Diversity Curves for Graph Representation Learning

TL;DR

This paper introduces diversity curves, a novel size-aware graph representation method that tracks structural diversity across coarsening levels, enabling interpretable and scalable graph analysis.

Contribution

The work proposes diversity curves for graph representations, improving interpretability and expressivity by tracking structural diversity through coarsening levels.

Findings

Diversity curves enable clustering and visualization of graphs across sizes.

They distinguish the geometry of single-cell and molecular graphs.

The method outperforms baseline approaches in various graph analysis tasks.

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.