Search Results (62)

Topological data analysis (TDA) has evolved into a flexible and robust paradigm for obtaining qualitative, geometry-inspired insights from high-dimensional, noisy, and complex data. Grounded in algebraic topology, geometry, statistics, and machine learning (ML), TDA provides multiscale descriptions through persistent homology, Mapper (a graph-based method that summarizes the shape of high-dimensional data), and related topological signatures that are often inaccessible to standard linear and metric methods. In recent years, and especially during 2024–2025, TDA has expanded rapidly across science, engineering, biomedical research, and socio-economic studies, while also being integrated with modern learning paradigms such as deep learning (DL) and graph learning. This survey summarizes recent developments in TDA using a carefully selected set of articles, with emphasis on 2024–2025. We first present the mathematical and computational foundations of TDA, covering simplicial complexes, filtrations, persistent homology, the Mapper algorithm, and computational advances such as data simplification, stability, and efficiency. We then review applications in time series and dynamical systems, biomedical imaging and precision medicine, engineering and physical sciences, finance and risk analysis, DL and interpretability, and security and critical infrastructure systems. Throughout, we highlight how TDA can extract informative features, function as a model component, and provide a conceptual lens for studying complex systems. However, the survey also emphasizes recurrent failure patterns: TDA performance is highly sensitive to filtration, embedding, and vectorization choices; aggressive simplification can dilute or remove informative topological signals; and integration into standard ML workflows still lacks uniform validation and reporting protocols. We conclude by outlining key challenges—including scalability, statistical foundations, interpretability, and compatibility with rapidly evolving artificial intelligence (AI) paradigms—and by identifying directions for future research. The survey also provides a unifying design perspective for TDA systems, highlighting methodological trade-offs and emerging research directions for integrating topology with modern ML. Full article

Many deterministic multi-objective optimization methods generate Pareto outcomes by repeatedly solving scalarized subproblems for different preference or reference vectors. When the number of objectives is $m\ge 3$, the resulting samples lie on an $(m-1)$-dimensional Pareto surface in objective space. For tasks such as visualization, trade-off exploration, interactive decision making, and sensitivity analysis, a finite cloud of non-dominated points may be insufficient; one often needs a continuous surrogate of the Pareto surface together with a quantitative control of its reconstruction error. This paper studies the corresponding outer-loop reconstruction problem: how should new reference vectors be selected so as to reconstruct the Pareto surface to a prescribed uniform accuracy while using as few scalarized solves as possible? We propose Certified Adaptive Triangulation Sampling (CATS), a curvature-aware adaptive triangulation method for reconstructing a Pareto surface from an oracle $u\mapsto z\left(u\right)$, $u\in {\Delta }_d$, where $d=m-1$. CATS builds a simplicial mesh over the reference simplex and refines the cell with the largest local interpolation quantity $\eta \left(\tau \right)=\frac{1}{2}\left({max}_k{M}_{\tau ,k}\right)\mathrm{diam}{\left(\tau \right)}^2$, where $M_{\tau ,k}$ is an upper bound on the Hessian norm of the kth component of the oracle-induced map over $\tau$. This quantity matches the natural error scale of affine interpolation for $C^2$ maps. The rigorous certified interpretation of CATS applies when the preference-to-Pareto map is single-valued, $C^2$, and equipped with reliable local Hessian-norm upper bounds. If such bounds are replaced by numerical curvature estimates, the same rule can still be used as an adaptive refinement indicator, but the resulting stopping test is not a formal certificate unless those estimates are themselves validated. Under the certified assumptions, we prove that the stopping condition ${max}_{\tau }\eta \left(\tau \right)\le \epsilon$ guarantees ${sup}_{u\in {\Delta }_d}{\parallel z\left(u\right)-\widehat{z}\left(u\right)\parallel }_{\infty }\le \epsilon$, and that the oracle complexity of certified simplicial piecewise-affine reconstruction is $\mathsf{\Theta }\left({\epsilon }^{-d/2}\right)$. On the rigorously certified core tests, CATS uses $2.7\times$–$3.8\times$ fewer oracle calls than uniform reference-direction sampling and $1.2\times$–$1.6\times$ fewer than an AWS-inspired patch-area refinement rule. Additional benchmark studies, evaluated with the same interpolation quantity as a practical stopping indicator, show the same qualitative advantage, especially on anisotropic and localized surface geometries. Full article

