Beyond Distance: Quantifying Point Cloud Dynamics with Persistent Homology and Dynamic Optimal Transport

Bey ond Distance: Quan tifying P oin t Cloud Dynamics with P ersisten t Homology and Dynamic Optimal T ransp ort Yixin W ang 1 , 2 , † , Ting Gao 1 , 2 , 3 ∗ , Jinqiao Duan 4 , 5 , ‡ 1 Sc ho ol of Mathematics and Statistics, Huazhong Universit y of Science and T ec hnology , W uhan, China 2 Cen ter for Mathematical Science, Huazhong Universit y of Science and T ec hnology , W uhan, China 3 Steklo v-W uhan Institute for Mathematical Exploration, Huazhong Universit y of Science and T ec hnology , China 4 Departmen t of Mathematics and Departmen t of Ph ysics, Great Bay Universit y , Dongguan, China 5 Guangdong Provincial Key Laboratory of Mathematical and Neural Dynamical Systems, Dongguan, China. Abstract W e introduce a framework for analyzing top ological tipping in time evolution- ary point clouds b y extending the recen tly proposed T op ological Optimal T ransp ort (T pOT) distance. While T pOT unies geometric, homological and higher‐order re- lations into one metric, its global scalar distance can obscure transient, lo calized structural reorganizations during dynamic phase transitions. T o ov ercome this lim- itation, w e presen t a hierarchical dynamic ev aluation framework driv en by a nov el top ological and h yp ergraph reconstruction strategy . Instead of directly interpo- lating abstract netw ork parameters, our method interpolates the underlying spa- tial geometry and rigorously re-computes the v alid topological structures, ensuring ph ysical delity . Along this geo desic, w e introduce a set of multi-scale indicators: macroscopic metrics (T op ological Distortion and Persistence Entrop y) to capture global shifts, and a no vel mesoscopic dual-p ersp ectiv e Hyp ergraph Entrop y (no de- p ersp ective and edge-p ersp ectiv e) to detect highly sensitive, asynchronous local rewirings. W e further propagate the cycle-level en tropy c hange on to individual v ertices to form a point-lev el topological eld. Extensiv e ev aluations on ph ysical dynamical systems (Rayleigh-V an der P ol limit cycles, Double-W ell cluster fusion), high-dimensional biological aggregation (D’Orsogna mo del), and longitudinal stroke fMRI data demonstrate the utility of combining transp ort‐based alignment with m ulti‐scale en tropy diagnostics for dynamic top ological analysis. Keyw ords: Dynamic T op ological Optimal T ransp ort, Multiscale Tipping, Per- sistence entrop y , Hyp ergraph entrop y , Medical imaging 1 In tro duction Complex systems (e.g., climate systems, biological netw orks, nancial markets) often exhibit nonlinear phase transitions and critical phenomena [1]. Detecting their tipping 1 p oin ts and structural changes is crucial for predicting systemic collapse and designing early-w arning mec hanisms [2]. T raditional statistical metrics, such as v ariance and au- to correlation co ecients, are employ ed to monitor critical transitions and their driving net works [3]; ho w ever, they o verlook potential top ological structural c hanges in dynam- ical systems. In con trast, dynamical mo dels like bifurcation theory require predened dieren tial equations and p erform p o orly on real-world non-stationary systems with un- kno wn couplings [4]. In [5], the authors in tro duce a no vel Schrödinger bridge approac h for early warning signals (EWS) in probabilit y measures that align with the entrop y pro- duction rate. Nev ertheless, aligning disparate distributions in a manner that is b oth mathematically rigorous and computationally tractable remains a fundamental challenge in in terdisciplinary science. Originating from Monge’s 1781 optimal transport problem, optimal transp ort (OT) theory seeks to nd couplings b etw een probabilit y distributions as eciently as p ossible with resp ect to a giv en cost function [6]. A w ell-known distance measure, the W asserstein distance, ev aluates the geometric separation b et w een supp orts, where the cost function is induced by the distance function in a metric space. Recen tly , extensions of the opti- mal transp ort problem, such as the Gromo v–W asserstein (GW) distance and its v ariant, the F used Gromo v–W asserstein (F GW) distance, hav e garnered increasing attention due to their applicability in settings where probabilit y distributions are dened on dierent metric spaces [7–9]. Motiv ated b y the goal of representing relations in complex systems through higher-order net works, recen t work has applied a Gromo v–W asserstein v ariant kno wn as co-optimal transp ort [10] to hypergraph mo deling, demonstrating its eective- ness in b oth theoretical framew orks and practical applications [11]. T op ological Data Analysis (TDA) is a mathematical framew ork rooted in algebraic top ology that extracts multiscale top ological features from high-dimensional complex data [12]. Giusti and Lee dev elop ed a computable feature map for paths of p ersistence diagrams [13]. Li et al. introduced a exible and probabilistic framework for trac king top ological features in time-v arying scalar elds by emplo ying merge trees and partial optimal trans- p ort [14]. Numerous researchers hav e developed inno v ativ e methodologies for topological feature trac king and even t detection b y leveraging diverse to ols from TD A, including p ersistence diagrams, merge trees, Reeb graphs, and Morse–Smale complexes. F or exam- ple, T an weer et al. in tro duced a TD A-based approach that uses sup erlev el p ersistence to mathematically quan tify P-t yp e bifurcations in sto c hastic systems through a ”homo- logical bifurcation plot,” which illustrates the c hanging ranks of 0th and 1st homology groups via Betti v ectors [15]. Shamir et al. prop osed a progressiv e isosurface algorithm that predicts the con tour at time step t + 1 based on the contour at time step t [16]. Doraisw amy et al. describ ed a framework for the exploration of cloud systems at multi- ple spatial and temp oral scales using infrared(IR) brightness temp erature images, which automatically extracts cloud clusters as contours for a given temp erature threshold[17]. A comprehensive review of metho dologies for top ological feature trac king and structural c hange detection is presented in [18]. In recent y ears, there has b een growing in terest in extending classical graph entrop y[19– 22] to hypergraphs, as h yp ergraphs are capable of capturing higher-order in teractions that con ven tional pairwise netw orks cannot. The concept of h yp ergraph en tropy was rst in- tro duced b y Simon yi (1996) [23] within an information-theoretic framew ork. Subsequen t studies ha ve prop osed alternative formulations, including en trop y v ectors deriv ed from 2 partial hypergraphs [24] and tensor-based en tropy for uniform hypergraphs [25]. A ddi- tionally, en tropy-maximization mo dels ha ve b een emplo yed to generate random h yp er- graphs, serving as useful n ull mo dels for real-w orld systems [26]. These developmen ts underscore the versatilit y of h yp ergraph en tropy as a to ol for quantifying uncertaint y and complexity in higher-order netw orks, motiv ating its application to the analysis of h yp ergraph structures deriv ed from p ersisten t homology . In this work, w e propose a set of dynamic distortion and entrop y indicators that in tegrate optimal transp ort, top ological data analysis, and information theory for h yp er- graphs. Our approac h builds up on the recen tly introduced T op ological Optimal T rans- p ort (T pOT) framew ork [27], whic h aligns point clouds while join tly optimizing geometric corresp ondence and topological delit y through a principled coupling of their p ersistent homology classes. By incorporating this trade-o betw een geometric and top ological preserv ation, our metrics enable robust detection of m ultiscale structural shifts. T o sys- tematically capture these m ultiscale dynamics, our methodology proceeds in four fun- damen tal stages (Figure 1): computing the initial T pOT spatial coupling, p erforming geometric in terp olation along the underlying geo desic, rigorously reconstructing v alid top ological and hypergraph structures to ensure ph ysical delity , and extracting m ulti- scale early warning indicators. Driv en by this pip eline, our main contributions are listed as follo ws. First, w e prop ose a dynamic h yp ergraph reconstruction strategy that in tegrate in terp olation of underlying geometric information. Second, w e prop ose dynamic distor- tion and entrop y as early warning indicators for m ulti-scale tipping detection, notably a nov el mesoscopic dual-p ersp ective h yp ergraph en trop y (HE V and HE E ) that uniquely captures async hronous structural decoupling. W e further propagate the cyle-lev el en tropy c hange on to individual v ertices to form a p oin t-lev el top ological eld that iden ties key lo cal transformations. Third, w e pro vide rigorous theoretical guaran tees for these metrics, whic h are precisely v alidated across physical dynamical systems, biological aggregations, and clinical strok e fMRI data. The structure of this pap er is organized as follows: Section 2 describ es our prop osed top ological & hypergraph reconstruction and dynamic distortions & entropies framework for m ultiscale tipping detection. Esp ecially , Sections 2.3 to 2.5 detail the dynamic re- construction strategy , the dual-p ersp ective entrop y form ulations with their theoretical pro ofs, and the point-lev el lo calized mapping, resp ectively . Section 3 presen ts compre- hensiv e exp erimental v alidations, nally w e summarize conclusions and future w ork in Section 4. 2 Metho dology In man y applications—from neuroscience and materials science to mac hine learning— p ersisten t homology (PH) has prov en eective for dimension reduction, feature generation, and hypothesis testing. How ev er, standard diagram-based comparisons often disregard the underlying geometry of represen tativ e cycles. Recent extensions address this limita- tion b y extracting explicit cycles and organizing them in to higher-order structures such as PH-h yp ergraphs, where vertices corresp ond to data p oints and hyperedges enco de cy- cle memberships [28]. It has also b een sho wn that the T opological Optimal T ransp ort (T pOT) framew ork not only provides a p ow erful distance for comparing complex data but also endo ws the resulting space with ric h geometric structure, enabling in terp ola- 3 tion, extrap olation, and barycenter computations among measure–top ological netw orks [27]. Detailed mathematical denitions and properties are pro vided in the Supplemen tary Material A.1. Before in tro ducing our extensions, we critically assess the prop erties of the T op olog- ical Optimal T ransp ort (T pOT) framework. This analysis motiv ates the design c hoices presen ted in this section. T pOT sim ultaneously aligns p oint–point anities via a Gromov–W asserstein term, p ersistence diagram co ordinates via a W asserstein term, and p oin t–cycle incidences via a co-optimal transp ort term, thereb y integrating geometric, homological, and higher- order relational information within a single optimization. The space of measure top o- logical net works  P / ∼ w , d TpOT ,p  admits geodesics that are realized by conv ex com- binations of kernels, birth–death em b eddings, and incidence functions. These geo desics enjo y non‐negativ e Alexandro v curv ature. As a pseudo-metric, d TpOT ,p satises symmetry and the triangle inequality . The non-negativ e curv ature prop erty of the