This paper illustrates a further application of topological data analysis to the study of self-organising models for chemical and biological systems. In particular, we investigate whether topological summaries can capture the parameter dependence of pattern topology in reaction diffusion systems, by examining the homology of sublevel sets of solutions to Turing reaction diffusion systems for a range of parameters. We demonstrate that a topological clustering algorithm can reveal how pattern topology depends on parameters, using the chlorite–iodide–malonic acid system, and the prototypical Schnakenberg system for illustration. In addition, we discuss the prospective application of such clustering, for instance in refining priors for detailed parameter estimation for self-organising systems.

While recent years have witnessed a fast growth in mathematical artificial intelligence (AI). One of the most successful mathematical AI approaches is topological data analysis via persistent homology (PH) that provides explainable AI by extracting multiscale structural features from complex datasets. Interpretability is crucial for world models, the new frontier in AI that can understand and simulate reality. This article investigates the interpretability and representability of three foundational mathematical AI methods, PH, persistent Laplacians (PL) derived from topological spectral theory, and persistent commutative algebra (PCA) rooted in Stanley–Reisner theory. We apply these methods to a set of data, including geometric shapes, synthetic complexes, fullerene structures, and biomolecular systems to examine their geometric, topological, and algebraic properties. PH captures topological invariants such as connected components, loops, and voids through persistence barcodes. PL extends PH by incorporating spectral information, quantifying topological invariants, geometric stiffness, and connectivity via harmonic and nonharmonic spectra. PCA introduces algebraic invariants such as graded Betti numbers, facet persistence, and ‐vectors, offering combinatorial, topological, geometric, and algebraic perspectives on data over scales. Comparative analysis reveals that while PH offers computational efficiency and intuitive visualization, PL provides enhanced geometric sensitivity, and PCA delivers rich algebraic interpretability. Together, these methods form a hierarchy of mathematical representations, enabling explainable and generalizable AI for real‐world data.

Topological data analysis and topological deep learning beyond persistent homology: a review

This study explores the feasibility and effectiveness of an interpretable machine learning model for assessing the pathological grading of pancreatic ductal adenocarcinoma (PDAC) using radiomics and topological features derived from contrast-enhanced CT habitat subregions. A retrospective study was conducted on a total of 306 patients with PDAC from two hospitals: a training cohort (n = 176), a validation cohort (n = 76), and a test cohort (n = 54). K-means clustering analysis was first used to segment portal venous phase CT images into three habitat regions. Radiomics features of the whole-tumour region, along with radiomics and topological features of each habitat region, were extracted respectively. LASSO regression was applied for feature dimensionality reduction to construct the radiomics score (Rad-score) for the whole-tumour region and the habitat score (H-score) for each habitat region. Meanwhile, logistic regression was used to identify statistically significant predictors from clinical and semantic features. Five machine learning algorithms were used to construct Habitat-TDA models, with interpretability analysis performed via SHAP analysis. Total volume, diabetes, and M staging were identified as independent risk factors for predicting the pathological grading of PDAC, and were used to construct the Clinical model. 6 radiomics features with non-zero coefficients were selected to calculate the Rad-score, which was further used to construct the WholeRad model. In the three habitat regions, 6, 5, and 6 topological and radiomics features were included to generate the H-score. The logistic regression algorithm performed best in the validation and test cohorts and was ultimately selected as the classifier for constructing the Habitat-TDA model. SHAP analysis showed that H-score1, derived from Habitat Region 1 (the habitat region with the lowest average CT value), has the most significant average impact on the model output intensity. The AUC values of the Habitat-TDA model in the training, validation, and test cohorts were 0.894, 0.872, and 0.829, all outperforming the clinical model (0.784, 0.765, 0.731) and WholeRad model (0.817, 0.810, 0.773). The Habitat-TDA model improves the accuracy and interpretability of preoperative predictions of PDAC grading, providing a promising tool for personalised management.