Cascading failures in complex networks occur when local node or edge failures propagate to trigger large-scale collapse. Traditional pairwise network models cannot adequately capture group coordination and multi-agent higher-order interactions. Higher-order networks incorporating simplicial structures more accurately represent group and multi-node interactions, providing a new framework to study cascading failures and network robustness. The paper proposes a higher-order coupled map lattice (CML) model to characterize cascading failures in simplicial complex networks and analyze the influence of higher-order structures on network robustness. Further experiments on fourth-order simplicial networks investigate robustness differences under various topologies and attack strategies. Results indicate that fourth-order simplicial networks are vulnerable to targeted attacks but robust against random failures, regardless of network type. Furthermore in single-order networks, the higher simplex dimensions, the greater robustness. The theoretical perturbation thresholds for third-order networks show a negative correlation between the critical perturbation and the sum of network coupling parameters. These results are validated by analysis of simplices added to ordinary networks, destructive experiments, and empirical networks. This study deepens the understanding of cascading failure mechanisms and robustness in higher-order networks, and provides theoretical guidance for designing resilient networks based on higher-order structures. Full article

Modeling and controlling multilayer complex network dynamics is challenging under coexisting crosslayer interactions, higher-order couplings, and decentralized strategic decisions. Most existing schemes focus on graph-based pairwise structures and overlook topological cavities, mesoscale loops, and layered self-interested actions. This paper presents TopoGame-MND, an algebraic-topological and game-theoretic framework for multilayer network dynamics. We first build a filtration-driven simplicial lifting to unify pairwise and higher-order interactions into a weighted multilayer simplicial complex. A topological state operator using generalized Hodge Laplacians and persistent homology is then constructed to characterize cross-scale diffusion, circulation, and structural inconsistency. A distributed potential-game mechanism is developed with a topology-aware utility, followed by a proximal mirror-best-response algorithm with consensus correction. We prove Nash equilibrium existence and uniqueness, global potential monotone descent, linear convergence, computational complexity, and input-to-state robustness. Simulations on multiplex and interdependent networks validate that TopoGame-MND outperforms baselines in regulation speed, oscillation energy, failure resilience, and robustness, providing a unified way to connect higher-order topology and distributed decision optimization. Full article

Fractals represent one of the fundamental manifestations of complexity, and fractal networks serve as tools for characterizing and investigating the fractal structures and properties of large-scale systems. Higher-order networks have emerged as a research hotspot due to their ability to express interactions among multiple nodes. This study proposes an iterative generation model for higher-order fractal networks. The iteration is controlled by three parameters: the dimension $K$ of the simplicial complex, the multiplier $m$, and the iteration count $t$. The constructed network is a pure simplicial complex. Theoretical analysis using the similarity dimension and experimental verification using the box-counting dimension demonstrate that the generated networks exhibit fractal characteristics. When the multiplier $m$ is large, the generalized degree distribution of the generated networks exhibits scale-free properties. Full article

Persistent Homology (PH) is a method of Topological Data Analysis that characterizes the topological structure of a space. Unfortunately, the computation of PH for high-dimensional and big data is not possible due to the exponential growth of the constructed complex. Fortunately, sparsification techniques can substantially reduce the size of the complex. This paper examines a sparsification technique ($\beta$-Sparsification) that produces a complex reduction capability that is scalable to a user-specified value $\beta$. At $\beta =0$ this scaling generates complexes that can have the same 1-Skeleton as the Vietoris–Rips complex; $\beta =1$ produces a Delaunay complex, and other values of $\beta$ produce a range of (unnamed) complexes. Experiments with $\beta$-Sparsification reveal that the topology of the sparsified simplicial complex is preserved for $0\le \beta \le 1$; for $\beta >1$, the complex begins to lose (potentially insignificant) topological features. Full article