geo desic further guaran tees con vexit y of squared‐distance functionals, which enables robust interpolation, barycen ter computations, and clustering in the netw ork space. How ever, the T pOT faces t wo limitations: • Static distortion loss masks ev olution details. The T pOT framew ork returns only a scalar distance that quan ties the cost of transp orting P to P ′ along the optimal geodesic. How ev er, this endp oint‐only v alue omits the information ab out the intermediate distortions L t geom , L t topo , and L t hyper for t ∈ (0 , 1) . As a result, any transien t or evolving geometric or top ological phenomena o ccurring b et w een P and P ′ are en tirely mask ed, prev enting the lo calization of when—and to what exten t —signican t structural changes o ccur along the in terp olation path. • Dicult y of detecting abrupt top ological c hanges. Constant-speed in terp ola- tion yields smo othly v arying loss curves. Consequently , abrupt top ological changes, suc h as the sudden birth or death of long-lived cycles, app ear only as mild inec- tions, making them dicult to detect within the aggregated T pOT ob jectiv e. • Sensitivit y of hyperparameters. The c hoice of the entropic regularization weigh t ε critically inuences the trade-o b etw een delity and smo othness. Inappropriate settings ma y result in ov erly diuse couplings that obscure salient structure, or in unstable transp ort plans that are highly sensitiv e to noise. T o address these inheren t limitations, we introduce tw o key extensions in this work: • Dynamic distortion curv es. Rather than report only the nal scalar distance d TpOT ,p , we extract, at eac h interpolation time t ∈ [0 , 1] , the three dynamic dis- tortion loss functions L t geom , L t topo , and L t hyper . These curves precisely lo calize the timing and magnitude of geometric, top ological, and incidence deformations along the geo desic. • En tropy‐based ev ent detectors. W e in tro duce p ersistenc e entr opy (PE) and hyp er gr aph entr opy (HE) as complementary diagnostics. PE quanties the Shannon en tropy of the distribution of barcode lengths, capturing sensitivit y to the abrupt app earance or disapp earance of signicant cycles. HE measures the uniformity 4 of vertex–h yp eredge incidence, highligh ting sudden reorganizations in higher-order connectivit y . By integrating these en tropy-based indicators, w e are able to contin uously monitor and detect anomalous top ological ev ents in temp orally ev olving p oin t clouds. In this section, w e pro vide a detailed description of the prop osed pip eline for quantifying m ulti-scale tipping phenomena with geometric/top ological/hypergraph distortions in time-evolving p oin t clouds, whic h is illustrated in Figure 1. The structure of this section is as follow: First, w e present the background kno wledge of Measure T op ology Net work and T pOT for optimal coupling in Section 2.1 and Section 2.2. Then, our main contributions on Hyp ergraph Reconstruction and early warning in- dicators with dynamic distortions are presen ted in Section 2.3, which explained tipping detection in four-steps aligned with Figure 1. F urthermore, we study v arious denitions of entrop y as early warning indicators in Section 2.4. Our con tribution also lies in nov el denitions of V ertex-P ersp ective entrop y , Hyp eredge-Perspective entrop y and Symmetric Hyp ergraph entrop y , with theoretical pro ofs on sensitivity of abrupt topological transi- tions and top ological upp er bound. In addition, w e also present p oint-lev el hypergraph en tropy in Section 2.5, as an important ev aluation metric in some real data exp erimen t. 2.1 Measure T op ological Net w ork Giv en a p oint cloud i.e., a nite set of p oin ts X = { x i } N i =1 ⊂ R d , w e build its asso ciated me asur e top olo gic al network [27] P =  ( X , k , µ ) , ( Y , ι, ν ) , ω  , as dened in Section A.3. Concretely: • Geometry: c ho ose a symmetric kernel k ( x, x ′ ) (e.g. Gaussian anity or Euclidean distance) and uniform measure µ ( x i ) = 1/ N . • T op ology: w e use a pow erful top ological to ol Ripser er.jl to compute persistent homology in the desired dimension (e.g. 0D or 1D) to obtain a set of cycle repre- sen tatives Y . Record each cycle’s birth–death co ordinates via ι : Y → Λ and assign equal mass ν ( y ) = 1/ | Y | . • Incidence: for each pair ( x i , y ) , set ω ( x i , y ) = 1 if x i lies on the chosen represen- tativ e cycle y , and 0 otherwise. Analogously , for a second p oint cloud X ′ , w e construct the corresp onding measure top o- logical net work P ′ = (( X ′ , k ′ , µ ′ ) , ( Y ′ , ι ′ , ν ′ ) , ω ′ ) follo wing the same pro cedure. 2.2 T pOT Distance and Optimal Coupling W e then compute the T pOT distance of order p = 2 b et w een P and P ′ (23) in which all geo desics in P / ∼ w are con vex. F or clarit y and ecien t implemen tation, we now express eac h distortion in purely tensor‐ or matrix‐based summation form. W e denote the cardinalities of the p oin t clouds 5 Figure 1: Sc hematic ov erview of the h yp ergraph reconstruction and dynamic distortion framework. Row 1 (Initial T pOT): The optimal spatial coupling π v ⋆ is computed b etw een the source X 0 and target X 1 p oin t clouds via solving optimal T pOT problem. Ro w 2 (Geometric Interpolation): Metric interpolation and Multidimen- sional Scaling (MDS) generate intermediate spatial congurations X t along the geo desic t ∈ [0 , 1] . Row 3 (T op ological Structure Reconstruction): P ersistent homology is computed based on the interpolated geometric p oint cloud to extract authentic in terme- diate features. Ro w 4 (Hyp ergraph Reconstruction: Measure top ological netw orks P t are assem bled, where no des represen t regions and colored h ulls represent top ological h yp eredges. Ro w 5 (Dynamic Distortion as Early W arning Indicators) The dy- namic distortions ( L t ) are rigorously computed b y solving the T pOT distances b etw een the global reference source P 0 and eac h intermediate state P t . 6 b y N = | X | and N ′ = | X ′ | , and the n umber of p ersisten t homology generators (cycles) b y M = | Y | and M ′ = | Y ′ | . With these notations, let: L geom ii ′ j j ′ :=   k ( x i , x j ) − k ′ ( x ′ i ′ , x ′ j ′ )   p , 1 ≤ i, j ≤ N , 1 ≤ i ′ , j ′ ≤ N ′ , L topo uv :=          ∥ ι ( y u ) − ι ′ ( y ′ v ) ∥ p , 1 ≤ u ≤ M , 1 ≤ v ≤ M ′ , ∥ ι ( y u ) − ι ′ ( ∂ Y ′ ) ∥ p , 1 ≤ u ≤ M , v = M ′ + 1 , ∥ ι ( ∂ Y ) − ι ′ ( y ′ v ) ∥ p , u = M + 1 , 1 ≤ v ≤ M ′ , 0 , u = M + 1 , v = M ′ + 1 L hyper ii ′ uu ′ :=                1 2   ω i,u − ω ′ i ′ ,u ′   p , 1 ≤ u ≤ M , 1 ≤ u ′ ≤ M ′ , 1 2   ω ′ i ′ ,u ′   p , u = M + 1 , 1 ≤ u ′ ≤ M ′ , 1 2   ω i,u   p , 1 ≤ u ≤ M , u ′ = M ′ + 1 , 0 , u = M + 1 , u ′ = M ′ + 1 . Let Π v ∈ R N × N ′ and Π e ∈ R ( M +1) × ( M ′ +1) b e the optimal coupling matrices. Then L geom (Π v ) = N ,N ′ X i,i ′ =1 N ,N ′ X j,j ′ =1 L geom ii ′ j j ′ Π v ii ′ Π v j j ′ , (1) L topo (Π e ) = M +1 X u =1 M ′ +1 X v =1 L topo uv Π e uv , (2) L hyper (Π v , Π e ) = N X i =1 N ′ X i ′ =1 M +1 X u =1 M ′ +1 X v =1 L hyper ii ′ uu ′ Π v ii ′ Π e uu ′ . (3) Giv en a 4-wa y tensor L and a matrix ( C ij ) ij , w e dene tensor-matrix m ultiplication[29] L ⊗ C := X kl L ij k l C kl ! ij . So the T pOT distance b etw een tw o measure top ological net works P and P ′ with tunable w eights can b e written as L TpOT ,p (Π v , Π e , α, β ) = α L geom (Π v ) + (1 − α ) L topo (Π e ) + β L hyper (Π v , Π e ) (4) = α ⟨ L geom ⊗ Π v , Π v ⟩ + (1 − α ) ⟨ L topo , Π e ⟩ + β ⟨ L hyper ⊗ Π v , Π e ⟩ (5) In practice w e solve the T pOT problem (23) in its entropic‐regularised form min π v ∈ Π( µ,µ ′ ) π e ∈ Π adm ( ν,ν ′ ) L TpOT ,p ( π v , π e , α, β ) + ε v KL  π v | µ ⊗ µ ′  + ε e KL  π e | ν ⊗ ν ′  . F ollo wing section3.4 of [27], we solv e the entropically regularised T pOT problem by alter- nately up dating the t w o couplings while k eeping the other xed. These tw o Sinkhorn-st yle up dates are utilized un til con vergence to ( π v ⋆ , π e ⋆ ) . 7 2.3 Hyp ergraph Reconstruction and Dynamic Distortion Theoretically , the space of measure topological netw orks admits geo desics dened b y con vex com binations of their comp onen ts [27]. Given an optimal coupling ( π v ⋆ , π e ⋆ ) , a the- oretical constan t‐sp eed geo desic P t for t ∈ [0 , 1] would b e dened b y linear in terp olation: P t =  ( X × X ′ , k t , π v ⋆ ) , (( Y × Y ′ ) ∪ ( Y × ∂ Y ′ ) ∪ ( ∂ Y × Y ′ ) , ι t , π e ⋆ ) , ω t  , where linear in terp olation yields k t = (1 − t ) k + t k ′ , ι t = (1 − t ) ι + t ι ′ , ω t = (1 − t ) ω + t ω ′ . Remark: A direct linear interpolation of the incidence matrix ω raises t w o funda- men tal issues: First, it could inevitably in tro duce fractional mem b ership v alues (e.g., ω t ( x, y ) ∈ (0 , 1) ), whic h are ill-dened in the context of binary h yp ergraphs. Second, and more critically , the top ological structures generated by such abstract algebraic in terp ola- tion may b ecome detac hed from the geometric realit y of the data manifold; specically , the in terp olated cycles my fail to corresp ond to actual lo ops formed b y the interpolated p oin ts. T o o vercome these limitations and preserv e ph ysical v alidit y , we prop ose a h yp er- graph reconstruction strategy . Instead of in terp olating the parameters of measure top ological netw orks directly , w e st in terp olate the underlying geometry , next re-compute the top ological structure and then reconstruct the measure top ological net w ork (Hyp er- graph). Specically , with the optimal geometric coupling π v ⋆ , we construct the contin uous tra jectory as follo ws: • Step 1: Geometric In terp olation: F ollowing the geodesic construction in tro- duced b y Han et al. [27, 30], w e rst obtain an optimal matc hing from the en tropic regularized spatial coupling π v ⋆ . F or the matc hed p oint pairs, w e linearly interpo- late their squared Euclidean distance matrices as k t = (1 − t ) k + tk ′ . The spatial co ordinates of the intermediate p oint cloud X t are then recov ered b y applying the Multidimensional Scaling (MDS) algorithm to k t , follo wed b y a rigid transformation step for spatial alignmen t. • Step 2: T op ological Structure Reconstruction: Rather than articially blend- ing abstract features, we re-compute the p ersistent homology directly on the newly generated p oin t cloud X t to extract the authentic top ological features (e.g., p ersis- tence barco des) that ph ysically exist at time t . • Step 3: Measure T op ological Netw ork (Hyp ergraph) Reconstruction: Us- ing the spatial co ordinates X t and the authentically extracted top ological features, w e construct the intermediate measure top ological net w ork P t =  ( X t , k t , µ t ) , ( Y t , ι t , ν t ) , ω t  follo wing the pip eline outlined in Section 2.1. 