  12. Experiments
  13. Verifying Computation Time on Synthetic Datasets
  14. Comparing to Existing Kernels: Graph Classification on Benchmark Datasets
  15. Urban Road Network Classification
  16. ...and 14 more sections

Key Result

Theorem 1

Let $G=(V,E)$ be a connected graph with a generic length function $\ell$. Under the canonical orientation, the matrix $Q$ computed by Algorithm alg:mat-Q is unique, and invariant under any refinement of $G$.

Figures (7)

  • Figure 1: An illustration of a metric graph. The leftmost figure presents a local road network near the Victoria underground (tube) station in London. A graph representing the road network is given in the middle. The rightmost figure presents a refinement by adding artificial landmarks.
  • Figure 2: Verifying computation time. From left to right: the first figure plots genus as a function of number of nodes; the second and third figures show the computation time of kernel matrices.
  • Figure 3: Classification accuracy of 10-fold cross-validation on benchmark datasets (partial).
  • Figure 4: Illustration of URN patterns. From left to right, URNs are taken from Chicago (gridiron), Nanchang (linear), Omdurman (chaotic), and London (tributary).
  • Figure 5: Dimension reduction by kernel PCA. The low dimensional representations indicate that TTE and TTW kernels are able classify different URNs.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1