TopER

Topological Embeddings in Graph Representation Learning

Astrit Tola, Funmilola Mary Taiwo, Cuneyt Gurcan Akcora, Baris Coskunuzer.
Advances in Neural Information Processing Systems (NeurIPS), 2025.

Overview

Graph embeddings play a critical role in graph representation learning, allowing ML models to explore and interpret graph-structured data. However, existing methods often rely on opaque, high-dimensional embeddings, limiting interpretability and practical visualization.

TopER — short for Topological Evolution Rate — is a novel, low-dimensional embedding approach grounded in topological data analysis. Inspired by Persistent Homology, TopER simplifies this classical topological tool by tracking how the shape of a graph evolves across a filtration, and summarizing this evolution into a compact, interpretable descriptor.

The method proceeds in three steps:

1. Filtration sequence — build a nested sequence of subgraphs $G_1 \subseteq G_2 \subseteq \dots \subseteq G_n = G$ using a chosen graph filtration (e.g., degree centrality, closeness, Forman–Ricci curvature).
2. Relative diagram — at each step, record (number of nodes, number of edges) of the subgraph.
3. Evolution fit — fit a line to this sequence via least squares; the two coefficients (pivot, growth) become the graph's 2D embedding.

The result is a remarkably simple yet expressive descriptor with several practical advantages:

• Interpretable — every embedding lives in 2D and can be directly visualized to reveal clusters, outliers, and structural patterns.
• Scalable — handles graphs with 100K+ nodes in minutes, avoiding the cubic complexity of persistent homology.
• Stable — robust under perturbations of graph structure.
• Competitive — TopER-based models match or surpass state-of-the-art results on graph classification and clustering across molecular, biological, and social network datasets (MUTAG, PROTEINS, COX2, IMDB-B, REDDIT-B, REDDIT-5K, and more).

TopER is available as a Python package: pip install toper

Links & Resources

Citation

If you use TopER in your research, please cite:

@inproceedings{tola2025toper,
  title     = {TopER: Topological Embeddings in Graph Representation Learning},
  author    = {Astrit Tola and Funmilola Mary Taiwo and Cuneyt Gurcan Akcora and Baris Coskunuzer},
  booktitle = {Advances in Neural Information Processing Systems (NeurIPS)},
  year      = {2025},
  url       = {https://openreview.net/pdf?id=JjTLq7SXaB}
}