8 • Step 4: Early W arning Indicator with Dynamic Distortion: Finally , the dynamic structural distortions L t are ev aluated b y solving the T pOT problem be- t ween the initial reference source P 0 and the reconstructed intermediate net work P t at each time step t ∈ [0 , 1] . That is, at each in terp olation step t , w e measure the optimal discrepancy b etw een the initial reference netw ork P 0 and the reconstructed h yp ergraph P t . The three dynamic distortion curves are dened as the minimized costs as follo ws. Let L geom ,t , L topo ,t , and L hyper ,t denote the cost tensors dened b etw een the static source P 0 and the dynamic reconstruction P t . Unlike the theoretical linear interpolation where the coupling remains xed, here we solv e the optimal transport problem at eac h step t . Let ( π v t , π e t ) b e the optimal couplings achieving the T pOT distance d TpOT ( P 0 , P t ) . L t geom = Z Z ( X × X t ) 2   k ( x, y ) − k t ( x ′ , y ′ )   p d π v t ( x, x ′ ) d π v t ( y , y ′ ) (6) L t topo = Z ¯ Y × ¯ Y t ∥ ι ( y ) − ι t ( y ′ ) ∥ p d π e t ( y , y ′ ) (7) L t hyper = Z X × X t × Y × Y t   ω ( x, y ) − ω t ( x ′ , y ′ )   p d π v t ( x, x ′ ) d π e t ( y , y ′ ) . (8) These curv es L t quan titatively track the ev olution of structural deviations. Specif- ically , π v t and π e t adaptiv ely up date the matc hing betw een the source and the ev olving manifold, ensuring that the distortion reects the true top ological dierence rather than artifacts of linear in terp olation. Similarly , let the time-dep enden t cost tensors L geom ,t ii ′ j j ′ , L topo ,t uv , L hyper ,t ii ′ uv b e computed b e- t ween the reference netw ork P 0 and the reconstructed netw ork P t (using the re-computed maps k t , ι t , and ω t ). Since the top ological structure of P t is re-generated at eac h step, the optimal corresp ondence may ev olve. Let Π v t and Π e t denote the time-sp e cic optimal couplings obtained b y solving the T pOT problem b etw een P 0 and P t . The three dynamic distortion curv es are then calculated as the minimized transp ort costs: L t geom = X i,i ′ ,j,j ′ L geom ,t ii ′ j j ′ Π v t, ii ′ Π v t, j j ′ =  L geom ,t ⊗ Π v t , Π v t  , (9) L t topo = X u,v L topo ,t uv Π e t, uv =  L topo ,t , Π e t  , (10) L t hyper = X i,i ′ ,u,v L hyper ,t ii ′ uv Π v t, ii ′ Π e t, uv =  L hyper ,t ⊗ Π e t , Π v t  . (11) Figure 1 illustrates this reconstruction-based T pOT inetrpolation and dynamic ev aluation framew ork. These three dynamic distortion curves L t capture the deformation of geometry , top ol- ogy , and cycle incidence as t increases. 2.4 En trop y as Early W arning Indicator T o detect abrupt top ological ev ents along the geo desic, we compute t wo en tropy metrics at eac h t : persistence entrop y and hypergraph entrop y . 9 First we apply the p ersistence en tropy whic h is in tro duced in [31]. Extract the barcode B t = { [ b t i , d t i ] } in ι t ( Y , Y ′ ) . Let l t i = d t i − b t i and L t = P i l t i . Dene the P ersistence En tropy on geo desic as PE ( B t ) = − X i l t i L t log  l t i L t  . (12) In tuitively , en trop y measures how dierent bars of the barco des are in length. A barco de with uniform lengths has small entrop y . Large c hanges in PE ( B t ) highligh t the birth or death of signican t homological features. In order to detect abrupt top ological transitions along the T pOT geo desic, one may consider Shannon‐t yp e entropies dened on the hypergraph. Let H = ( V , E ) denote the h yp ergraph asso ciated with a measure top ological netw ork, where V is the set of vertices (corresp onding to the data p oin ts in X ) and E is the set of hyperedges (corresp onding to the p ersistent cycles in Y ). A classical prop osal (e.g.[24]) pro ceeds by forming the matrix L ( H ) = I ( H ) I ( H ) T where I ( H ) is the incidence matrix and then computing the eigen v alues 0 ≤ λ 1 ( L ( H )) ≤ λ 2 ( L ( H )) ≤ . . . ≤ λ m ( L ( H )) . Then classicaly the Shannon en tropy of h yp ergraph H is dened as S ( H ) = − m X i =1 µ i log 2 ( µ i ) , (13) where µ i is dened as µ i = λ i ( L ( H )) P m i =1 λ i ( L ( H )) = λ i ( L ( H )) Tr ( L ( H )) . While this Shannon entr opy captures global connectivity patterns, it has tw o dra w- bac ks: • It requires eigen‐decomp osition of an m × m matrix (where m = | V | ), which can b e exp ensiv e for large h yp ergraphs. • It smo oths o ver lo cal v ertex–edge redistributions, and th us ma y fail to react sharply to small but top ologically signican t c hanges (e.g. the in tersection of sev eral cycles). T o address these issues and formulate a mathematically rigorous measure of struc- tural information, we adopt the information-theoretic framew ork for netw orks prop osed b y Dehmer (2008) [21, 22]. Dehmer’s paradigm assigns a probability v alue to each graph elemen t based on a lo cal structural functional f ( · ) , follo wed by the computation of Shan- non en tropy . Let H = ( V , E ) b e the hypergraph formed by the extracted p ersistent cycles, with incidence matrix ω ∈ { 0 , 1 } | V |×| E | . Let L ( v ) = P e ∈ E ω ( v , e ) be the degree of vertex v , and S ( e ) = P v ∈ V ω ( v , e ) b e the size of hyperedge e . T o rigorously a void singularities suc h as 0 · ln (0) or division b y zero, w e restrict our analysis to the active vertex set V ∗ = { v ∈ V | L ( v ) > 0 } and the active hyp er e dge set E ∗ = { e ∈ E | S ( e ) > 0 } . The total incidence of the h yp ergraph is I total = P v ∈ V ∗ L ( v ) = P e ∈ E ∗ S ( e ) . 10 Denition 1 ( V ertex-Perspective Entrop y ) . F ol lowing the Dehmer fr amework, we dene the structur al functional for a vertex as its de gr e e, f ( v ) = L ( v ) . Normalizing this functional over the active set yields a valid pr ob ability distribution p ( v ) = L ( v ) I total , which r epr esents the pr ob ability that a r andomly chosen incidenc e c onne ction b elongs to vertex v . The V ertex-Persp e ctive Entr opy is dene d as: HE V ( H ) = − X v ∈ V ∗ p ( v ) ln p ( v ) . (14) Denition 2 ( Hyperedge-Perspective Entrop y ) . Dual ly, dening the structur al func- tional for a hyp er e dge as its size f ( e ) = S ( e ) yields the pr ob ability distribution q ( e ) = S ( e ) I total . The Hyp er e dge-Persp e ctive Entr opy is dene d as: HE E ( H ) = − X e ∈ E ∗ q ( e ) ln q ( e ) . (15) By formulating the entropies as standard Shannon entropies o ver the discrete proba- bilit y spaces V ∗ and E ∗ , w e obtain the following rigorous structural prop erties: Prop ert y 1 ( Bounds and Maximal Assumption ) . The entr opies ar e b ounde d by 0 ≤ HE V ( H ) ≤ ln | V ∗ | and 0 ≤ HE E ( H ) ≤ ln | E ∗ | . The maximum HE V ( H ) = ln | V ∗ | is achieve d if and only if H is a regular h yp ergraph (i.e., L ( v ) is c onstant for al l v ∈ V ∗ ). Dual ly, HE E ( H ) = ln | E ∗ | is achieve d if and only if H is a uniform hypergraph (i.e., S ( e ) is c onstant for al l e ∈ E ∗ ). Pro ofs of this prop erty and subsequent theorems are provided in the App endix B. With this prop erty , w e could normalize ] HE V ( H ) = HE V ( H ) ln | V ∗ | , g HE E ( H ) = HE E ( H ) ln | E ∗ | ∈ [0 , 1] . Denition 3 ( Symmetric Hyp ergraph Entrop y ) . T o c aptur e b oth p ersp e ctives simul- tane ously, we intr o duc e HE sym ( G ) = α ] HE V + (1 − α ) g HE E , α ∈ [0 , 1] . (16) By adjusting α , one c an emphasize vertex‐level ( α ≈ 1 ) or hyp er e dge‐level ( α ≈ 0 ) changes, or tr e at b oth e qual ly ( α = 0 . 5 ). Theorem 1 ( Sensitivit y to Abrupt T op ological T ransitions ) . L et { H t } t ∈T b e a se quenc e of dynamic hyp er gr aphs p ar ameterize d by t . Supp ose at a critic al p ar ameter t c , an abrupt top olo gic al tr ansition o c curs via the emer genc e of a new active hyp er e dge e new of size k > 0 . Then the vertex-p ersp e ctive hyp er gr aph entr opy strictly changes at t c (i.e., lim t → t − c HE V ( H t )  = HE V ( H + ) ), exc ept p ossibly for a highly r estrictive set of hyp er gr aphs whose de gr e e se quenc es satisfy a sp e cic, rigid Diophantine e quation. Theorem 2 ( Dual Sensitivit y to T op ological T ransitions ) . The same c onclusion holds for hyp er e dge-p ersp e ctive entr opy HE E , exc ept for a highly r estrictive zer o-me asur e algebr aic c ondition governe d by the F undemental The or em of A rithmetic. 11 Theorem 3 ( Isomorphism In v ariance of T op ological En trop y ) . L et H 1 = ( V 1 , E 1 ) and H 2 = ( V 2 , E 2 ) b e two hyp er gr aphs derive d fr om two me asur e top olo gic al networks. If H 1 and H 2 ar e isomorphic (i.e., ther e exist bije ctions φ : V 1 → V 2 and ψ : E 1 → E 2 pr eserving the incidenc e r elations ω 1 ( v , e ) = ω 2 ( φ ( v ) , ψ ( e )) ), then HE V ( H 1 ) = HE V ( H 2 ) and HE E ( H 1 ) = HE E ( H 2 ) . Theorem 4 ( Algebraic T op ological Upp er Bound ) . L et D k denote the k -th p ersis- tenc e diagr am obtaine d fr om the V ietoris–Rips ltr ation of the network, and let | D k | b e the numb er of p ersistent gener ators (i.e., top olo gic al fe atur es with non-zer o p ersistenc e). A nd supp ose the hyp er gr aph H is c onstructe d using al l p ersistent gener ators acr oss al l dimensions. Then the hyp er e dge-p ersp e ctive entr opy is strictly b ounde d by the top olo gic al c omplexity of the manifold: HE E ( H ) ≤ ln X k | D k | ! . In man y practical data analysis scenarios, the measure topological netw ork is con- structed explicitly fo cusing on a sp ecic homological dimension k (e.g., k = 1 to analyze lo ops, or k = 2 for v oids). Corollary 1 ( Dimension-Sp ecic T op ological Bound ) . If the hyp er gr aph H is c on- structe d exclusively using the k -dimensional p ersistent gener ators, the hyp er e dge-p ersp e ctive entr opy is strictly b ounde d by the numb er of k-dimensional top olo gic al fe atur es : HE E ( H ) ≤ ln ( | D k | ) . (17) Remark: This corollary has signican t practical implications. F or instance, when trac king the structural dynamics of a 1-dimensional functional net work, the maxim um p ossible hyperedge en trop y is fundamentally b ottleneck ed by the total n umber of k- dimensional p ersistent features ln ( | D 1 | ) . This implies that our entrop y metric is not only a statistical measure of incidence, but a direct pro xy for the 1-dimensional algebraic top ological capacit y of the system. Our newly prop osed h yp ergraph entropies hav e follo wing adv antages • Computational eciency: eac h requires only O ( | V | · | E | ) operations, a v oiding costly eigen‐decomp ositions. • The or etic al Sensitivity: As prov en in Theorem 1, the abrupt app earance or disap- p earance of a top ological feature instantaneously forces a non-zero displacement in the probability simplex. This guarantees a mathematically rigorous en tropy spik e or inection, making it highly sensitiv e to critical top ological ev ents. • Structur al Interpr etability: The metrics provide a clear macroscopic interpretation of top ological uniformit y . F urthermore, their theoretical upp er bounds are deeply ro oted in algebraic top ology: the h yp eredge en tropy is fundamentally b ottleneck ed b y the top ological capacity of the system (i.e., b ounded b y the p ersistent features, HE E ≤ ln β k , as established in Theorem 3 and Corollary 1), serving as a direct quan titative proxy for the homological complexity of the data manifold. 12 W e computeHE sym ( t ) on the interpolated h yp ergraph G t along the T pOT geo desic, and use sudden deviations in these curv es to ag discrete top ological even ts. Sudden deviations in HE sym mark structural reorganizations of the cycle hypergraph. In practice, w e ev aluate the symmetric h yp ergraph en tropy HE sym ( H t ) = α HE V ( H t ) ln | V ∗ | + (1 − α ) HE E ( H t ) ln | E ∗ | on the dynamically reconstructed h yp ergraph H t at eac h in terp olation step. W e then use the sharp discontin uities and sudden deviations in these curv es to detect discrete top ological phase transitions and structural reorganizations within the ev olving p oin t cloud. F or clarit y , we provide pseudo co de b elow for our metho d (Algorithm 1). 