ABSTRACT We present cosmological constraints from Dark Energy Survey Year 3 (DES Y3) weak lensing data using persistent homology, a topological data analysis technique that tracks how features like clusters and voids evolve across density thresholds. For the first time, we apply spherical persistent homology to galaxy survey data through the algorithm TopoS2, which is optimized for curved-sky analyses and healpix compatibility. Employing a simulation-based inference framework with the Gower Street simulation suite – specifically designed to mimic DES Y3 data properties – we extract topological summary statistics from convergence maps across multiple smoothing scales and redshift bins. After neural network compression of these statistics, we estimate the likelihood function and validate our analysis against baryonic feedback effects, finding minimal biases (under $0.3\sigma$) in the $\Omega _\mathrm{m}-S_8$ plane. Assuming the wCold Dark Matter model, our combined Betti numbers and second moments analysis yields $S_8 = 0.821 \pm 0.018$ and $\Omega _\mathrm{m} = 0.304\pm 0.037$ – constraints 70 per cent tighter than those from cosmic shear two-point statistics in the same parameter plane. Our results demonstrate that topological methods provide a powerful and robust framework for extracting cosmological information, with our spherical methodology readily applicable to upcoming Stage IV wide-field galaxy surveys.

Heart rate variability (HRV) is a well-established marker of autonomic regulation and undergoes profound maturation during early human development. In this study, topological data analysis (TDA) is applied to investigate the evolving geometric complexity of HRV across pediatric developmental stages. Using persistent homology in homological dimension 1, we extracted topological descriptors from time-delay embedded RR interval series of 127 individuals aged 1 month to 17 years. We identified statistically significant, age-dependent transformations in the topological structure of HRV signals. Neonates and infants exhibited a greater number and strength of persistent features, reflecting highly heterogeneous cardiac control dynamics during early autonomic maturation. In contrast, adolescents displayed reduced topological complexity and increased entropy, suggesting a shift toward more uniform and structured physiological control. Topological measures correlated with conventional HRV indices, confirming their physiological relevance. Furthermore, pairwise distances between persistence landscapes revealed an inverse relationship between intra-group topological variability and classical HRV measures. Collectively, our findings demonstrate that persistent homology provides a powerful, multiscale-aware framework to capture developmental trajectories in cardiac autonomic regulation, with potential applications in pediatric monitoring, developmental physiology, and early detection of dysautonomia.

To solve the optimal transport problem between two uniform discrete measures of the same size, one seeks a bijective assignment that minimizes some matching cost. For this task, exact algorithms are intractable for large problems, while approximate ones may lose the bijectivity of the assignment. We address this issue and the more general cases of non-uniform discrete measures with different total masses, where partial transport may be desirable. The core of our algorithm is a variant of the Quicksort algorithm that provides an efficient strategy to randomly explore many relevant and easy-to-compute couplings, by matching BSP trees in loglinear time. The couplings we obtain are as sparse as possible, in the sense that they provide bijections, injective partial matchings or sparse couplings depending on the nature of the matched measures. To improve the transport cost, we propose efficient strategies to merge k sparse couplings into a higher quality one. For k = 64, we obtain transport plans with typically less than 1% of relative error in a matter of seconds between hundreds of thousands of points in 3D on the CPU. We demonstrate how these high-quality approximations can drastically speed-up usual pipelines involving optimal transport, such as shape interpolation, intrinsic manifold sampling, color transfer, topological data analysis, rigid partial registration of point clouds and image stippling.

Unveiling complex nonlinear dynamics in stock markets through topological data analysis

An enhancement of multi-scope topological graph pooling and representation learning with attention for molecular graph classification.

This study investigates the structural organization of an email communication network constructed from the SNAP Enron dataset, where nodes represent individual email addresses and edges correspond to communication links between them. Communities within the network were identified using the Label Propagation Algorithm (LPA), yielding 35 distinct groups. To evaluate the structural coherence and significance of these communities, we integrated two complementary analytical frameworks: Persistent Homology, from Topological Data Analysis (TDA), and Modularity, a key metric in network theory. Persistent homology was utilized to detect enduring topological features—such as connected components, loops, and voids—that characterize the intrinsic structure of each community across varying filtration scales. Modularity analysis, in turn, quantified the relative density of intra- and inter-community connections. Combining these approaches enabled the classification of communities as non-significant, significant, influential, or highly influential. The findings reveal a strong correlation between persistent topological features and high modularity scores, offering deeper insights into the stability, cohesion, and influence of communities in large-scale social communication networks.

