Entropy doi: 10.3390/e28060617

Authors: Zixun Liu Tianle Yang Yongjie Wang

The spatial heterogeneity of crowd distributions poses significant challenges for density estimation. Dense regions exhibit high local information entropy due to severe occlusion and feature ambiguity, while sparse regions and backgrounds carry progressively lower informational complexity. To address this, we propose an entropy-inspired crowd density estimation framework that allocates computational attention in proportion to the local information complexity of crowd regions. A Density-Guided Map (DGMap), constructed from nearest-neighbor distance statistics of head annotations, serves as a proxy for local information entropy, enabling the model to differentiate among dense, sparse, and isolated pedestrian regions. The proposed network, termed DGCC-Net, comprises four components: a Twins-Transformer backbone for hierarchical feature extraction, a Local Attention Module (LAM) that enhances high-resolution features through multi-scale receptive fields and rotational attention, a Multi-Level Feature Fusion Module (MLFM) with cross-scale dense connectivity and learnable branch weights for integrating semantic and spatial information, and a Density Guidance Module (DGM) supervised by the entropy-inspired DGMap to achieve density-adaptive feature refinement. Extensive experiments on four benchmark datasets (ShanghaiTech PartA, UCF-QNRF, UCF_CC_50, and JHU-Crowd++) demonstrate that DGCC-Net achieves competitive or state-of-the-art performance, validating the effectiveness of entropy-inspired attention allocation in heterogeneous crowd scenarios.

Entropy doi: 10.3390/e28060616

Authors: Yun Dang Xiaoran Yan Li Zhou Dongxi Li

Multimodal omics data portray biological processes across molecular layers, yet their heterogeneity and high dimensionality hinder a unified representation. Existing integrative approaches either focus on local feature interactions or adopt static fusion, often overlooking the complementary global sequential context and the dynamic relevance among omics sources. Consequently, clinically critical tasks such as accurate cancer-subtype classification and therapy selection still lack sufficient accuracy and robustness. We introduce the Dual-Branch Contrastive Learning for Multi-Omics Dynamic Fusion Network (DBCL-DFNet), a dual-branch contrastive-learning framework that simultaneously encodes local heterogeneous graphs and global omics sequences, distills key features via contrastive objectives, and employs a dynamic attention mechanism for adaptive, data-driven fusion. Benchmarked on three public cancer multi-omics datasets, DBCL-DFNet outperforms both conventional machine-learning models and state-of-the-art deep-integration methods, establishing a competitive and reliable framework for multi-omics integration and demonstrating potential for precision-oncology decision-making. From an information-theoretic perspective, the framework integrates Copula-entropy-guided feature selection with mutual-information-maximizing contrastive alignment, providing a principled foundation for robust multi-omics integration.

Entropy doi: 10.3390/e28060615

Authors: Zoe Lee-Youngzie Naotsugu Tsuchiya Michael Robinson Donna Dietz Martin M. Monti

Recent work in the structural approach to consciousness has shown great promise as a research paradigm for the formal and empirical study of the phenomenal qualities of experience, i.e., qualia. In this paradigm, qualia are characterized by modeling the internal organization of parts within an experience, or by modeling external relations between instances of experience. A major next step for the structural approach is to integrate these two perspectives into an account of phenomenally unified global experience. In this paper, we describe these two types of structural models and how their category-theoretic formalizations contribute to the task of identifying the physical bases of phenomenal consciousness. We then propose a sheaf-theoretic framework that integrates these two approaches by mapping mereological parts of experience to empirical measures of their qualia. Through an application to the experience of visual space, we demonstrate that this framework enables a formal description of the structure of experience and conditions for phenomenal unity. We discuss how this integrative approach supports an empirical research program for investigating the relationship between local and global phenomenal qualities, and outline directions for future work toward a structural characterization of global phenomenal consciousness.