2.5 P oin t-Level Hyp ergraph En trop y Change Complex hypergraph netw orks often contain multiple in terdep enden t cycles that en- co de structural relationships at dieren t spatial and top ological scales. When such net works evolv e—for instance, under temp oral deformation, diusion, or connectivity reorganization—the glob al hypergraph entrop y tends to av erage out the lo cal transforma- tions, thereb y obscuring where the most signicant structural c hanges o ccur. T o resolv e this limitation, we in tro duce a cycle-level entr opy de c omp osition that projects en tropy do wn to the vertex level. This formulation enables identication of lo cal top ological v ari- ations within complex h yp ergraph systems. Incidence transp ort via optimal coupling. W e dene this decomp osition generally b et w een a r efer enc e h yp ergraph (with incidence matrix ω ∈ R n × m ≥ 0 ) and a tar get hyper- graph (with incidence matrix ω ′ ∈ R n ′ × m ′ ≥ 0 ). Here, ω [ i, j ] indicates the participation weigh t of vertex i in the j -th reference cycle. Because the ordering of top ological features gener- ally diers b et w een states, the target matrix ω ′ is column-aligned to the reference ω using a p erm utation (or assignmen t) matrix A ∈ { 0 , 1 } m ′ × m : b ω = ω ′ A. This matrix multiplication explicitly permutes the columns of the target incidence matrix, yielding a column-aligned matrix b ω ∈ R n ′ × m ≥ 0 where the j -th column directly corresp onds to the j -th reference cycle. Column normalization. T o compute the en tropy , the columns of the incidence ma- trices are normalized to form probability distributions o ver the v ertices. F or the reference matrix ω and the aligned target matrix b ω , we explicitly dene their normalized counter- parts P and b P as: P [ i, j ] = ω [ i, j ] P n i ′ =1 ω [ i ′ , j ] + ε , b P [ i, j ] = b ω [ i, j ] P n ′ i ′ =1 b ω [ i ′ , j ] + ε , where ε is a sucien tly small constant to ensure numerical stability . Cycle-lev el en tropy . Let p j = P [: , j ] denote the normalized vertex distribution of the j -th reference cycle, and b p j = b P [: , j ] denote the corresp onding distribution from the 13 aligned target matrix. W e dene the cycle-lev el en tropy for the reference and target states, resp ectiv ely , as: H j = − 1 log n n X i =1 p ij log p ij , b H j = − 1 log n ′ n ′ X i =1 b p ij log b p ij , (18) whic h measures ho w spatially diuse (large entrop y) or concentrated (small en tropy) the participation is within that sp ecic top ological cycle. En tropy dierence and propagation. F or eac h reference cycle j , the change in struc- tural en tropy b et w een the tw o states is dened as: ∆ H j = b H j − H j . (19) T o lo calize this structural v ariation on to the target v ertex set, the absolute en tropy c hange | ∆ H j | is propagated back to the target vertices according to their membership weigh ts in the aligned cycles: s i = m X j =1 b P [ i, j ] | ∆ H j | . (20) By dening the column v ector s = [ s 1 , . . . , s n ′ ] ⊤ and the en tropy dierence vector ∆ H = [∆ H 1 , . . . , ∆ H m ] ⊤ , this bac k-pro jection can b e compactly written in matrix form as s = b P | ∆ H | . The v ector s forms a vertex-level hyp er gr aph-entr opy eld , whic h highlights lo calized regions undergoing the strongest structural transformations. In terpretation. Equations (18)–(20) transform the complex structural comparison of high-dimensional h yp ergraphs into an interpretable scalar eld o v er the vertex domain. Visualizing the eld s on the spatial p oint cloud rev eals lo calized topological deformations that global en tropy measures inherently av erage out. This completes the detailed description of our metho d. In Section 3 w e v alidate its eectiv eness on synthetic and real‐world datasets. 3 Exp erimen ts W e ev aluate our metho d on four distinct settings: tw o syn thetic phenomenological bi- furcation mo dels, a high-dimensional biological aggregation model (D’Orsogna), and a real-w orld longitudinal fMRI dataset. In each case, we report: (1) the three dynamic distortion curv es ( L geom , L topo , L hyper ), whic h trac k the hierarc hical structural ev olution, and (2) the entrop y indicators, namely Persistence En tropy (PE), Symmetric Hyp er- graph En tropy (HE sym ), V ertex-P ersp ective Entrop y(HE V ) and Hyp eredge-P ersp ectiv e En tropy(HE E ). These en tropy measures pro vide complemen tary , parallel c haracteriza- tions of top ological evolution: PE captures v ariations in the p ersistence-diagram domain, while HE sym , HE V , HE E quan tify changes in the higher-order hypergraph incidence struc- ture. In addition, the point-lev el h yp ergraph-entrop y eld is used to capture local v ertices that con tribute most to the observed structural reorganization in the real fMRI data. 14 Algorithm 1 Early warning detection via dynamic distortions and Entrop y Require: Time-series p oint clouds { X ( i ) } T i =1 , distance k ernel k . Require: T pOT trade-o w eights ( α , β ) , regularization weigh ts ( ε v , ε e ) , and symmetric en tropy weigh t γ . Require: Lo cal geodesic resolution L , forming a uniform partition of the in terv al [0 , 1] denoted b y { τ ℓ } L ℓ =1 . Ensure: A contin uous global tra jectory of structural ev aluations: distortion curves L geom ( τ ∗ ) , L topo ( τ ∗ ) , L hyper ( τ ∗ ) and entrop y curves PE ( τ ∗ ) , HE sym ( τ ∗ ) parameterized b y the global contin uous timeline τ ∗ ∈ [0 , 1] . 1: Initialization: 2: Construct the global reference netw ork P (1) =  ( X (1) , k (1) , µ (1) ) , ( Y (1) , ι (1) , ν 1 ) , ω (1)  . 3: for i = 1 to T − 1 do 4: Construct net works P ( i ) and P ( i +1) via p ersisten t homology . 5: Compute T pOT b etw een P ( i ) and P ( i +1) using w eigh ts ( α, β ) and regularization w eights ( ε v , ε e ) to obtain the optimal spatial coupling π v ⋆ . 6: for  = 1 to L do 7: τ ← τ ℓ // L o c al ge o desic p ar ameter τ ∈ [0 , 1] 8: τ ∗ ← ( i + τ − 1)/( T − 1) // Mapping to glob al c ontinuous timeline τ ∗ ∈ [0 , 1] 9: // Step 1: Ge ometric Displac ement Interp olation 10: Compute in terp olated spatial p ositions based on matched pairs ( x, x ′ ) ∼ π v ⋆ : 11: X τ ∗ ← Embed   k τ ∗ = (1 − τ ) k ( i ) + τ k ( i +1)   π v ⋆ > 0   12: // Step 2: T op olo gic al R e c onstruction 13: Re-compute p ersistent homology strictly on X τ ∗ to construct the intermediate net work P τ ∗ . 14: // Step 3: Dynamic Curve Evaluation (R elative to Glob al R efer enc e P (1) ) 15: Solv e T pOT b etw een P (1) and P τ ∗ using weigh ts ( α, β , ε v , ε e ) to obtain ev alu- ation couplings (Π v τ ∗ , Π e τ ∗ ) . 16: Obtain dynamic distortion curv es by computing the tensor inner pro ducts: 17: L geom ( τ ∗ ) =  L τ ∗ geom , Π v τ ∗ ⊗ Π v τ ∗  18: L topo ( τ ∗ ) =  L τ ∗ topo , Π e τ ∗  19: L hyper ( τ ∗ ) =  L τ ∗ hyper , Π v τ ∗ ⊗ Π e τ ∗  20: Obtain structural en tropy curves from the active top ology of P τ ∗ : 21: PE ( τ ∗ ) = − P l j L log l j L // fr om b ar c o de lengths of Y τ ∗ 22: HE sym ( τ ∗ ) ← HE sym ( P τ ∗ ; γ ) // Evaluate d with entr opy weight γ 23: Compute HE V ( τ ∗ ) and HE E ( τ ∗ ) iden tically following denition equations. 24: end for 25: end for 15 A cross all experiments, the en tropic T pOT problem is consistently optimized using the Sinkhorn algorithm. W e congure the optimization with trade-o parameters α = 0 . 5 and β = 1 . 0 , along with en tropic regularization weigh ts ε s = 0 . 003 for the spatial coupling and ε f = 0 . 01 for the feature coupling. The symmetric en tropy weigh t is set to γ = 0 . 5 . 3.1 Exp eriment 1: T op ological Phase T ransition in Sto c hastic Oscillators Data Generation. T o ev aluate the proposed dynamic T pOT and structural entrop y framew ork on systems exhibiting complex top ological phase transitions, w e generate a syn thetic dataset based on the stochastic Rayleigh-V an der Pol (R VP) oscillator [15]. F orced b y additiv e white Gaussian noise, the stationary joint probabilit y density function (PDF) of the R VP oscillator’s state space ( x 1 , x 2 ) is prop ortional to the exp onential of its p oten tial energy: p ( x 1 , x 2 ) ∝ exp  − V ( x 1 , x 2 ) T  , where V ( x 1 , x 2 ) = 1 2 ( x 2 1 + x 2 2 ) 2 + h ( x 2 1 + x 2 2 ) , (21) where h is the critical bifurcation parameter and T controls the eectiv e noise intensit y of the system. As theoretically established in stochastic dynamical systems [15], this oscillator un- dergo es a phenomenological bifurcation (P-bifurcation) exactly at h = 0 . F or h < 0 , the system exhibits limit-cycle oscillations (LCO), and its PDF forms a crater-like geome- try . T op ologically , this corresp onds to a prominent 1-dimensional p ersistent lo op (i.e., a Betti-1 feature). As h increases past 0 , the system shifts to a monostable state, and the geometry structurally collapses in to a single dense 0-dimensional connected comp onen t. T o replicate a real-w orld discrete data acquisition scenario, w e generated a time- v arying sequence of p oint clouds b y sampling from this analytical PDF. Since direct sampling from this unnormalized distribution is non-trivial, we employ ed the Metrop olis- Hastings Marko v Chain Mon te Carlo (MCMC) algorithm. W e sim ulated the dynamical ev olution by discretizing the bifurcation parameter h ∈ [ − 1 , 1] in to 51 uniformly spaced snapshots. F or each snapshot, w e set the eectiv e noise T = 0 . 001 and extracted N = 200 samples. This rigorous pro cedure yields a dynamic p oint cloud sequence { X ( i ) } 51 i =1 that accu- rately captures b oth the con tin uous geometric deformation and the abrupt topological phase transition from a ring to a single cluster, serving as the ground truth for our dy- namic ev aluation framew ork. A visualization of the sampled point clouds is sho wed in the top ro w of gure 2. 16 Figure 2: Ground truth ev olution of the stochastic R VP oscillator. (T op) Scatter plots of the state space showing the transition from a limit cycle to a monostable p oin t. F our sparse snapshots are c hosen as our training densit y samples for top ological optimal transp ort task (illustrated in red b o xes ). (Bottom) The corresp onding p ersistence dia- grams trac king the birth and death of the 1-dimensional homological feature. Figure 3: Baseline(ground truth) sequen tial ev aluation directly computed betw een the ref- erence state ( h = − 1 ) and subsequent empirical snapshots. (a) The top ological distortion p eaks and attens exactly as the limit cycle collapses. (b) Both p ersistence en tropy and the prop osed symmetric h yp ergraph en trop y exhibit a discontin uous jump at the critical p oin t h = 0 . (c) The jump in symmetric en tropy is primarily driv en by the h yp eredge- p ersp ective comp onen t (HE E ).T o facilitate a direct visual comparison, b oth p ersp ective en tropies are normalized by their resp ectiv e theoretical upp er b ounds. Baseline Sequential Ev aluation Before ev aluating our prop osed geo desic interpola- tion metho d, w e rst apply our framework directly to the fully sampled empirical sequence o ver the parameter range h ∈ [ − 1 , 1] to establish a ground truth baseline as presen ted in gure 3. As the bifurcation parameter h increases, the system strictly follows the theo- retical dynamics of the R VP oscillator: the crater-lik e limit cycle collapses in to a dense, 17 monostable cluster at the critical point h = 0 . T o quan tify this P-bifurcation, we compute the structural distortions relativ e to the global reference state h = − 1 . The distortion dynamics show a m ulti-scale temp oral decoupling, corresp onding to a ’macro-meso-micro’ structural relaxation sequence. A t the macro-scale , the top ological distortion ( L topo ) p eaks drastically just b efore h = 0 as the global limit cycle contracts, incurring a substan tial W asserstein p enalty . Once the macroscopic lo op v anishes, L topo attens in to a constan t plateau, as