To address the challenge of balancing economic growth with carbon emission reduction, improving regional Carbon Emission Efficiency (CEE) has emerged as a central pathway to achieving the “dual carbon” goals. While most existing studies focus on inter-regional CEE linkages through pairwise interaction networks, such approaches fall short in capturing the high-order mechanisms of multi-regional collaboration. This study integrates the Super-SBM model with a modified gravity model to construct a CEE correlation network across 30 provincial administrative regions in China from 2007 to 2023. To overcome the limitations of traditional pairwise networks, simplicial complex theory is introduced to establish a high-order topological representation framework. Furthermore, by applying the multiorder Laplacian to assess the synchronization stability of the network, a directed second-order degree swap strategy is proposed to optimize its high-order structure. The findings reveal that the CEE correlation network has evolved from a single-pole aggregation pattern toward a multi-center equilibrium. Provinces with high connectivity play a dominant role in both pairwise and triadic synergies, though their collaborative advantages are gradually diffusing to central and western regions. Notably, with only a limited number (approximately five) of second-order degree swaps among key node pairs, the network’s synchronization stability can be substantially improved. When first-order and second-order interaction strengths reach comparable levels (coupling strength ${\alpha }^{*}\approx 0.5$), the system achieves optimal resistance to external perturbations. This study highlights the pivotal role of high-order collaboration in shaping regional CEE linkages and offers a practical optimization pathway for structurally enhancing CEE through coordinated efforts in pursuit of the “dual carbon” goals. Full article

To address the topological mismatch between signal space and physical space caused by uneven signal feature distribution in indoor non-line-of-sight and complex topological environments, this paper proposes an indoor positioning algorithm based on Energy-domain Fingerprint Manifold Graph Total Variation (EFM-GTV). To mitigate neighborhood distortion caused by uneven high-dimensional signal feature distribution, a UMAP manifold topology graph construction method based on fuzzy simplicial sets is designed to establish a graph basis consistent with physical space topology. To reduce false matching risks in global search, a physical topology pruning strategy combining Jaccard similarity is proposed, effectively eliminating pseudo-connections. Building upon this foundation, we introduced an optimization model based on graph total variation, reformulating the positioning problem as a graph signal recovery task. This approach effectively overcomes signal fluctuation interference in complex topologies like U-shaped corridors, achieving robust position estimation. Experiments demonstrate that this algorithm effectively leverages manifold structure constraints to correct NLOS errors. On real-world field test datasets, compared to traditional weighted algorithms, the average positioning accuracy improves to 1.4267 m, with maximum positioning error reduced by over 50%, achieving high-precision robust positioning. Full article

The transition of the mining industry towards Industry 5.0 demands predictive models capable of strictly adhering to physical laws and modeling complex, non-Euclidean geometries—capabilities often lacking in standard graph neural networks. This systematic review, conducted under the PRISMA 2020 protocol, analyzes the emergence of “Era 5” architectures by synthesizing 96 high-impact studies from 2019 to 2026, focusing on Clifford (geometric algebra) GNNs, simplicial and cell complex neural networks, symplectic/Hamiltonian GNNs, and generative flow networks (GFlowNets). The analysis demonstrates that Clifford architectures provide superior rotational equivariance for robotic control; Simplicial networks capture high-order topological interactions critical for geomechanics; Symplectic GNNs ensure energy conservation for stable long-term simulation of structural dynamics; and GFlowNets offer a novel paradigm for generative mine planning. We conclude that shifting from data-driven approximations to these mathematically rigorous, structure-preserving architectures is fundamental for developing reliable, physics-informed digital twins that optimize structural integrity and operational efficiency in complex industrial environments. Full article

Vaccination behavior and epidemic spreading are strongly intertwined processes, and their coevolution is often shaped by both individual decision-making and social interactions. However, most existing studies model such interactions at the pairwise level, overlooking the potential impact of higher-order social influence arising from group interactions. In this work, we develop a coupled vaccination–epidemic spreading model on multilayer higher-order networks, where vaccination behavior evolves on a simplicial complex and epidemic propagation occurs on a physical contact network. The model incorporates imperfect vaccine efficacy, allowing vaccinated individuals to become infected, and introduces a hybrid vaccination strategy that combines rational cost–benefit evaluation with social influence from both pairwise and higher-order interactions, as well as negative effects induced by vaccine failure. By constructing the coupled dynamical equations, we analytically derive the epidemic outbreak threshold and elucidate how higher-order interactions, behavioral responses, and vaccine-related parameters jointly affect epidemic dynamics. Numerical simulations on networks with different structural properties validate the theoretical results and reveal pronounced structure-dependent effects. The results show that higher-order social interactions can significantly reshape vaccination behavior and epidemic prevalence, while network heterogeneity and vaccine imperfection play crucial roles in determining the outbreak threshold and steady-state infection level. These results emphasize the necessity of incorporating higher-order interactions together with realistic vaccination behavior into epidemic modeling and offer new insights for the design of effective vaccination strategies. Full article