Entropy doi: 10.3390/e28060614

Authors: Weijun Li Rui Han Jiangwei Shang Hui Khoon Ng Berthold-Georg Englert

Random samples of quantum states with specific properties are useful for various applications, such as Monte Carlo integration over the state space. In the high-dimensional situations that one already encounters when working with a few qubits, the quantum state space has a very complicated boundary, and it is challenging to incorporate the specific properties into the sampling algorithm. In this paper, we present the Sequentially Constrained Monte Carlo (SCMC) algorithm as a practical and versatile method for sampling quantum states in accordance with properties that can be stated as inequalities. We apply the SCMC algorithm to the generation of samples of bound entangled states; for example, we obtain nearly ten thousand bound, entangled, two-qutrit states in a few minutes, compared with less than ten such states per day from independence sampling in our implementation. In the second application, we draw samples of high-dimensional quantum states from a narrowly peaked target distribution and observe, for the system sizes investigated, that SCMC sampling remains computationally manageable as the dimensions grow. In yet another application, the SCMC algorithm produces uniformly distributed quantum states in regions bounded by values of the problem-specific target distribution; such samples are needed when estimating parameters from the probabilistic data acquired in quantum experiments.

Entropy doi: 10.3390/e28060613

Authors: Meng Zhang Jian Kang Bing Han Qian Wu

To address the problems of the limited feature extraction capability and insufficient training efficiency of the traditional Continuous Bag-of-Words (CBOW) model in Natural Language Processing (NLP), the Quantum Neural Network-enhanced CBOW model (QNN-CBOW) integrates Quantum Neural Networks (QNN) with the CBOW model, effectively enhancing training performance. This work aims to systematically investigate the sensitivity and influence patterns of key factors (activation function type, number of quantum feature extraction layers, context window size, and quantum gate noise level) on model behavior under controlled small-scale simulation conditions. Comparative experiments are carried out using the control variable method to clarify the influence mechanism of each factor. This paper presents a NISQ-era proof-of-concept study, which provides a theoretical basis and practical reference for the fusion and optimization of quantum neural networks and traditional NLP models.

Entropy doi: 10.3390/e28060612

Authors: Yasuko Kawahata

In the era of big data, the Law of Large Numbers is often treated as an absolute guarantee that increasing sample size (N) leads to a more accurate representation of truth. However, this study challenges this paradigm by demonstrating that in social systems characterized by conformity pressure and systemic bias, the maximization of N paradoxically triggers a structural shift in the selection and filtration of information. Using a sociophysical framework based on statistical mechanics and opinion dynamics, we identify a critical threshold—the “Signal Cliff”—where the diversity of information plummets and minority signals are irreversibly discarded as statistical noise. By executing large-scale simulations up to N=1010 via macro-dynamic approximations, we observe a phase transition from a stochastic phase of informational diversity to a deterministic phase. This collapse of Shannon entropy serves as a mathematical demonstration of “Epistemic Injustice,” where the sheer scale of data acts as a mechanism for silencing minority perspectives. We propose “Informational Health Diagnostics” as a necessary framework for evaluating the integrity of decision-making processes in digital public opinion and democratic elections. This approach provides a vital benchmark for distinguishing between a healthy consensus and a distorted convergence, ensuring robust information judgment in increasingly complex data-driven environments.

Entropy doi: 10.3390/e28060611

Authors: Zhengmian Hu

Can artificial intelligence discover, from raw experience and without human supervision, concepts that humans have discovered? One challenge is that human concepts themselves are fluid: conceptual boundaries can shift, split, and merge as inquiry progresses (e.g., Pluto is no longer considered a planet). To make progress, we need a definition of “concept” that is not merely a dictionary label, but a structure that can be revised, compared, and aligned across agents. We propose an algorithmic information viewpoint that treats a concept as an information object defined only through its structural relation to an agent’s total experience. The core constraint is determination: a set of parts forms a reversible consistency relation if any missing part is recoverable from the others (up to the standard logarithmic slack in Kolmogorov complexity). This reversibility prevents “concepts” from floating free of experience and turns concept existence into a checkable structural claim. To judge whether a decomposition is natural, we define excess information, measuring the redundancy overhead introduced by splitting experience into multiple separately described parts. On top of these definitions, we formulate dialectics as an optimization dynamics: as new patches of information appear (or become contested), competing concepts bid to explain them via shorter conditional descriptions, driving systematic expansion, contraction, splitting, and merging. Finally, we formalize low-cost concept transmission and multi-agent alignment using small grounds that allow another agent to reconstruct the same concept under a shared protocol, making communication a concrete compute-bits trade-off.