the reference loop can only b e matc hed to diagonal features. Subsequen tly , at the meso-scale , the incidence distortion ( L hyper ) p eaks and reorganizes. The mem b ership incidence matrix m ust dissolv e and reorganize to accommo- date the collapsed top ology . Finally , at the micro-scale , the geometric distortion ( L geom ) settles last, reecting the ph ysical diusion time required for individual stochastic par- ticles to cluster at the new local density peak. A ccompan ying this multi-scale cascade, the structural en tropies (Persistence Entrop y and the prop osed HE sym ) exhibit a surge near the topological tipping p oint (h=0). This behavior suggests their p otential utilit y as indicators of the phase transition without relying on predened thresholds. Dynamic Reconstruction of T op ological Phase T ransitions. T o demonstrate the predictive pow er of our metho dology on temp orally sparse observ ations, we arti- cially subsampled the dataset in to merely four equidistan t k eyframes across the param- eter space (simulating a lo w-resolution data acquisition scenario). W e then applied our reconstruction-based T pOT interpolation framew ork (Algorithm 1) to generate the con- tin uous tra jectory parameterized b y τ ∈ [0 , 1] . The results are illustrated in gure 4. The dynamic ev aluation along the reconstructed geodesic tra jectory approximates the unobserv ed P-bifurcation, closely aligning with the unobserved ground truth dynamics. As visualized in the interpolation results, the generated intermediate states P τ capture b oth the geometric collapse and the progressiv e deterioration of the Betti-1 barco de. More imp ortantly , the reconstructed distortion and en tropy curves along the in terp ola- tion parameter τ closely aligning with the unobserv ed ground truth dynamics previously observ ed along the true parameter h . Our metho d successfully repro duces the abrupt en tropic jumps at the critical transition phase, strictly v erifying the theoretical guarantees of Theorem 1(2) on in terp olated data. F urthermore, the generated trajectory explicitly preserves the macro-meso-micro temp oral decoupling: the in terp olated L topo curv e p eaks and attens well b efore the stabilization of the geometric component L geom . This conrms that our in terp olation strategy do es not merely blend co ordinates linearly , but rigorously reconstructs the authentic, multi-scale pro cess of the underlying sto chastic dynamical system from highly sparse observ ations. 18 (a) Reconstructed spatial geometry and corresp onding p ersistence diagrams (b) Reconstructed structural dynamics Figure 4: Exp erimental v alidation via T pOT geo desic in terp olation. W e sub- sampled the empirical dataset into merely four equidistant keyframes and reconstructed the con tinuous top ological ev olution parameterized b y τ ∈ [0 , 1] . (a) The MDS-based isometric em b edding interpolates the intermediate spatial geometries b etw een the sparse k eyframes, approximating the collapse of the limit cycle. (b) The dynamic ev aluation along the reconstructed geodesic repro duces the hierarc hical macro-meso-micro distortion sequence and the abrupt en tropic jumps at the critical transition p oint, closely aligning with the unobserv ed ground truth dynamics. 19 3.2 Exp eriment 2: Dimension-Sp ecicit y and Negativ e Con trol in a Double-W ell P oten tial Figure 5: Phenomenological bifurcation in the double-w ell p otential mo del. (T op) Scatter plots of the state space sho wing the transition from a bistable regime(t wo distinct clusters) to a monostable regime(a single fused cluster). F our sparse snapshots are c hosen as our training density samples for topological optimal transp ort task (illustrated in red b o xes ). (Bottom) The corresp onding p ersistence diagrams trac king the birth and death of the 1-dimensional homological feature. Data Generation and Ph ysical Mo del. T o further rigorously ev aluate our frame- w ork, particularly its dimension-specicity and the decoupling of its distortion comp o- nen ts, w e construct a second synthetic dataset inspired b y the phenomenological bifurca- tions discussed in App endix A of T an weer et al. [15]. W e simulate a sto chastic system go verned by a parameterized double-well p oten tial: V ( x 1 , x 2 ; h ) = ( x 2 1 − h ) 2 + x 2 2 , (22) where the state space is sampled using a Metrop olis-Hastings MCMC sampler with a generalized noise temp erature T = 0 . 04 . The parameter h decreases from 1 . 00 to − 1 . 00 o ver 51 uniform snapshots. As visualized in the scatter plots of Figure 5, when h > 0 , the system is in a bistable r e gime , forming tw o distinct connected components ( β 0 = 2 ). As h ≤ 0 , the tw o wells merge into a single global minimum, and the p oin t cloud top ologically fuses into a single monostable cluster ( β 0 = 1 ). Unlik e the R VP oscillator in Exp erimen t 1, this fusion pro cess strictly in volv es a 0-dimensional top ological transition, completely devoid of any macroscopic 1-dimensional homological features (i.e., no authentic lo ops or β 1 generators are created or destro yed). Baseline Ev aluation: Sp ecicit y and Dimension-Selectivit y . T o demonstrate the targeted selectivit y of our prop osed metrics, we explicitly constrained the p ersistent ho- mology feature extraction exclusively to 1-dimensional homology ( H 1 ) while ev aluating this β 0 -driv en dataset. The fully separated state at h = 1 . 00 serves as the global reference P (1) . The baseline dynamics (computed directly on the full sequence) support the specicity of our framew ork, as shown in the ground truth curves of Figure 6. Because the t wo distinct probabilit y masses physically migrate and conv erge to w ard the center ov er time, 20 Figure 6: Ground truth baseline dynamics o ver parameter h . Because the system only undergo es a 0-dimensional transition, the 1-dimensional structural metrics ( L topo , L hyper , and Entropies) remain completely suppressed, while the geometric distortion L geom rises to capture the spatial con vergence. the pure geometric distortion ( L geom ) exhibits a contin uous, monotonic rise, successfully capturing the macroscopic spatial fusion. Ho wev er, since our top ological h yp ergraph w as strictly congured to monitor H 1 fea- tures, it acts as a precise theoretical lter. Because no macroscopic lo ops exist during the cluster merging, the H 1 -based top ological distortion ( L topo ) and incidence distortion ( L hyper ) remain completely suppressed at near-zero levels throughout the en tire bifurcation in terv al. Consequen tly , the structural entropies (P ersistence Entrop y and the prop osed HE sym ) do not exhibit the stark step-function discon tinuities seen in Experiment 1; in- stead, they merely uctuate stably around baseline noise levels. This serv es as a p ow erful ne gative c ontr ol , proving that our entrop y indicators are not spuriously triggered by mere spatial displacemen t, but are strictly sensitiv e only to the designated algebraic top ological dimensions. Exp erimen tal V alidation of Dimension-Selectiv e T rac king. W e then subsampled this tra jectory into four highly sparse keyframes and applied our reconstruction-based T pOT in terp olation (Algorithm 1) o ver τ ∈ [0 , 1] . The interpolated tra jectory captures b oth the spatial dynamics and the dimensional decoupling. As demonstrated in Figure 7(a), the dynamic curves computed along the geo desic τ closely aligning with the ground truth: the interpolated geometric curve L geom rises to capture the spatial fusion, while the H 1 -sp ecic top ological comp onents and en- trop y indicators remain correctly inv ariant. This conrms that our in terp olation frame- w ork reliably reconstructs b oth the presence and the absenc e of top ological phenomena, preserving the strict decoupling b etw een micro-scale geometric diusion and macro-scale homological p ersistence across arbitrarily sparse temp oral observ ations. 21 (a) Reconstructed structural dynamics (b) Reconstructed spatial geometry and corresp onding p ersistence diagrams Figure 7: Dimension-sp ecicity and negativ e control ev aluation. (a)Dynamic reconstruction via geo desic in terp olation from only four keyframes successfully reproduces this exact dimension-sp ecic decoupling. (b)The MDS-based isometric embedding and corresp onding 1-D p ersistence diagram. 3.3 Exp eriment 3: Self-Organization and Dimensionalit y in Bi- ological Aggregations Data Generation and Physical Mo del. T o demonstrate the capabilit y of our frame- w ork in analyzing higher-dimensional, real-world biological phenomena, w e turn to the w ell-known D’Orsogna mo del of biological aggregations (e.g., bird o cks and sh sc ho ols)[32– 35]. W e utilized the publicly av ailable sim ulation dataset from T opaz’s study , whic h tracks the complex top ological self-organization of N = 500 self-prop elled in teracting particles. Unlik e the previous purely spatial mo dels, the state of this system must b e describ ed in a 4-dimensional phase space ( x, y , v x , v y ) , as the particles’ orien tations and v elo cities in trinsically dictate the collectiv e top ological state (e.g., distinguishing a single mill from a double mill). The agen ts ob ey Newtonian dynamics driven by self-propulsion, friction, and a pairwise attractive-repulsiv e in teraction potential. Over time, the system sp on taneously self-organizes from a relatively disorganized, disk-like sw arm into a highly structured “mill” or v ortex state, c haracterized b y particles rotating around a hollo w core. In the 22 4-dimensional phase space, the emergence of this hollow core and the rotational o w corresp onds to the birth of prominen t 1-dimensional homological features ( H 1 top ological circles). W e extracted a uniformly spaced sequence of 61 snapshots from T = 1 . 00 to T = 60 . 00 to serve as our empirical sequence. Figure 8: Ground truth ev olution of the D’Orsogna biological aggregation mo del. (T op) The 2-dimensional spatial pro jection ( x, y ) of the 4-dimensional phase space shows parti- cles self-organizing from a disorganized state into a mill with a hollow core. F our sparse snapshots are chosen as our training density samples for top ological optimal transport task (illustrated in red boxes ). (Bottom) The corresponding 1-dimensional p ersistence diagrams track the emergence of a robust Betti-1 feature represen ting the rotational v or- tex. Baseline Ev aluation: Self-Organization and En tropy Drop. W e rst ev aluate the baseline dynamics computed directly on the full 61-frame sequence, using the initial unstructured state at T = 1 . 00 as the global reference P (1) . As visualized in Figure 8, the system gradually dev elops a prominent p ersisten t lo op in H 1 . Quan titatively , the distortion curves in Figure 9(a) eectively capture the multi-scale hierarc hical relaxation of this self-organization. At the macro-scale, the topological dis- tortion ( L topo ) rises substantially as the initial trivial top ology develops a robust Betti-1 v ortex. A t the meso- and micro-scales, the incidence ( L hyper ) and geometric ( L geom ) dis- tortions track the con tin uous physical rearrangement of particles en tering the annular o w. The structural entropies correlate with the system’s dynamical shifts, suggesting their utilit y as e arly-warning indic ators for self-organization. As sho wn in Figure 9(b) and (c), unlik e the top ological distortion ( L topo ) which exhibits a con tinuous, gradual rise as the physical optimal transp ort costs accumulate, the Symmetric Hyp ergraph Entrop y exp erience a sharp, discon tinuous drop from a highly en tropic state ( ≈ 1 . 