This paper introduces a novel categorical framework that unifies classical algebraic topology with modern topological data analysis through the lens of category theory. We develop the theory of persistence categories as a natural generalization of persistence modules, establishing functorial relationships between classical topological invariants and their persistent counterparts. Our approach reveals deep connections between sheaf cohomology, spectral sequences, and multi-parameter persistence, providing a rigorous mathematical foundation for understanding the stability and structure of topological features in data. We prove that persistent homology can be viewed as a particular instance of a more general categorical construction that encompasses both classical and computational topology. Furthermore, we establish new stability theorems for categorical persistence and demonstrate how classical results in algebraic topology can be lifted to the persistent setting through appropriate functorial constructions. We present practical applications in data science, computational biology, and machine learning, demonstrating the effectiveness of our theoretical framework through concrete implementations and computational experiments.

Data-model matching, typically achieved through direct contact, is critical to digital markets. However, when data and models belong to different owners, the direct contact-based form faces some security threats, including data security, privacy disclosure, and model reverse engineering attacks. A natural question emerges: Can effective data-model matching be achieved without direct contact ? Previous methodologies can partially alleviate but not eliminate the necessity of direct contact between data and models, making security and privacy challenges persist throughout the matching process. In this paper, our research findings indicate that, despite the essential differences between data and models, both can be represented using topological spaces. Therefore, we establish a unified metric of data complexity and model expressivity from a topological perspective. The unified metric satisfies three conditions toward contactless data-model matching. Then, we develop a contactless matching paradigm, circumventing the necessity for direct contact between data and models and addressing privacy and security concerns. Specifically, we use topological data analysis to generate the data complexity topological descriptors (DCTD) and use topological simplification to generate the model expressivity topological descriptors (METD). We compute the matching degree and return the matching result. Through theoretical proof and experimental analysis, we validate the feasibility of the proposed contactless data-model matching paradigm in real-world scenarios.

Abstract Interactions of the maxima and minima of the univariate functions appear in combinatorics as Dyck paths, and in topological data analysis as persistent homology. We study these descriptors for Brownian motions with drift, deriving the intensity measure and correlation functions for the persistence diagram point process $$\textbf{PH}_0$$ PH 0 , and quantifying the intrinsic asymmetries in the coupling of maxima and minima.

Early detection of breast cancer by mammography scans is crucial for improving treatment outcomes. However, low image resolutions, size, and location of lesions in dense breast tissue prove to be challenges in mammography, underscoring the importance of accurate and efficient computer-aided diagnostic systems. This article introduces a novel classification framework that utilizes histogram of oriented gradients (HOG) as a feature extractor and principal component analysis (PCA) for dimensionality reduction. Classification is implemented using the persistent homology classification algorithm (PHCA), which leverages persistent homology (PH) to capture topological properties of mammography images. The framework was evaluated on 7,632 images from the INbreast dataset with an extensive use of grid-search cross-validation to optimize preprocessing parameters. Two optimal combinations of HOG parameters and scaler were identified, with the best configuration (16 × 16 pixels per cell, 3 × 3 cells per block, and Minmax scaler) achieving strong performance. Validating on the test set, PHCA achieved an overall accuracy, precision, recall, F1-score, and specificity of 97.31%, 96.86%, 97.09%, 96.97%, and 96.86%, respectively. Clinically, the high precision (98.23%) and high recall (97.75%) for malignant cases highlight PHCA’s sensitivity in identifying malignancies, ensuring that very few malignant cases go undetected with highly trustworthy predictions. These results are shown to be competitive with existing state-of-the-art models, even exceeding in some cases, while requiring lower computational cost than deep learning-based approaches. Although the proposed method trails advanced deep models by 3–4% in some metrics, it offers a computationally efficient alternative and a potential for deployment in large-scale screening systems, demonstrating the promise of topological data analysis for breast cancer classification.

