We propose a unified framework based on persistent homology (PH) to characterize both local and global structures in disordered systems. It can simultaneously generate local and global descriptors using the same algorithm and data structure, and has shown to be highly effective and interpretable in predicting particle rearrangements and classifying global phases. We also demonstrated that using a single variable enables a linear SVM to achieve nearly perfect three-phase classification. Inspired by this discovery, we define a non-parametric metric, the Separation Index (SI), which not only achieves this classification without sacrificing significant performance but also establishes a connection between particle environments and the global phase structure. Our methods provide an effective framework for understanding and analyzing the properties of disordered materials, with broad potential applications in materials science and even wider studies of complex systems.

翻译：我们提出了一种基于持久同调的统一框架，用于表征无序体系中的局部和全局结构。该框架能够使用相同的算法和数据结构同时生成局部与全局描述符，并在预测粒子重排和分类全局相态方面表现出高效性和可解释性。我们还证明，仅使用单一变量即可使线性支持向量机实现近乎完美的三相分类。受此发现启发，我们定义了一种非参数度量——分离指数，该指数不仅能在不显著牺牲性能的前提下完成分类，还建立了粒子环境与全局相结构之间的联系。我们的方法为理解和分析无序材料的特性提供了一个有效框架，在材料科学乃至更广泛的复杂系统研究中具有广阔的应用潜力。