0 ) to a low- en tropy state ( ≈ 0 . 7 ). This discrete step-function b ehavior clearly demonstrates a rapid state-to-state phase transition. F urthermore, a close examination of the temp oral timeline rev eals a distinct an tic- ipatory sensitivity: the prop osed Symmetric Hyp ergraph En tropy (HE sym ) triggers the abrupt en tropy drop visibly earlier than b oth the topological distortion and the Persis- tence En tropy (PE). This temp oral precedence is theoretically w ell-founded within our macro-meso-micro framew ork. While L topo and PE require the global macroscopic lo op ( H 1 ) to fully mature and ac hieve a measurable barco de lifespan, HE sym con tinuously ev al- uates the instantaneous mesosc opic incidence distribution. Consequen tly , it successfully detects the initial lo calized structural reorganizing—as agents just b egin to align into the 23 ann ular ow—w ell b efore the global vortex fully forms. This mathematical b eha vior reects the ph ysical transition from a c haotic swarm to a deterministic mill state. Figure 9: Baseline dynamic ev aluation of the 4D D’Orsogna mo del. (a) The top ological distortion L topo rises signicantly to capture the formation of the homological lo op. (b, c) Unlike previous bifurcation exp eriments, the structural entropies undergo a signican t drop, rigorously quantifying the thermo dynamic transition from a disorganized chaotic sw arm into a highly self-organized, low-en tropy vortex state. Dynamic Reconstruction of T op ological Phase T ransitions. T o test our frame- w ork’s capacity to handle high-dimensional phase spaces with extreme temp oral sparsity , w e aggressively subsampled the 61-frame sequence in to merely four k eyframes. W e then applied Algorithm 1 to reconstruct the 4-dimensional topological geo desic parameterized b y τ ∈ [0 , 1] . 24 Figure 10: Reconstructed spatial geometry and p ersistence diagrams from only four k eyframes. While the 2D visual pro jection exhibits minor alignment artifacts due to the rigid 4-dimensional optimal transp ort coupling, the 1-dimensional p ersisten t homol- ogy accurately reconstructs the birth of the v ortex core. Figure 11: Exp erimental v alidation of the reconstructed structural dynamics. The T pOT geo desic interpolation tracks the ground truth dynamics from Figure 9, recov ering b oth the the increase in topological distortion and the subsequen t en trop y drop associated with biological self-organization. As shown in Figure 10, our MDS-based geometric reconstruction interpolates the 4- dimensional phase space. It is worth noting that while the 2-dimensional ( x, y ) spatial pro jections of the interpolated states may exhibit slight visual alignment artifacts, these are purely cosmetic. This visual discrepancy is a natural consequence of projecting and rigidly aligning the full 4-dimensional ( x, y , v x , v y ) optimal transport coupling in to a 2D viewing plane. Th us, these alignment artifacts ha ve no impact on the quan titative re- sults of our framew ork. Because our dynamic ev aluation—including b oth the persistent 25 homology reconstruction and the T pOT cost computation—relies strictly on the in trinsic pairwise distance function k τ , the mathematical outcomes are in v arian t to rigid rotational alignmen ts used for visualization. The underlying topological structure and structural en- tropies are highly preserv ed. This is denitiv ely pro ven by the dynamic curv es computed along the in terp olated geo desic τ (Figure 11). The reconstructed tra jectories mirror the ground truth sequence: the framework exactly recov ers the contin uous rise in top ological distortion L topo and the precise tra jectory of the en tropy drop. This v alidates that our algorithm robustly captures complex, high-dimensional biological self-organization from highly sparse observ ations, main taining strict mathematical accuracy ev en when lo w-dimensional visual pro jections b ecome c hallenging. 3.4 Real‐W orld Data: Stroke fMRI 3.4.1 Data description and prepro cessing Let the fMRI data at tw o time p oints b e denoted as X ( M ) ∈ R n x × n y × n z × T M , X ( Y ) ∈ R n x × n y × n z × T Y , where ( i, j, k ) indexes v o xel co ordinates in the spatial domain Ω , and t indexes time. T o obtain a stable v oxel representation, we compute the mean of the BOLD signal: ¯ X ( M ) ( i, j, k ) = 1 T M T M X t =1 X ( M ) ( i, j, k , t ) , ¯ X ( Y ) ( i, j, k ) = 1 T Y T Y X t =1 X ( Y ) ( i, j, k , t ) . Eac h scan is a 4D volume of size 64 × 64 × 34 × 60 . F or eac h time p oint, we compute the v oxel-wise temp oral a v erage, pro ducing tw o 3D volumes of size 64 × 64 × 34 . The human cerebral cortex exhibits highly heterogeneous patterns of functional con- nectivit y . T o eliminate v oxel-lev el noise and enhance in terpretability , we utilize the widely adopted Y eo7 atlas [36] to partition the spatial domain Ω into seven functionally coher- en t regions. After aligning the atlas to the sub ject’s native space, w e obtain a discrete partition Ω = S 7 v =1 Ω v , where v ∈ { 1 , . . . , 7 } indexes the large-scale functional net works (e.g., Visual, Somatomotor, Default Mo de). Eac h vo xel is represented by a 4-dimensional feature vector incorp orating spatial co- ordinates and the BOLD signal: x ( M ) i,j,k = ( i, j, k , ¯ X ( M ) ( i, j, k ) ) ⊤ , x ( Y ) i,j,k = ( i, j, k , ¯ X ( Y ) ( i, j, k ) ) ⊤ . The region-sp ecic p oint clouds for each functional netw ork v are thus dened as: P ( M ) v = { x ( M ) i,j,k | ( i, j, k ) ∈ Ω ( M ) v } , P ( Y ) v = { x ( Y ) i,j,k | ( i, j, k ) ∈ Ω ( Y ) v } . Finally , w e form the full vo xel-wise datasets: X M = 7 [ v =1 P ( M ) v ∈ R N M × 4 , X Y = 7 [ v =1 P ( Y ) v ∈ R N Y × 4 , with corresp onding lab el vectors  M ∈ { 1 , ..., 7 } N M ,  Y ∈ { 1 , ..., 7 } N Y . 26 (a) Raw fMRI v olume (3-month) (b) UMAP pro jections of parcellated net w orks Figure 12: Visualization of the strok e patien t’s fMRI data and dimensionality reduction. (a) An exemplary 3D spatial slice of the patien t’s ra w fMRI volume. (b) 2D UMAP em- b eddings of the vo xel feature v ectors at 3-mon th (left) and 1-y ear (righ t) p ost-strok e time p oin ts. Poin ts are colored b y their corresp onding Y eo7 functional netw ork assignments. 3.4.2 Em b edding and netw ork construction T o visualize and analyse the strok e fMRI volumes at 3 mon ths and 12 months, w e applied the follo wing dimensionality-reduction pip eline: Standar dization and UMAP emb e dding. The v o xel-wise datasets X M and X Y are standardized to ha ve zero mean and unit v ariance. Uniform Manifold Approximation and Pro jection (UMAP) is then applied to map the standardized data from R 4 to R 2 , yielding the resp ectiv e low-dimensional em b eddings Z M , Z Y ⊂ R 2 . The result is sho wn in Figure 12(b). Within the em b eddings, eac h Y eo7 netw ork corresp onds to a regional subset: Z M ,v = { z i ∈ Z M |  M [ i ] = v } , Z Y ,v = { z i ∈ Z Y |  Y [ i ] = v } . These subsets are essen tial for visualizing inter-net w ork geometry and for constructing h yp ergraphs o ver the embedded regions in subsequent en tropy-based analyses. These 2D embeddings are b oth resampled to 600 points and then serve as the input p oin t clouds for our T pOT analysis. W e then construct 1D p ersistent homology (retaining the top 20 p ersistence pairs to represen t the dominant top ological features), binary incidence matrices, and measure top ological net w orks. 3.4.3 Multi-Scale T op ological Analysis and Findings The en tropic T pOT problem is solved as b efore, and distortions are computed on an in terp olation grid of 51 p oints spanning the geo desic b etw een the 3-month and 12-mon th p ost-strok e states. T o characterize the dynamic reconguration of functional brain organization, we ev al- uated the em b edded vo xel sets through a hierarchical set of indicators: the macroscopic metrics (T op ological Distortion L topo and Persistence Entrop y PE), our prop osed meso- scopic dual-persp ective framew ork (HE V , HE E , and the aggregated HE sym ), and the base- line Geometric Distortion ( L geom ). 27 Figure 13 displa ys the heatmaps of these six indicators across the seven Y eo brain areas (v ertical axis) and the interpolated temp oral parameter τ ∈ [0 , 1] (horizon tal axis). A comparativ e analysis across these heatmaps illustrates the m ulti-scale structural dynamics of our framew ork. Macroscopic and Mesoscopic Dynamics. The macroscopic indicators, PE and L topo (Figures 13d, e), follow contin uous tra jectories and sho w similar global trends. Both metrics capture the distinct top ological ev olution (sharply rise at τ ≈ 0 . 4 ) in Netw ork 2 (Somatomotor), c haracterizing structural shifts at a macroscopic scale. A t the mesoscopic scale, the three hypergraph en tropies (Figures 13a-c) are more sensitiv e to lo cal and transient top ological reorganizations. These metrics rev eal region- sp ecic reorganization patterns that are t ypically obscured b y macroscopic measures. Unlik e syn thetic datasets where transitions are often uniform, the real-world fMRI data sho ws a clear de c oupling b etw een HE V and HE E : • In certain brain net works, suc h as Netw ork 2 (Somatomotor) and Net w ork 3 (Dorsal A ttention), HE V and HE E displa y opp osite temp oral trends. • In Netw ork 4 (V entral A ttention), HE V remains stable while HE E decreases around τ ≈ 0 . 9 . • Conv ersely , in others suc h as Net w ork 7 (Default Mo de), HE V and HE E follo w sync hronized tra jectories. This asynchronous b eha vior suggests that brain functional reorganization is complex, with no de participation (HE V ) and functional loop uniformit y (HE E ) evolving indep en- den tly during strok e recov ery . Symmetric Hyp ergraph En tropy (HE sym ) integrates these v ariations. As shown in Figure 13a, HE sym com bines features from b oth vertex- and edge-p ersp ective entropies to represent mesoscopic structural transitions. Microscopic Geometric Distortion. It is worth noting that the Geometric Distortion ( L geom , Figure 13f) displa ys an almost uniform linear gro wth across all brain regions. This indicates that while pure geometric optimal transp ort distances reliably trac k the o verall spatial displacement of the p oint clouds, they are naturally less sensitive to the complex top ological phase transitions o ccurring within the functional net works during the recov ery pro cess. V ertex-Level Lo calization. T o estimate the spatial distribution of these structural transitions, Figure 14a illustrates the v ertex-level hypergraph entrop y eld on the Dor- sal Atten tion Net work. W e mapped the absolute cycle-lev el entrop y v ariation | ∆ HE | = | HE τ − HE 0 | back to the v ertex domain (as dened in Eq. 20). Eac h p oin t represen ts a v ertex in the resampled UMAP embedding, and its color encodes the propagated en- trop y change. Regions highlighted in red corresp ond to vertices that participate in cycles exhibiting the strongest en trop y v ariations, indicating lo calized structural reorganization within the hypergraph. This spatial map, complemented by the exact embeddings at 3 mon ths and 1 year (Figures 14b and 14c) , demonstrates how our cycle-to-p oin t entrop y propagation eectively identies key lo cal transformations. A dditionally , the sp ecic dis- tortion curv es on the Dorsal Atten tion Netw ork is presen ted in Figure 15 28 (a) Symmetric En trop y (HE sym ) (b) V ertex-p ersp ectiv e En- trop