The microstructure of porous electrodes of solid oxide cells (SOCs) significantly impacts their electrochemical performance. Therefore, various attempts have been made to analyze their complex porous microstructure in three dimensions. The obtained microstructures have been quantitatively evaluated using structural metrics, such as volume fraction, surface area density, and tortuosity factor of the constituent phases, as well as double- and triple-phase boundaries. These structural characteristics are useful to correlate the electrode performance with their microstructure, thereby determining optimal electrode design. However, these intuitive metrics often fails to explain the electrode performance. In fact, structural changes during long term operation of SOCs are subtle so that the performance degradation of the electrodes cannot be fully detected by these intuitive characteristics. Therefore, it is necessary to find a way to extract hidden structural metrics that characterize the electrode microstructure and to clarify their correlation with electrode performance.One of the promising candidates for such structural metrics is topological information. Topology is a structural property that is invariant to successive deformation operations, such as connected structures and the number of holes in the structures. Topology-based structural analysis has many applications in material science, where macroscopic properties of materials are correlated with topological information about their internal structures (e.g., crystal structures). For example, Wang et al. [1] performed a topological analysis on the crystal structure of double-phase steel to characterize its structure and to predict its macroscopic properties, such as stress-strain curves. Application of topological data analysis is also found in the analysis of SOFC electrodes; Pawlowski et al. [2] attempted to capture the structural changes in SOFC anodes during long-term operation from a topological perspective.In addition, topological analysis is regarded as a tool for dimensional reduction of complex structural information. Therefore, the information extracted from the topological analysis can be used to evaluate conventional metrics of porous microstructures, such as surface area and triple-phase boundary. If these metrics can be accurately evaluated from a limited number of information from the topological analysis, the numerical cost is expected to be significantly reduced.Therefore, this study investigates the applicability of the topological analysis to the electrode microstructure analysis. Fig. 1 shows the schematic diagram of the structural analysis in this study. First, persistent homology analysis [3] is employed to extract the topological information of the electrodes, where the birth and death of topological features are detected during the filtration process within the structure datasets to produce the persistent diagram (PD). The persistent diagram is then discretized and concatenated to obtain the persistent image (PI). Subsequently, principal component analysis (PCA) is conducted to further reduce the dimensionality of the structural information extracted in the persistent image. The obtained values of the principal components are correlated with the conventional structural metrics of the porous electrodes, such as volume fraction, surface area density, and triple-phase boundary density.Moreover, neural networks (NNs) consisting of fully-connected layers are constructed to quantify the structural metrics from the principal components. The constructed neural networks are trained using the real electrode microstructure datasets obtained using the focused ion beam and scanning electron microstructure [4]. To validate the developed neural networks, they are compared with the convolutional neural network (CNN), which directly quantifies the microstructural metrics from the three-dimensional structures, in terms of quantification accuracy, the number of training datasets, and required training time.The values obtained from the topological analysis followed by the principal component analysis are found to contain essential structural information in the porous electrodes. This is implied by the fact that the principal component values obtained from the real structure datasets are significantly different from those from the artificial sphere-packing structures, even though the conventional structural metrics, i.e., volume fraction and surface density, are identical between the real and artificial structures. In addition, the principal component values have sensitivity to the conventional statistical metrics, because the structures with different solid compositions form clusters in different locations in a principal component space. These suggest that the topological information will be useful not only to detect subtle structural changes in the porous electrodes undetectable in conventional structural metrics, but also to reduce the dimensionality of the complex porous electrodes without losing the information quality.Since the information extracted using the persistent homology and principal component analysis possess essential information about the electrode structures, conventional microstructural metrics are accurately quantified by the constructed neural network. It should be emphasized that the number of parameters in the neural network that need to be adjusted during the training process is significantly reduced compared with the convolutional neural network. As a result, the number of required datasets for the training of the network is significantly reduced. These clearly indicate the effectiveness of the dimensional reduction in the quantification of the electrode microstructures.