Entropy doi: 10.3390/e28060610

Authors: Barak Kol

This article offers a broad-brush account of the Newtonian three-body problem, from its origins with Newton to its vibrant present, emphasizing its enduring influence on theoretical physics. It unfolds through a series of self-contained episodes that illuminate the scientific fields and the paradigm shift that have grown out of this problem.

Entropy doi: 10.3390/e28060609

Authors: Junrui Wang

This study investigates statistical inference for a heterogeneous competing risks model under an improved adaptive Type-II progressive censoring scheme, which effectively controls testing time while ensuring sufficient failure data. Assuming the latent lifetimes of distinct failure causes are independent and follow Chen and Weibull distributions, we develop both frequentist and Bayesian approaches to derive point and interval estimates for the unknown parameters. Interval estimators include approximate confidence intervals, bootstrap confidence intervals, and highest posterior density credible intervals. Under the Bayesian framework, Markov Chain Monte Carlo techniques are utilized to obtain numerical solutions under the squared error loss function, assuming independent gamma priors. Extensive Monte Carlo simulations and a real-world data application are presented to demonstrate the efficacy and practical utility of the proposed methodologies.

Entropy doi: 10.3390/e28060607

Authors: Jiaozi Pu Yaxin Shi

Background: Perception-based evaluation using Likert-scale survey data is widely applied in tourism and transport research, yet conventional point-valued encoding imposes artificial precision and overlooks ambiguity between adjacent ordinal categories. This limitation is particularly relevant in experiential contexts, where subjective judgments often involve transitional evaluations. Methods: This study develops a parameterized fuzzy–entropy exploratory factor analysis (FE-EFA) framework for uncertainty-aware analysis of ordinal perception data. The approach transforms ordinal responses into parameterized fuzzy membership distributions governed by a single effective uncertainty ratio, constructs a correlation structure in the five-dimensional membership space, and incorporates Shannon entropy and Jensen–Shannon divergence to characterize distributional dispersion and representation differences. The framework is applied to survey data from Chengdu Tramway Line 2 (N = 1242; 32 indicators). Results: Under the Kaiser criterion (eigenvalues > 1), conventional EFA yields a seven-factor structure, whereas FE-EFA identifies an additional eighth factor located near the retention boundary. Under a unified factor specification, both approaches preserve a consistent high-level structure, while FE-EFA shows fewer cross-loadings and a more differentiated loading pattern in this empirical case under the adopted exploratory specification. From an information-theoretic perspective, FE-EFA produces higher entropy (average = 0.8688) and low Jensen–Shannon divergence (average = 0.0133), suggesting a limited redistribution of ordinal information without substantially altering the overall distributional structure. Entropy-adjusted weighting further reveals systematic shifts in indicator importance across key dimensions. Conclusions: The FE-EFA framework extends conventional Likert-scale analysis by introducing an uncertainty-aware representation layer prior to factor extraction. It preserves overall structural stability while suggesting a more differentiated organization of latent constructs and indicator-level representations in this empirical context. The proposed approach provides an exploratory representation-level extension for perception-based evaluation and decision support in tramway cultural tourism development and related contexts.

Entropy doi: 10.3390/e28060608

Authors: Juan Yu Lei Li Ming Li Shu-Qian Shen

Nonlinear entanglement witnesses constructed from multiple linear entanglement witnesses and multiple copies of quantum states have recently been proposed as a powerful tool for entanglement detection. In this work, we show, via an explicit counterexample, that the fineness of linear witnesses generally fails to transfer to their tensor-product nonlinear counterparts. For the canonical family of nonlinear witnesses in the form of (αI−L)⊗(βI−T), we rigorously prove that the optimal nonlinear witness is uniquely attained with weakly optimal parameters of α=λmax(L) and β=λmax(T). Meanwhile, we analytically demonstrate that the self-tensor products of two representative linear witnesses fail to detect any entangled state. The question of whether a nonlinear entanglement witness capable of detecting entanglement can be constructed by tensoring a linear witness with itself remains open.

