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On the Expressive Power of Sparse Geometric MPNNs

Yonatan Sverdlov, Nadav Dym

TL;DR

This work analyzes the expressive power of geometric graph neural networks under realistic sparsity, distinguishing between rotation-equivariant (E-GGNN) and invariant (I-GGNN) variants. It shows that generic separation by E-GGNNs holds exactly for connected graphs, while I-GGNNs achieve generic separation only on generically globally rigid graphs, linking separability to rigidity theory. The authors introduce EGenNet, a simple, maximally expressive E-GGNN architecture with theoretical guarantees and strong empirical performance on synthetic benchmarks and chemical-property tasks. They further leverage power graphs to boost I-GGNN expressivity and demonstrate that EGenNet can achieve competitive or superior results with lower architectural complexity. Overall, the paper provides a rigorous, architecture-guided view of when sparse geometric MPNNs can uniquely identify geometric graphs and offers a practical, scalable model for real-world applications in chemistry and beyond.

Abstract

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.

On the Expressive Power of Sparse Geometric MPNNs

TL;DR

This work analyzes the expressive power of geometric graph neural networks under realistic sparsity, distinguishing between rotation-equivariant (E-GGNN) and invariant (I-GGNN) variants. It shows that generic separation by E-GGNNs holds exactly for connected graphs, while I-GGNNs achieve generic separation only on generically globally rigid graphs, linking separability to rigidity theory. The authors introduce EGenNet, a simple, maximally expressive E-GGNN architecture with theoretical guarantees and strong empirical performance on synthetic benchmarks and chemical-property tasks. They further leverage power graphs to boost I-GGNN expressivity and demonstrate that EGenNet can achieve competitive or superior results with lower architectural complexity. Overall, the paper provides a rigorous, architecture-guided view of when sparse geometric MPNNs can uniquely identify geometric graphs and offers a practical, scalable model for real-world applications in chemistry and beyond.

Abstract

Motivated by applications in chemistry and other sciences, we study the expressive power of message-passing neural networks for geometric graphs, whose node features correspond to 3-dimensional positions. Recent work has shown that such models can separate generic pairs of non-isomorphic geometric graphs, though they may fail to separate some rare and complicated instances. However, these results assume a fully connected graph, where each node possesses complete knowledge of all other nodes. In contrast, often, in application, every node only possesses knowledge of a small number of nearest neighbors. This paper shows that generic pairs of non-isomorphic geometric graphs can be separated by message-passing networks with rotation equivariant features as long as the underlying graph is connected. When only invariant intermediate features are allowed, generic separation is guaranteed for generically globally rigid graphs. We introduce a simple architecture, EGENNET, which achieves our theoretical guarantees and compares favorably with alternative architecture on synthetic and chemical benchmarks. Our code is available at https://github.com/yonatansverdlov/E-GenNet.
Paper Structure (44 sections, 16 theorems, 53 equations, 2 figures, 8 tables)

This paper contains 44 sections, 16 theorems, 53 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Contributions
  3. Appendix
  4. Setup: Geometric Graph Neural Networks and Rigidity Theory
  5. Generic separation and Rigidity Theory
  6. E-GGNN
  7. I-GGNNs
  8. Maximally expressive E-GGNN and I-GGNNs
  9. Expressive power of I-GGNN
  10. I-GGNN generic separation by power graph preprocessing
  11. Expressive power of E-GGNN
  12. Building a generically maximally expressive E-GGNN
  13. The $\textsc{EGenNet}$ architecture
  14. Aggregation
  15. Construction of $f_{\mathrm{inv}}$ and ReadOut
  16. ...and 29 more sections

Key Result

Theorem 3.1

[expressive power of I-GGNN] Let $d$ be a natural number. Let $F$ be an I-GGNN. Let ${\bm{A}}$ be a graph not generically globally rigid on $\mathbb{R}^d$. Then, $F$ generically fails to identify ${\bm{A}}$. Conversely, if ${\bm{A}}$ is generically globally rigid on $\mathbb{R}^d$ and $F$ is a maxim

Figures (2)

  • Figure 1: (A) and (B) are pairs of non-isomorphic geometric graphs with the same distances along edges. I-GGNN cannot distinguish such pairs, while E-GGNN can (generically) if the graph is connected, as in (B). Subplot (C) depicts an example of globally rigid graphs, where non-isomorphic geometric graphs do not share the same distances across edges (in the figure, the edge between pink and light green). Such examples can be separated by both I-GGNN and E-GGNN.
  • Figure 2: From Left to Right: $k$-chain pair graphs that are non-generic and require $\sim k/2$ blocks for separation, Pair a in which $2$-power graph is enough for I-GGNN to distinguish and Pair b in which $3$-power graph is necessary and sufficient for separation.

Theorems & Definitions (30)

  • Definition 2.1: Globally rigid
  • Definition 2.2
  • Theorem 3.1
  • proof : Proof idea
  • Theorem 4.1
  • proof : Proof idea
  • Theorem 5.1
  • proof : Proof idea
  • Corollary 5.2
  • Theorem D.1
  • ...and 20 more