Standardness is a popular assumption in the literature on set estimation. It also appears in statistical approaches to topological data analysis, where it is common to assume that the data were sampled from a probability measure that satisfies the standardness assumption. Relevant results in this field, such as rates of convergence and confidence sets, depend on the standardness parameter, which in practice may be unknown. In this paper, we review the notion of standardness and its connection to other geometrical restrictions. We prove the almost sure consistency of a plug-in type estimator for the so-called standardness constant, already studied in the literature. We propose a method to correct the bias of the plug-in estimator and corroborate our theoretical findings through a small simulation study. We also show that it is not possible to determine, based on a finite sample, whether a probability measure satisfies the standardness assumption.

Suspense is an affective state that is ubiquitous in human life, from art to quotidian events. However, little is known about the behavior of large-scale brain networks during suspenseful experiences. To address this question, we examined the continuous brain responses of participants watching a suspenseful movie, along with reported levels of suspense from an independent set of viewers. We employ sliding window analysis and Pearson correlation to measure functional connectivity states over time. Then, we use Mapper, a topological data analysis tool, to obtain a graphical representation that captures the dynamical transitions of the brain across states; this representation enables the anchoring of the topological characteristics of the combinatorial object with the measured suspense. Our analysis revealed changes in functional connectivity within and between the salience, fronto-parietal, and default networks associated with suspense. In particular, the functional connectivity between the salience and fronto-parietal networks increased with the level of suspense. In contrast, the connections of both networks with the default network decreased. Together, our findings reveal specific dynamical changes in functional connectivity at the network level associated with variation in suspense, and suggest topological data analysis as a potentially powerful tool for studying dynamic brain networks.

Proactive and privacy-Preserving defense for DNS over HTTPS via federated AI attestation (PAFA-DoH).

Topological data analysis (TDA) has emerged as a powerful framework for extracting robust, multiscale, and interpretable features from complex molecular data for artificial intelligence (AI) modeling and topological deep learning (TDL). This review provides a comprehensive overview of the development, methodologies, and applications of TDA in molecular sciences. We trace the evolution of TDA from early qualitative tools to advanced quantitative and predictive models, highlighting innovations such as persistent homology, persistent Laplacians, and topological machine learning. The paper explores TDA's transformative impact across diverse domains, including biomolecular stability, protein-ligand interactions, drug discovery, materials science, topological sequence analysis, and viral evolution. Special attention is paid to recent advances in integrating TDA with machine learning and AI, enabling breakthroughs in protein engineering, solubility, and toxicity prediction, and the discovery of novel materials and therapeutics. We also discuss the limitations of current TDA approaches and outline future directions, including the integration of TDA with advanced AI models and the development of new topological invariants. This review aims to serve as a foundational reference for researchers seeking to harness the power of topology in molecular sciences.

Advanced chemical profiling of complex samples, suchas botrytizedwines, requires advanced analytical techniques capable of capturingsubtle compositional variations. In this study, we introduce a statisticallyrobust framework that leverages a topological data analysis (TDA)tool, Ball Mapper, in the context of comprehensive two-dimensionalgas chromatography (GC × GC) with high-resolution time-of-flightmass spectrometry (HR-TOF-MS) to obtain untargeted identificationof sample-specific chemical markers. A key design element of the proposedapproach is its ability to numerically process the immense data volumegenerated per sample, whose statistical and chemical significanceis often difficult to interpret using conventional methods. Each ofthe 34 wine samples yielded over 470,000 mass spectral functions,which were discretized, normalized, and clustered to obtain representativeand relatively unique discrete mass spectral vectors in high-dimensionalspace. With only two interpretative parameters, the proposed frameworkuncovered 2,792 extracted mass spectral distributions, from which1191discriminative features were identified, including 334 compounds assignedto known volatile organic compound classes. The resulting chemicalsignatures reflected regional differences in fermentation style, grapevariety, botrytization conditions, and microbial activity. Moreover,statistically robust framework of using Ball Mapper revealed consistentgrouping patterns both within and between wines. These findings demonstratethat the proposed framework can support chemical characterizationcomplex natural matrices and serve as a general strategy for analyzingany domains where GC × GC with HR-TOF-MS data are collected.