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Topological Neural Networks go Persistent, Equivariant, and Continuous

Yogesh Verma, Amauri H Souza, Vikas Garg

TL;DR

TopNets unify topological neural networks with persistent-homology descriptors to enrich representations on simplicial complexes. By coupling general message passing with PH vectorizations and topological aggregations, and by extending to $E(n)$-equivariant and continuous-time (Neural ODE) settings, the approach yields enhanced expressivity and strong empirical results across graph-classification, molecular-property prediction, antibody design, and molecular dynamics. The work also provides theoretical insights, including a discretization-error analysis that connects discrete TopNets to their continuous-depth analogues. While offering a powerful framework, it acknowledges computational costs from PH and scope limitations to simplicial complexes, suggesting directions for broader geometric completions and scalable PH embeddings.

Abstract

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.

Topological Neural Networks go Persistent, Equivariant, and Continuous

TL;DR

TopNets unify topological neural networks with persistent-homology descriptors to enrich representations on simplicial complexes. By coupling general message passing with PH vectorizations and topological aggregations, and by extending to -equivariant and continuous-time (Neural ODE) settings, the approach yields enhanced expressivity and strong empirical results across graph-classification, molecular-property prediction, antibody design, and molecular dynamics. The work also provides theoretical insights, including a discretization-error analysis that connects discrete TopNets to their continuous-depth analogues. While offering a powerful framework, it acknowledges computational costs from PH and scope limitations to simplicial complexes, suggesting directions for broader geometric completions and scalable PH embeddings.

Abstract

Topological Neural Networks (TNNs) incorporate higher-order relational information beyond pairwise interactions, enabling richer representations than Graph Neural Networks (GNNs). Concurrently, topological descriptors based on persistent homology (PH) are being increasingly employed to augment the GNNs. We investigate the benefits of integrating these two paradigms. Specifically, we introduce TopNets as a broad framework that subsumes and unifies various methods in the intersection of GNNs/TNNs and PH such as (generalizations of) RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in) geometric complexes, extending the scope of TNNs and PH to spatial settings. Theoretically, we show that PH descriptors can provably enhance the expressivity of simplicial message-passing networks. Empirically, (continuous and E(n)-equivariant extensions of) TopNets achieve strong performance across diverse tasks, including antibody design, molecular dynamics simulation, and drug property prediction.
Paper Structure (37 sections, 4 theorems, 51 equations, 2 figures, 11 tables)

This paper contains 37 sections, 4 theorems, 51 equations, 2 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Graph neural networks (GNNs).
  4. Topological neural networks
  5. Persistent homology.
  6. Combining PH and GNNs.
  7. $\mathbf{E(n)}$-Equivariant networks.
  8. Graph ODEs.
  9. A unified framework: Topological persistent neural networks (TopNets)
  10. $\mathbf{E(n)}$ Equivariant TopNets
  11. Continuous (Equivariant) TopNets
  12. Discretization Error Bound
  13. Experiments
  14. Graph Classification
  15. Molecular data - QM9
  16. ...and 22 more sections

Key Result

Proposition 1

There are pairs of non-isomorphic clique complexes that SWL cannot distinguish but persistence diagrams from color-based filtrations can.

Figures (2)

  • Figure 1: Comparison of representative PH-based architectures for graph learning.
  • Figure 2: Two non-isomorphic simplicial complexes.

Theorems & Definitions (6)

  • Proposition 1: SWL + PH $\succ$ SWL
  • Definition 1: $i$-simplex-color filtrations
  • Definition 2: Geometric $i$-simplex-color filtrations
  • Proposition 2: Invariant persistence diagrams
  • Proposition 3: Discretization error for TOGL
  • Proposition 4: Discretization error for RePHINE