Entropy doi: 10.3390/e28060606

Authors: Olivier Nusbaumer

This work formulates semiclassical gravity within a causal-diamond framework where a finite-resolution boundary provides the edge structure for a local Wheeler–DeWitt description. Because the diffeomorphism-invariant Hilbert space does not factorize, each diamond is equipped with a boundary-completed algebra AO, ensuring the operational state ρO and the semiclassical reference family σO[Λ] share identical operator content. Dynamics are posed as local statistical inference: the relative-entropy functional Srel(ρO∥σO[Λ]) quantifies the mismatch between data and reference. This yields the minimal operational axioms defining subsystems, intrinsic clocks, and regulated observables in a finite-resolution, background-independent setting. The topology-locked boundary capacity budget fixes an effective channel multiplicity N≈1.23×1011. Calibrating its coherent fraction to Newton’s constant determines a matching scale Ms≈3.02×1013GeV. In the modular/KMS regime, the relative-entropy Hessian (Kubo–Mori metric) block-diagonalizes into orthogonal tensor, vector, and scalar response sectors. A heat-kernel expansion on the fixed S3×S1 history manifold maps this near-equilibrium response to a matching-scale effective field theory, yielding the Einstein–Hilbert tensor structure, Yang–Mills susceptibilities, and leading mass deformations. Vector and scalar responses remain intensive, while the tensor response scales extensively with coherent channel multiplicity. The fixed modular protocol and quantized boundary currents imply α−1(Ms)=4πk at integer levels k, while the reduced R2 plateau sector yields linked cosmological targets: ns≃0.965, r≃0.0038, and As≃2.1×10−9. Translations between causal diamonds act as completely positive trace-preserving (CPTP) updates. The resulting open-modular Walsh filtration selects the three-dimensional degree-one sector as the algebraic basis for family structure. Treating continuum fields as the structured response of a finite boundary, the framework yields correlated, falsifiable relations for gravitational stiffness, gauge response, plateau cosmology, and threefold matter-sector organization from one minimal operational architecture.

Entropy doi: 10.3390/e28060605

Authors: Yuan Peng Xiongbo Zheng Zhiyong Cheng

To reduce the hardware cost of massive multiple-input multiple-output (MIMO) systems, low-bit analog-to-digital converters (ADCs) and sparse arrays are widely used. Compared with traditional 1-bit and 2-bit quantization techniques, 1.5-bit quantization uses two symmetric non-zero thresholds to quantize signal power into three levels, thereby balancing quantization complexity against system performance. However, the quantization loss introduced by 1.5-bit quantization is still significant and leads to degradation in DOA estimation performance. To improve the DOA estimation accuracy of 1.5-bit sparse massive MIMO systems, a covariance matrix estimation method is proposed. This method exploits the Toeplitz property of the covariance matrix of sparse arrays and the relationship between 1.5-bit quantized signals and their unquantized counterparts to transform the covariance matrix estimation problem for 1.5-bit sparse arrays into a non-convex optimization problem with equality constraints. We then further exploit the properties of 1.5-bit quantized signals to relax this problem into a convex problem and solve it via semidefinite programming. Once the covariance is estimated, the DOAs can be recovered by subspace-based methods. Numerical results show that the proposed method achieves higher estimation accuracy than 1.5B-MUSIC and 1-bit covariance-fitting baselines on 1.5-bit sparse arrays, and is competitive with structured covariance-fitting baselines applied to unquantized data, especially on coprime arrays in low-snapshot scenarios.