y (HE V ) (c) Edge-p ersp ective En tropy (HE E ) (d) Persistence En tropy (PE) (e) T op ological Distortion ( L topo ) (f ) Geometric Distortion ( L geom ) Figure 13: Dynamic en trop y and distortion heatmaps across sev en functional brain netw orks (Y eo7) along the T pOT geo desic. The mesoscopic h yp ergraph en tropies (a-c) capture async hronous lo cal rewirings and region-sp ecic v ariations. In con trast, the macroscopic Persistence Entrop y (d) and T op ological Distortion (e) charac- terize global structural shifts (e.g., in Net work 2). The Geometric Distortion (f ) exhibits uniform linear gro wth, tracking o verall spatial displacemen t rather than topological phase transitions. (a) Visualization of v ertex-lev el h yp ergraph en trop y change in Dorsal Atten tion Netw ork de- ned as (20) (b) Embeddings of the Dorsal A ttention net work at 3 month (c) Em b eddings of the Dorsal A ttention net work at 1 year Figure 14: V ertex-lev el h yp ergraph en tropy analysis of the Dorsal Atten tion netw ork. 29 Figure 15: Distortion and en tropy tra jectories along the T pOT geo desic of the Dorsal A ttention netw ork. Computational details. All exp eriments were implemen ted in Python and using Ripserer.jl for p ersistent homology . Syn thetic runs required approximately 20 seconds p er frame on an M3 Pro CPU; the real‐w orld exp erimen t to ok appro ximately 300 seconds. Conclusion. Exp erimen ts on sto c hastic mo dels and biological systems sho w that the T pOT framew ork recov ers top ological phase transitions from sparse temp oral observ a- tions. The dynamic distortion comp onents ( L geom , L hyper , L topo ) distinguish contin uous ph ysical deformations from discrete structural jumps, identifying a hierarc hical critical transition across macro-, meso-, and micro-scales. Our structural entrop y indicators serve as threshold-free markers for v arious top olog- ical even ts. While Persistence En trop y tracks the lifespan of global homological features, Symmetric Hyp ergraph En tropy (HE sym ) acts as an early-w arning indicator for b oth bi- furcations (indicated by en tropy jumps) and biological self-organization (indicated by en- trop y drops). F urthermore, negativ e con trol ev aluations conrm the dimension-selectivit y and sp ecicit y of these metrics. Finally , application to strok e fMRI data illustrates the utility of the dual-p ersp ective framew ork. The decoupling b etw een v ertex- and edge-p ersp ective en tropies rev eals asym- metric cortical reorganization. Com bined with the p oin t-level h yp ergraph-en tropy eld to localize the structural origins of these transitions, our metho d pro vides a m ulti-scale approac h for analyzing dynamic top ological phenomena in complex systems. 4 Summary and F uture W ork In this pap er, w e in tro duced a rigorous mathematical framew ork for tracking dynamic structural transitions in time-v arying p oint clouds. By utilizing a top ological and hy- p ergraph reconstruction strategy instead of direct abstract netw ork in terp olation, our 30 metho d yields contin uous top ological trajectories from sparse temp oral snapshots. Along these trajectories, we prop osed a hierarchical ev aluation framew ork. W e demonstrated that macroscopic metrics (such as PE and T op ological Distortion) are w ell-tted for cap- turing global ev olutions, whereas our prop osed mesoscopic dual-p ersp ective Hyp ergraph En tropy (HE V and HE E ) provides a highly sensitive lens for detecting transien t, asyn- c hronous lo cal rewirings. Our v alidation across ph ysical, biological, and neuroimaging datasets conrms the sp ecicit y and complemen tarity of these multi-scale indicators. The real-world stroke fMRI exp erimen t in this study serv ed primarily as a metho dolog- ical pro of-of-concept to demonstrate the computational sensitivity of our mathematical framew ork. Lo oking ahead, our future w ork will focus on extending this framework to large-scale longitudinal clinical cohorts, statistically correlating our dynamic h yp ergraph en tropy curves with cognitive and motor reco very scores to establish robust top ological biomark ers for p ost-stroke rehabilitation. Our future w ork will explore the underlying bio- logical mechanisms and information geometry driving the entrop y decoupling observed in these functional brain netw orks. W e aim to mathematically explain how geometric defor- mations of probability supp ort sets drive this async hronous structural evolution. Finally , dev eloping scalable appro ximations for T pOT and top ological extraction on massiv e, un- parcellated graphs remains a crucial computational direction. A c kno wledgemen ts W e would lik e to thank Professor Xiaosong Y ang for the helpful discussions. This work w as supp orted b y the National Natural Science F oundation of China (12401233), NSFC In ternational Creative Researc h T eam (W2541005), National Key Research and Develop- men t Program of China (2021ZD0201300), Guangdong-Dongguan Join t Researc h F und (2023A1515140016), Guangdong Provincial Key Laboratory of Mathematical and Neu- ral Dynamical Systems (2024B1212010004), Guangdong Ma jor Pro ject of Basic Researc h (2025B0303000003), and Hub ei Key Laboratory of Engineering Mo deling and Scien tic Computing. A Supplemen tary material A.1 P ersisten t Homology Giv en a nite point cloud X ⊂ R d , P ersistent Homology(PH) constructs a nested sequence of simplicial complexes (e.g., the Vietoris–Rips or Čec h complexes) parameterized b y a scale parameter ε [37]. As ε increases, simplices are added whenever all pairwise distances among their v ertices fall b elo w ε , yielding a ltration K ε ( X ) : K ε 0  → K ε 1  → · · ·  → K ε M , where K ε i ⊆ K ε i +1 . By trac king the appearance (“birth”) and disappearance (“death”) of homology classes (connected comp onen ts, lo ops, etc.) throughout this ltration, one obtains a p ersistence diagram—a m ultiset of p oints { ( b i , d i ) } in the plane, eac h recording the in terv al ( b i , d i ) o ver which a top ological feature exists. 31 The multiset of lifespans { d i − b i } serv es as a succinct “signature” of the data’s top ol- ogy: longer interv als corresp ond to more prominent features, while short-liv ed inter- v als often reect top ological noise. P ersistence diagrams are stable under p erturbations of the input, and admit w ell-studied metrics suc h as the b ottleneck and W asserstein distances[38, 39]. A.2 Gromo v-W asserstein and Co-Optimal T ransp ort Distances In this subsection we review three fundamen tal optimal-transp ort-based metrics that form the building blo c ks of the T op ological Optimal T ransp ort framework. Let ( X , d ) b e a Polish metric space and µ, µ ′ t wo Borel probabilit y measures supp orted on X. F or p ∈ 1 , ∞ ) , the p-W asserstein distance is dened by d W,p ( µ, µ ′ ) =  inf π ∈ Π( µ,µ ′ ) Z X × X d ( x, x ′ ) p d π ( x, x ′ )  1/ p , where Π( µ, µ ′ ) denotes the set of couplings (joint measures) with marginals µ and µ ′ , and an optimal coupling π realises this inmum. When comparing p ersistence diagrams D and D ′ , one augmen ts the plane with a “diagonal” p oin t ∂ Λ to allo w unmatc hed features, and replaces Π( µ, µ ′ ) by the set of admissible partial matc hings Π( D , D ′ ) . The resulting diagram-W asserstein distance is d PD W,p ( D , D ′ ) p = min π ∈ Π( D ,D ′ )  X ( a,b ) ∈ π ∥ a − b ∥ p p + X s ∈ U π ∥ s − Proj ∂ Λ( s ) ∥ p p  , where U π are unmatc hed p oin ts and Proj ∂ Λ pro jects on to the diagonal[39]. When the t w o measures liv e on dieren t metric spaces ( X, d, µ ) and ( X ′ , d ′ , µ ′ ) , the Gromo v–W asserstein (GW) distance aligns their relational structures b y minimizing dif- ferences of pairwise distances. F or a coupling π ∈ Π( µ, µ ′ ) , the p-distortion dis GW ,p ( π ) =  Z Z ( X × X ′ ) 2   d ( x, y ) − d ′ ( x ′ , y ′ )   p d π ( x, x ′ ) d π ( y , y ′ )  1/ p . The GW distance is then d GW ,p  ( X , d, µ ) , ( X ′ , d ′ , µ ′ )  = inf π ∈ Π( µ,µ ’ ) dis GW ,p ( π ) , whic h denes a pseudo-metric on the metric-measure spaces[8]. T o compare t wo measure hypernetw orks H = ( X , µ, Y , ν, ω ) and H ′ = ( X ′ , µ ′ , Y ′ , ν ′ , ω ′ ) , where ω enco des v ertex–h yp eredge incidences, one seeks couplings π v ∈ Π( µ, µ ′ ) (v ertices) and π e ∈ Π( ν, ν ′ ) (hyperedges) that minimise dis COOT ,p ( π v , π e ) =  Z X × X ′ × Y × Y ′   ω ( x, y ) − ω ′ ( x ′ , y ′ )   p d π v ( x, x ′ ) d π e ( y , y ′ )  1/ p . The co-optimal transp ort distance is then d COOT ,p ( H , H ′ ) = inf π v ∈ Π( µ,µ ′ ) π e ∈ Π( ν,ν ′ ) dis COOT ,p ( π v , π e ) , inducing a pseudo-metric on the space of measure h yp ernet works[11]. 32 A.3 Measure T op ological Net w ork A me asur e top olo gic al network is dened as the triple P =  ( X , k , µ ) , ( Y , ι, ν ) , ω  , whic h in tegrates geometric, top ological, and incidence information in to a unied mea- sure‐theoretic framew ork[27]. Belo w we describ e each comp onent in detail. Geometric Component ( X , k , µ ) . • X = { x 1 , . . . , x N } ⊂ R d is a nite p oin t cloud represen ting the raw data samples. • k : X × X → R is a symmetric kernel or similarity function; for example, one may tak e k ( x, x ′ ) = exp  −∥ x − x ′ ∥ 2 / σ 2  or k ( x, x ′ ) = ∥ x − x ′ ∥ , lo cal geometric anities or pairwise distances. • µ is a probabilit y measure supp orted on X , often c hosen to b e the uniform distribu- tion µ ( { x i } ) = 1/ N . This measure allows us to sp eak of “mass” at eac h data p oin t and to transp ort mass in later constructions. T op ological Comp onen t ( Y , ι, ν ) . • Y is a lo cally compact Polish space whose points corresp ond to homology generators (e.g. cycles) extracted via p ersisten t homology . • ι : Y → Λ is a con tinuous map in to the persistence‐diagram domain Λ = { ( b, d ) ∈ R 2 | d > b ≥ 0 } . Under ι , eac h generator y ∈ Y is sen t to its birth–death pair ι ( y ) = ( b y , d y ) . • ν is a Radon measure on Y such that the push‐forw ard ι # ν coincides with the usual p ersistence‐diagram measure on Λ . In practice one may tak e ν to assign equal mass to eac h cycle representativ e in a given homological dimension. Incidence F unction ω : X × Y → R . The function ω records the binary mem b ership of p oin ts in cycles: ω ( x, y ) = ( 1 , x is a v ertex of the cycle represented by y , 0 , otherwise . By treating ω as a measurable k ernel, w e couple the geometric and top ological parts: mass transp orted b et w een tw o p oin t clouds in X can b e coheren tly matched with transp ort of their asso ciated cycles in Y . The measure top ological net w ork P simultaneously captures: • Metric structur e through ( X , k , µ ) , enabling geometry‐a ware transp ort; • T op olo gic al fe atur es via ( Y , ι, ν ) , preserving the birth–death statistics of homology classes; • Higher‐or der r elation through ω , enforcing consistency b etw een p oin ts and the cycles they generate. 