发现拓扑数据分析中多数信号来自密度而非拓扑，仅高阶统计量具真正拓扑意义。
How much of persistent homology is topology? A quantitative decomposition for spin model phase transitions
- 通过对比真实与密度匹配的随机模型，量化拓扑贡献比例。
- 零维统计量94-100%由密度驱动，一维统计量随系统尺寸增长呈现拓扑特征。
- 建议用随机置换空模型作为标准流程，关注高阶拓扑信号。
点云持久同调（PH）被广泛用于检测经典自旋模型相变，以往研究归因于持久图中的拓扑内容。本文提出 f_topo 量化分解方法，通过比较真实自旋构型与密度匹配的随机空模型，分离密度与拓扑对PH统计量的贡献。在2D伊辛模型（系统尺寸 L = 16-128，十温度）和庞茨模型（q = 3, 5）中，发现 H_0 统计量（总持久性、持久熵、特征数）有94-100%由密度驱动（f_topo < 0.07）。密度匹配的随机空模型与真实构型在临界温度位置和峰值高度完全一致，表明密度已足够用于相变检测。而 H_1 统计量具有部分拓扑性：拓扑分数随系统尺寸增长满足 delta(TP_{H_1}) ~ L^{0.53}，并符合有限尺寸标度坍缩形式 δ(T, L) = L^{0.53} g(tL^{1/ν})，坍缩质量系数 CV = 0.27。最长持久条带具有强拓扑性（f_topo > 1），且与关联长度相关。尺度解析分析显示，拓扑超额从大尺度向小尺度转移。建议将随机空模型纳入相变分析的标准流程，并优先使用 H_1 统计量获取真实拓扑信息。
原文摘要 · Abstract (English)
Point-cloud persistent homology (PH) -- computing alpha or Rips complexes on spin-position point clouds -- has been widely applied to detect phase transitions in classical spin models since Donato et al. (2016), with subsequent studies attributing the detection to the topological content of the persistence diagram. We ask a simple question that has not been posed: what fraction of the PH signal is genuinely topological? We introduce f_topo, a quantitative decomposition that separates the density-driven and topological contributions to any PH statistic by comparing real spin configurations against density-matched shuffled null models. Across the 2D Ising model (system sizes L = 16-128, ten temperatures) and Potts models (q = 3, 5), we find that H_0 statistics -- total persistence, persistence entropy, feature count -- are 94-100% density-driven (f_topo < 0.07). The density-matched shuffled null detects T_c at the identical location and with comparable peak height as real configurations, showing that density alone is sufficient for phase transition detection. However, H_1 statistics are partially topological: the topological fraction grows with system size as delta(TP_{H_1}) ~ L^{0.53} and follows a finite-size scaling collapse delta(T, L) = L^{0.53} g(tL^{1/nu}) with collapse quality CV = 0.27. The longest persistence bar is strongly topological (f_topo > 1) and scales with the correlation length. A scale-resolved analysis reveals that the topological excess shifts from large-scale to small-scale features as L increases. We propose that the TDA-for-phase-transitions community adopt shuffled null models as standard practice, and that H_1 rather than H_0 statistics be used when genuine topological information is sought.
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