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Multi-scale symmetry analysis in molecular structures

Jing-Wen Gao, Yunan He, Jian Liu

TL;DR

The paper addresses the challenge of capturing symmetry across multiple scales in data by introducing multi-scale symmetry analysis (MSA) based on persistent automorphism modules. It formalizes a graph-to-module functor to convert automorphism information from graphs into genuine persistence modules, enabling a scalable, symmetry-aware extension of persistent topology. The framework is demonstrated on fullerene structures, yielding a strong correlation (0.979) between predicted and observed stability across 12 molecules and providing interpretable invariants such as symmetry order and symmetry degree curves. This work links algebraic symmetry with topological persistence, offering a new tool for symmetry-driven analysis in materials science and molecular structures.

Abstract

Topological data analysis (TDA), as a relatively recent approach, has demonstrated great potential in capturing the intrinsic and robust structural features of complex data. While persistent homology, as a core tool of TDA, focuses on characterizing geometric shapes and topological structures, the automorphism groups of Vietoris-Rips complexes can capture the structured symmetry features of data. In this work, we propose a multi-scale symmetry analysis approach that leverages persistent automorphism modules to quantify variations in symmetries across scales. By modifying the category of graphs and constructing a suitable functor from the graph category to the category of modules, we ensure that the persistent automorphism module forms a genuine persistence module. Furthermore, we apply this framework to the structural analysis of fullerenes, predicting the stability of 12 fullerene molecules with a competitive correlation coefficient of 0.979.

Multi-scale symmetry analysis in molecular structures

TL;DR

The paper addresses the challenge of capturing symmetry across multiple scales in data by introducing multi-scale symmetry analysis (MSA) based on persistent automorphism modules. It formalizes a graph-to-module functor to convert automorphism information from graphs into genuine persistence modules, enabling a scalable, symmetry-aware extension of persistent topology. The framework is demonstrated on fullerene structures, yielding a strong correlation (0.979) between predicted and observed stability across 12 molecules and providing interpretable invariants such as symmetry order and symmetry degree curves. This work links algebraic symmetry with topological persistence, offering a new tool for symmetry-driven analysis in materials science and molecular structures.

Abstract

Topological data analysis (TDA), as a relatively recent approach, has demonstrated great potential in capturing the intrinsic and robust structural features of complex data. While persistent homology, as a core tool of TDA, focuses on characterizing geometric shapes and topological structures, the automorphism groups of Vietoris-Rips complexes can capture the structured symmetry features of data. In this work, we propose a multi-scale symmetry analysis approach that leverages persistent automorphism modules to quantify variations in symmetries across scales. By modifying the category of graphs and constructing a suitable functor from the graph category to the category of modules, we ensure that the persistent automorphism module forms a genuine persistence module. Furthermore, we apply this framework to the structural analysis of fullerenes, predicting the stability of 12 fullerene molecules with a competitive correlation coefficient of 0.979.

Paper Structure

This paper contains 14 sections, 10 theorems, 88 equations, 9 figures, 3 tables.

Table of Contents

  1. Abstract
  2. Keywords
  3. Introduction
  4. Combinatorial automorphism group
  5. Automorphism group of graphs
  6. Automorphism group of simplcial complexes
  7. Persistent automorphisms
  8. Functorial construction of automorphism groups
  9. Persistent automorphism module
  10. Two criteria
  11. Structure and stability analysis of fullerene molecules
  12. Structure analysis of fullerene $\mathrm{C}_{60}$
  13. Stability prediction of small fullerene molecules
  14. Conclusion

Key Result

Theorem 2.1

Let $G$ be a finite simple graph, and let $F(G)$ denote its clique complex (i.e., the flag complex generated by the cliques of $G$). Then there is a natural group isomorphism

Figures (9)

  • Figure 1: Illustration of the filtration of graphs in Example \ref{['example:graph_filtration']}.
  • Figure 2: The barcode for persistence module $\{\mathbb{F}{\rm Aut}(G_i), \Phi(f_i)\}_{i\geq 0}$.
  • Figure 3: a Graph $G$ with three automorphism-invariant cycles; b Graph $G^{\prime}$ no automorphism-invariant cycle.
  • Figure 4: a Graph $G$ which is cycle-stable; b Graph $G'$ with only one cycle.
  • Figure 5: a Illustration of the structure of fullerene $\mathrm{C}_{60}$; b The symmetry order curve and the symmetry degree curve for fullerene $\mathrm{C}_{60}$.
  • ...and 4 more figures

Theorems & Definitions (39)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1
  • proof
  • Example 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 29 more