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While TpOT unifies geometrical, homological and higher-order relations into one metric, its global scalar distance can obscure transient, localized structural reorganizations during dynamic phase transitions. To overcome this limitation, we present a hierarchical dynamic evaluation framework driven by a novel topological and hypergraph reconstruction strategy. Instead of directly interpolating abstract network parameters, our method interpolates the underlying spatial geometry and rigorously re-computes the valid topological structures, ensuring physical fidelity. Along this geodesic, we introduce a set of multi-scale indicators: macroscopic metrics (topological distortion and persistence entropy) to capture global shifts, and a novel mesoscopic dual-perspective hypergraph entropy (node-perspective and edge-perspective) to detect highly sensitive, asynchronous local rewirings. We further propagate the cycle-level entropy change onto individual vertices to form a point-level topological field. Extensive evaluations of physical dynamical systems (Rayleigh\u2013van der Pol limit cycles, double-well cluster fusion), high-dimensional biological aggregation (D\u2019Orsogna model) and longitudinal stroke fMRI data demonstrate the utility of combining transport-based alignment with multi-scale entropy diagnostics for dynamic topological analysis.<\/jats:p>\n                  <jats:p>This article is part of the theme issue \u2018Critical transitions and intelligent control in complex systems\u2019.<\/jats:p>","DOI":"10.1098\/rsta.2025.0375","type":"journal-article","created":{"date-parts":[[2026,6,18]],"date-time":"2026-06-18T10:47:56Z","timestamp":1781779676000},"source":"Crossref","is-referenced-by-count":1,"title":["Beyond distance: quantifying point cloud dynamics with persistent homology and dynamic optimal transport"],"prefix":"10.1098","volume":"384","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-9881-4683","authenticated-orcid":false,"given":"Yixin","family":"Wang","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Huazhong University of Science and Technology 1 , Wuhan, Hubei, People\u2019s Republic of China"},{"name":"Center for Mathematical Sciences, Huazhong University of Science and Technology 2 , Wuhan, Hubei, People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7840-7060","authenticated-orcid":false,"given":"Ting","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Huazhong University of Science and Technology 1 , Wuhan, Hubei, People\u2019s Republic of China"},{"name":"Center for Mathematical Sciences, Huazhong University of Science and Technology 2 , Wuhan, Hubei, People\u2019s Republic of China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2077-990X","authenticated-orcid":false,"given":"Jinqiao","family":"Duan","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Department of Physics, Great Bay University 3 , Dongguan, Guangdong, People\u2019s Republic of China"},{"name":"Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems 4 , Dongguan, People\u2019s Republic of China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"175","published-online":{"date-parts":[[2026,6,18]]},"reference":[{"key":"2026092303411977800_bib1","doi-asserted-by":"crossref","first-page":"031101","DOI":"10.1063\/5.0195042","article-title":"Early warning indicators via latent stochastic dynamical systems","volume":"34","author":"Feng","year":"2024","journal-title":"Chaos"},{"key":"2026092303411977800_bib2","doi-asserted-by":"crossref","first-page":"e4","DOI":"10.1017\/S1446181124000178","article-title":"Early warning prediction with automatic labelling in epilepsy patients","volume":"67","author":"Zhang","year":"2025","journal-title":"ANZIAM J"},{"key":"2026092303411977800_bib3","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s40484-013-0008-0","article-title":"Dynamical network biomarkers for identifying critical transitions and their driving networks of biologic processes","volume":"1","author":"Liu","year":"2013","journal-title":"Quant. 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