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.

翻译：我们通过热带代数几何首次引入了度量图的图核方法。与基于节点、边和子图等图组合结构的传统图核不同，我们的度量图核完全基于底层度量空间的几何与拓扑性质。我们构造方法的一个关键特征性质是其对边细分的恒定性，这使得该核函数本质上非常适合比较代表不同底层度量空间的图。我们开发了高效算法来计算这些核并分析了其复杂度，该复杂度主要取决于输入图的亏格而非其规模。通过对合成数据及选定现实数据集的实验，我们证明所提出的核能够捕捉标准组合方法所忽略的互补几何与拓扑信息，尤其在无标签场景下表现突出。我们进一步通过城市道路网络分类任务展示了其实用价值。