A polarity of an exceptional geometry of type ${\mathsf{E}}_6$ is called regularif its fixed structure, viewed as a simplicial complex, is a building. Polarities that do not act trivially on the underlying field were classified a long time ago by Jacques Tits. In the present paper, we classify the regular polarities of exceptional geometries of type ${\mathsf{E}}_6$ that act trivially on the underlying (arbitrary) field. As a result, we discover new subgeometries of the exceptional geometry of type ${\mathsf{E}}_6$. Full article

Polypharmacology in Amyotrophic lateral sclerosis (ALS) demands models that capture irreducible higher-order drug co-action. We introduce a categorical–topological pipeline that encodes regimens as truncated multicomplexes with a hypergraph–simplicial envelope; irreducible effects are identified by Möbius inversion, and CatMixNet predicts dose-response under monotone calibration while aligning multimodal omics via sheaf constraints. Under face-disjoint evaluation, omics fusion reduced RMSE from 0.164 to 0.149 (≈9%), increased PR-AUC from 0.38 to 0.44, and lowered calibration error to 2.6–3.1% with <10 dose-monotonicity violations per ${10}^3$ surfaces. Triad-irreducible signal strengthened (95th percentile ${\mathsf{\Delta }}^{★}=0.151$; antagonism retained at 24%). A risk-sensitive selector produced triads with toxicity headroom and projected ALSFRS-R slope gains of +0.04–0.05 points/month. Ablations confirmed the necessity of Möbius consistency, sheaf regularization, and monotone heads. Distilled monotone splines yielded compact titration charts with mean error 0.023. The framework supplies reproducible artifacts and actionable shortlists for iPSC and SOD1 validation. Full article

In the domain of Shape Encoding, the approximation of the Medial Axis of a solid region in ${\mathbb{R}}^3$ with Boundary Representation M, is relevant because the Medial Axis is an efficient encoding for M in Design, Manufacturing, and Shape Learning. Existing Medial Axis approximations include (a) full Voronoi and (b) and partial Shape Diameter Function (SDF)-based ones. Methods (a) produce large high-frequency data, which must then be pruned. Methods (b) reduce computing expenses at the price of not handling some shapes (e.g., prismatic), and currently, they only synthesize 1D Medial Axes. To partially overcome these limitations, this investigation performs a direct synthesis of a 1D and 2D simplex-based Medial Axis approximation by a combination of stochastic geometric reasoning and graph operations on the SDF-originated point cloud. Our method covers one- and two-dimensional Simplicial Complex Medial Axes, thus improving on 1D Medial Axes approximation methods. Our approach avoids the expensive full computing plus pruning of Medial Axis based on Voronoi methods. Future work is needed in the synthesis of Medial Axis approximation for high-frequency neighborhoods of mesh M. Full article

The present study proposes a multiscale analysis of the simplicial complex approximate entropy (MS-SCAE) applied to cardiac interbeat series. The MS-SCAE method is based on quantifying the changes in the simplicial complex associated with the time series when a coarse-grained procedure is performed. Our findings are consistent with those of previously reported studies, which indicate that the complexity of healthy interbeat dynamics remains relatively stable over different scales. However, these dynamics undergo changes in the presence of certain cardiac pathologies, such as congestive heart failure and atrial fibrillation. The method we present here allows for effective differentiation between different dynamics and is robust in its ability to characterize both real and simulated sequences. This makes it a suitable candidate for application to a variety of complex signals. Full article

The analysis of the synchronization of oscillator systems based on simplicial complexes presents some interesting features. The transition to synchronization can be abrupt or smooth depending on the substrate, the frequency distribution of the oscillators and the initial distribution of the phase angles. Both partial and complete synchronization can be seen as quantified by the order parameter. The addition of interactions of a higher order than the usual pairwise ones can modify these features further, especially when the interactions tend to have the opposite signs. Cluster synchronization is seen on sparse lattices and depends on the spectral dimension and whether the networks are mixed, sparse or compact. Topological effects and the geometry of shared faces are important and affect the synchronization patterns. We identify and analyze factors, such as frustration, that lead to these effects. We note that these features can be observed in realistic systems such as nanomaterials and the brain connectome. Full article