Diversity Curves for Graph Representation Learning

2026-05-07

AI Summary

This work addresses the challenge of interpretable, scalable, and reliable comparison of cross-scale graph structures in unsupervised settings by proposing a diversity curve embedding method based on graph coarsening. The approach introduces an isometric invariant termed “graph diffusion capacity” and integrates an edge-contraction coarsening strategy to track structural diversity across multiple scales, yielding an efficient and interpretable one-dimensional embedding representation. By capturing both topological and geometric characteristics, the method substantially enhances the expressiveness and geometric awareness of graph representations. Empirical evaluations demonstrate its superior performance over existing baselines in diverse tasks, including synthetic graph clustering, geometric discrimination of single-cell graphs, molecular graph comparison, and geometric shape characterization.

Abstract

Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging. Unsupervised tasks in particular require interpretable, scalable, and reliable size-aware graph representations. Our work addresses these issues by tracking the structural diversity of a graph across coarsening levels. The resulting graph embeddings, which we denote diversity curves, are interpretable by construction, efficient, and directly comparable across coarsening hierarchies. Specifically, we track the spread of graphs, a novel isometry invariant that is inherently well-suited for encoding the metric diversity and geometry of graphs. We utilise edge contraction coarsening and prove that this improves expressivity, thus leading to more powerful graph-level representations than structural descriptors alone. Demonstrating their utility over a range of baseline methods in practice, we use diversity curves to (i) cluster and visualise simulated graphs across varying sizes, (ii) distinguish the geometry of single-cell graphs, (iii) compare the structure of molecular graph datasets, and (iv) characterise geometric shapes.

Problem

• graph representation learning
• graph-level representations
• structural diversity
• size-aware graphs
• unsupervised graph comparison

Innovation

• diversity curves
• graph coarsening
• isometry invariant
• graph-level representation
• structural diversity