33 A.4 T op ological Optimal T ransp ort (T pOT) Giv en tw o measure top ological netw orks P =  ( X , k , µ ) , ( Y , ι, ν ) , ω  and P ′ =  ( X ′ , k ′ , µ ′ ) , ( Y ′ , ι ′ , ν ′ ) , ω ′  , The T op olo gic al Optimal T r ansp ort (T pOT) distance of order p then is dened by d TpOT ,p ( P , P ′ ) = inf π v ∈ Π( µ,µ ′ ) π e ∈ Π adm ( ν,ν ′ ) h L geom ( π v ) + L topo ( π e ) + L hyper ( π v , π e ) i 1/ p , (23) where the inmum is taken ov er all v ertex–v ertex couplings π v and admissible top ology couplings π e [27]. In tuitively , π v matc hes data p oints in X with those in X ′ , while π e matc hes homology generators in Y with those in Y ′ . The three distortion terms quan tify mismatc hes of geometry , top ology , and incidence structure, resp ectiv ely . Geometric distortion L geom . This term generalises the Gromov–W asserstein dis- crepancy to our k ernelized setting: L geom ( π v ) = Z Z ( X × X ′ ) 2   k ( x 1 , x 2 ) − k ′ ( x ′ 1 , x ′ 2 )   p d π v ( x 1 , x ′ 1 ) d π v ( x 2 , x ′ 2 ) . (24) By comparing pairwise anities k versus k ′ , this term ensures that the global metric relationships among p oin ts are preserv ed under the optimal coupling. T op ological distortion L topo . The T pOT measures distance betw een p ersistence diagrams via a classical W asserstein cost: L topo ( π e ) = Z ¯ Y × ¯ Y ′   ι ( y ) − ι ′ ( y ′ )   p p d π e ( y , y ′ ) , (25) where ¯ Y = Y ∪ { ∂ Y } and lik ewise for ¯ Y ′ , with ∂ Y the diagonal “n ull” feature. This term aligns birth–death pairs, p enalizing large shifts in feature lifetimes. Hyp ergraph incidence distortion L hyper . Finally , to couple points and cycles consisten tly , we hav e L hyper ( π v , π e ) = Z X × X ′ × Y × Y ′   ω ( x, y ) − ω ′ ( x ′ , y ′ )   p d π v ( x, x ′ ) d π e ( y , y ′ ) . (26) Since ω and ω ′ are binary incidence functions, this term enforces that matc hed points participate in matc hed cycles, thereby preserving higher‐order top ological connectivit y . In summary , T pOT simultaneously optimizes ov er corresp ondences of p oin ts and cy- cles, striking a balance b etw een geometric delit y , top ological consistency , and cycle mem b ership preserv ation. The resulting distance d TpOT ,p denes a pseudo-metric on the space of measure topological net w orks, suitable for comparing complex data with b oth geometry and top ology . 34 A.5 Geo desic In terp olation in T pOT Space An imp ortant prop ert y of the T pOT framework is that the space of measure top ological net works endo wed with the distance d TpOT ,p is a (non‐negatively curved) geo desic space. In particular, giv en t w o netw orks P and P ′ and an optimal coupling, one can explicitly construct a constant‐speed geo desic b et w een them via conv ex combinations of their data. The follo wing result summarises this construction. Let P =  ( X , k , µ ) , ( Y , ι, ν ) , ω  , P ′ =  ( X ′ , k ′ , µ ′ ) , ( Y ′ , ι ′ , ν ′ ) , ω ′  , and let π v ∈ Π( µ, µ ′ ) , π e ∈ Π adm ( ν, ν ′ ) be optimal couplings achieving the inm um in (23). Then for each t ∈ [0 , 1] , the in terp olated netw ork is dened as P t =  ( e X , k t , π v ) , ( e Y , ι t , π e ) , ω t  , (27) where: • e X = X × X ′ is the pro duct of the tw o point clouds, endo wed with the coupling measure π v . • e Y = ( Y × Y ′ ) ∪ ( Y × { ∂ Y ′ } ) ∪ ( { ∂ Y } × Y ′ ) augmen ts the cycle space with diagonal placeholders to accommo date unmatc hed generators, carrying the coupling π e . • Geometric k ernel in terp olation: k t  ( x 1 , x ′ 1 ) , ( x 2 , x ′ 2 )  = (1 − t ) k ( x 1 , x 2 ) + t k ′ ( x ′ 1 , x ′ 2 ) . A t t = 0 , this recov ers the original kernel k , and at t = 1 it recov ers k ′ , while for in termediate t it provides a linear blend of anities. • T op ological coordinate in terp olation: ι t ( y , y ′ ) = (1 − t ) ι ( y ) + t ι ′ ( y ′ ) . Here ι ( y ) and ι ′ ( y ′ ) lie in the p ersistence diagram plane, and their con vex combina- tion traces a straigh t line segment b et w een birth–death pairs. • Hyp eredge incidence in terp olation: ω t  ( x, x ′ ) , ( y , y ′ )  = (1 − t ) ω ( x, y ) + t ω ′ ( x ′ , y ′ ) . This in terp olation main tains fractional mem b ership v alues, ensuring that the binary incidence structure of cycles deforms con tinuously along the geo desic. Besides, w e hav e the following prop erties: • Metric geo desicity . The space  P / ∼ w , d TpOT ,p  is a geo desic metric space. • Conv exit y of geo desics for p = 2 . When p = 2 , ev ery geodesic in this space is c onvex . • Non‐negative curv ature. The metric space  P / ∼ w , d TpOT ,p  has curv ature bounded b elo w b y zero. This guaran tees con v exit y of the cost functional and stabilit y of in terp olation. 35 B Pro ofs for Section 2.4 Pr o of of Pr op erty 1. W e prov e the prop erty for the v ertex-p ersp ectiv e entrop y HE V ( H ) ; the pro of for HE E ( H ) follo ws symmetrically . By Denition 1, p ( v ) = L ( v ) I total constitutes a discrete probability distribution o ver the nite active vertex set V ∗ , satisfying p ( v ) > 0 for all v ∈ V ∗ and P v ∈ V ∗ p ( v ) = 1 . By Gibbs’ inequality , giv en tw o discrete probability distributions P = { p ( v ) } v ∈ V ∗ and Q = { q ( v ) } v ∈ V ∗ , then − X v ∈ V ∗ p ( v ) ln p ( v ) ≤ − X v ∈ V ∗ p ( v ) ln q ( v ) with equalit y if and only if p ( v ) = q ( v ) , for v ∈ V ∗ . Substitute q ( v ) = 1 | V ∗ | in to the inequalit y , we obtain that the vertex-persp ectiv e entrop y is p ositively strictly b ounded: 0 < − X v ∈ V ∗ p ( v ) ln p ( v ) ≤ ln | V ∗ | The upp er b ound ln | V ∗ | is achiev ed if and only if the probabilit y distribution is uniform, i.e., p ( v ) = 1 | V ∗ | for all v ∈ V ∗ . Substituting the denition of p ( v ) , this equalit y holds if and only if L ( v ) I total = 1 | V ∗ | = ⇒ L ( v ) = I total | V ∗ | ∀ v ∈ V ∗ This implies that the degree L ( v ) is a constan t for all active v ertices. By denition in graph theory , a h yp ergraph where all v ertices hav e the iden tical degree is a r e gular hyp er gr aph . Dually , HE E ( H ) = ln | E ∗ | if and only if S ( e ) is constant for all e ∈ E ∗ , which denes a uniform hyp er gr aph . Pr o of of The or em 1. F ollo wing the algebraic framew ork of Homan and Singleton[40] for constraining graph top ologies via Diophan tine equations, our pro of analyzes the entrop y transition using a n um b er-theoretic approac h rather than contin uous-limit appro xima- tions. Let H − = ( V , E − ) denote the h yp ergraph strictly b efore t c , with total incidence I . Let H + = ( V , E + ) denote the h yp ergraph at t c with the new edge e new connecting a v ertex subset V new ( | V new | = k ). The new total incidence is I + k . F or notational simplicity , let Σ = P v ∈ V ∗ L ( v ) ln L ( v ) . Using Denition 1, the v ertex- p ersp ective entrop y b efore the transition is: HE V ( H − ) = − X v ∈ V ∗ L ( v ) I ln  L ( v ) I  = ln I − Σ I . After the transition, the degrees up date to L ( v ) + 1 for v ∈ V new , and remain L ( v ) for v / ∈ V new . The lo cal v ariation term is dened as ∆Σ = P v ∈ V new  ( L ( v ) + 1) ln ( L ( v ) + 1) − L ( v ) ln L ( v )  . The new entrop y is: HE V ( H + ) = ln ( I + k ) − Σ + ∆Σ I + k . W e pro ceed b y analyzing the exact condition under whic h the en tropy remains un- c hanged. Assume HE V ( H − ) = HE V ( H + ) . Equating the tw o expressions and rearranging yields: ln ( I + k ) − ln I = Σ + ∆Σ I + k − Σ I = I · ∆Σ − k Σ I ( I + k ) . 36 Multiplying both sides by I ( I + k ) to isolate the terms with integer co ecients, w e obtain: I ( I + k ) ln ( I + k ) − I ( I + k ) ln I = I · ∆Σ − k Σ . W e expand Σ and ∆Σ into their explicit v ertex summations b y partitioning V ∗ in to aected v ertices ( v ∈ V new ) and unaected v ertices ( v / ∈ V new ): I · ∆Σ − k Σ = I X v ∈ V new  ( L ( v ) + 1) ln ( L ( v ) + 1) − L ( v ) ln L ( v )  − k X v ∈ V new L ( v ) ln L ( v ) + X v / ∈ V new L ( v ) ln L ( v ) ! . Grouping the terms for v ∈ V new and v / ∈ V new separately , and utilizing the logarithmic iden tity x ln y = ln ( y x ) , the righ t-hand side b ecomes: X v ∈ V new ln  ( L ( v ) + 1) I ( L ( v )+1) L ( v ) ( I + k ) L ( v )  − X v / ∈ V new ln  L ( v ) kL ( v )  . Applying the same logarithmic iden tities to the left-hand side and equating both sides as single logarithms of pro ducts , w e obtain: ln  ( I + k ) I ( I + k ) I I ( I + k )  = ln Q v ∈ V new ( L ( v ) + 1) I ( L ( v )+1) Q v ∈ V new L ( v ) ( I + k ) L ( v ) · Q v / ∈ V new L ( v ) kL ( v ) ! . Since the logarithmic function is strictly monotonic, w e can remov e the logarithms, whic h yields a strict integer multiplicativ e identit y: ( I + k ) I ( I + k ) Y v / ∈ V new L ( v ) kL ( v ) Y v ∈ V new L ( v ) ( I + k ) L ( v ) = I I ( I + k ) Y v ∈ V new ( L ( v ) + 1) I ( L ( v )+1) . (28) Equation (28) represen ts a highly constrained non-linear Diophan tine equation[41, 42]. By the F undamen tal Theorem of Arithmetic, b oth sides must yield the exact same prime factorization. Notice that the addition of e new acts as a lo cal top ological p erturbation, y et it in tro- duces a massiv e global m ultiplier shift via the terms ( I + k ) I ( I + k ) and I I ( I + k ) . Because I and I + k generally possess distinct prime factors (e.g., they are coprime if k = 1 ), satisfying this equality demands that the exact missing prime factors b e supplied by the degree sequences L ( v ) of the vertices. In the com binatorial space of h yp ergraphs, the degree sequence cannot arbitrarily absorb suc h macroscopic algebraic shifts without fundamentally restructuring the en tire graph. A solution requires an exact matc hing of prime factors b etw een the global size I and local degrees L ( v ) . Consequen tly , equality holds only for a trivially small, mathemat- ically contriv ed class of degree sequences. F or generic structural transitions, the prime factorizations strictly div erge, guaranteeing HE V ( H − )  = HE V ( H + ) . Pr o of of The or em 2. Let H − = ( V , E − ) denote the h yp ergraph strictly b efore t c , with total incidence I . Let H + = ( V , E + ) denote the h yp ergraph at t c , where E + = E − ∪ { e new } and the size of the new h yp eredge is S ( e new ) = k . The new total incidence is I + k . 37 Note that unlik e vertex degrees which up date lo cally , the sizes of the existing hyper- edges remain strictly unchanged: S ( e ) is constant for all e ∈ E − . Let Σ E = P e ∈ E − S ( e ) ln S ( e ) . Using Denition 2, the h yp eredge-p ersp ective entrop y b efore the transition is: HE E ( H − ) = − X e ∈ E − S ( e ) I ln  S ( e ) I  = ln I − Σ E I . After the top ological transition, the summation expands to include the new hyperedge e new , while the global denominator up dates to I + k : HE E ( H + ) = − X e ∈ E − S ( e ) I + k ln  S ( e ) I + k  − k I + k ln  k I + k  = ln ( I + k ) − Σ E + k ln k I + k . W e analyze the condition for entrop y stagnation b y assuming HE E ( H − ) = HE E ( H + ) . Equating the t wo forms and rearranging yields: ln ( I + k ) − ln I = I ( k ln k ) − k Σ E I ( I + k ) . Similar to the pro of of Theorem 1, w e deriv e a Diophantine equation ( I + k ) I ( I + k ) Y e ∈ E − S ( e ) kS ( e ) = I I ( I + k ) k I k . (29) This iden tity reveals a mathematically rigid algebraic dep endency . The righ t-hand side of Equation (29) is completely determined by the macroscopic v ariables: the initial total incidence I and the p erturbation size k . Con versely , the left-hand side relies heavily on the microscopic top ological distribution of all prior existing h yp eredge sizes S ( e ) . By the F undamen tal Theorem of Arithmetic, this equalit y holds if and only if both sides share the exact same prime factorization. This condition requires the term Q S ( e ) kS ( e ) to exactly oset the dierence in prime factors. Suc h a prime factor alignment b e- t ween the prior top ological state and the global p erturbation is com binatorially improb- able. Therefore, for any generic top ological transition, the equality fails, conrming that HE E ( H − )  = HE E ( H + ) . Pr o of of The or em 3. Since the isomorphism preserv es incidence, the degree mapping is preserv ed: L 1 ( v ) = L 2 ( φ ( v )) for all v ∈ V 1 . S 1 ( e ) = S 2 ( ψ ( e )) for all e ∈ E 1 . 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