Entropy doi: 10.3390/e28060604

Authors: Ying Chen Jianrong Bao Yanhai Shang Chao Liu Shenji Luan

To obtain efficient channel codes with high power efficiency at moderate signal-to-noise ratios (SNRs), an efficient high-rate protograph quasi-cyclic (QC) low-density parity-check (LDPC) codes is optimally constructed. By an optimized protograph template, the code framework is firstly produced by the extensions of the variable nodes. By enlarging the dimension of the sub-matrices related to the protograph framework, the base QC matrix template is generated with required code rate and length by the extrinsic information transfer (EXIT) chart for better decoding threshold. Then, the elements in the base matrix are split with even smaller square sub-matrices of the same row and column weights. In this procedure, a progressive-edge-growth (PEG) algorithm is employed to find the optimized positions of the QC sub-matrices to obtain larger girth for better error floor performance. Moreover, an asymptotic weight distribution (AWD) is employed to keep a low-code-error floor for the code. Also the circulant offsets in all QC sub-matrices are optimally searched by a QC oriented nested loop approximated cycle extrinsic message degree (QC-NLACE) algorithm, which improves the relationship of the unavoidable loops in the code’s Tanner graph to cut the error floor. Simulation results show that the codes produced by the proposed method show quite good bit-error-rate (BER) performance. In addition, they exhibit good properties of high spectrum efficiency brought by the high code rate, and the low complexity by the short code length. Moreover, a series of different rate-compatible LDPC codes can be generated from the same protograph framework with some variable node extensions, which significantly eases the code design. Therefore, the proposed code construction can be efficiently applied in the optimal construction of high-rate and short-length rate-compatible QC-LDPC codes with a high data rate and rational complexity, which makes the codes extremely suited for use in new-generation power-constrained wireless communications.

Entropy doi: 10.3390/e28060603

Authors: Armand M. Makowski Siddharth Pal

We discuss several limiting degree distributions for a class of homogeneous random graphs, known as random threshold graphs, in the many node regime. This analysis is carried out under a weak assumption on the distribution of the underlying fitness variable. This assumption, which is satisfied by the exponential distribution, determines a natural scaling under which the following limiting results are shown: the nodal degree distribution, i.e., the distribution of any node, converges in distribution to a limiting pmf. However, for each d=0,1,…, the fraction of nodes with degree d converges only in distribution to a non-degenerate random variable Π(d) (whose distribution depends on d), and not in probability to the aforementioned limiting nodal pmf as is customarily expected. The distribution of Π(d) is identified only through its characteristic function. Implications of this result include the following: (i) the empirical node distribution may not be used either as a proxy for or estimate of the limiting nodal pmf; (ii) even in homogeneous graphs, the network-wide degree distribution and the nodal degree distribution may capture vastly different information; and (iii) random threshold graphs with exponential distributed fitness do not provide an alternative scale-free model to the Barabási–Albert model as was argued by some authors; the two models cannot be meaningfully compared in terms of their degree distributions!

Entropy doi: 10.3390/e28060602

Authors: Smail Bougouffa Kamal Berrada

We investigate the non-Markovian dynamics of quantum steering in a tripartite photonic system subject to dephasing noise. By developing a theoretical framework based on the single-photon dephasing model extended to three independent photons, we analyze the temporal evolution of steering measures SA−BC and SAB−C for two distinct classes of initial states: W-type entangled states and GHZ-type mixed entangled states. The system is studied under various environmental configurations, ranging from fully Markovian to fully non-Markovian regimes, with asymmetric distributions of memory effects across the three photons. Our results reveal that the dynamics of tripartite steering are highly sensitive to both the number of photons coupled to non-Markovian environments and the specific partition of the system being considered. For W-states, non-Markovian effects induce oscillatory behavior with death–revival cycles, where the intervals of sudden death and revival amplitudes depend critically on the distribution of memory effects. For GHZ-states, we observe multiple death–revival cycles in some configurations and prolonged preservation of steering without complete sudden death in others. Notably, we find that non-Markovian environments significantly influence the dynamics of quantum steering through information backflow effects, with their impact depending sensitively on the subsystem to which the environment is coupled and on the roles of the steering and steered parties. These findings demonstrate that non-Markovian effects can significantly influence the preservation and degradation of directional quantum correlations, with their impact depending strongly on the coupling configuration and the choice of steering and steered subsystems. This behavior provides useful insight into the control of quantum steering in photonic networks and related quantum information processing tasks.

Entropy doi: 10.3390/e28060601

Authors: Zoe Leyva-Acosta Eduardo Acuña Yeomans Francisco Hernández-Quiroz

Algorithmic complexity is a foundational notion in theoretical computer science, but its incomputability has led to two families of practical estimators: compression-based and program-execution-based (e.g., the Coding Theorem Method, CTM). Despite widespread use, the correspondence between these paradigms remains poorly understood. We present a systematic comparative framework that uses the Block Decomposition Method (BDM) to extend CTM-based estimates to longer strings, enabling direct comparison with compression-based estimators across multiple computational models. A control estimator (BDMId) isolates the contribution of block structure from algorithmic information, providing a rigorous baseline for interpreting correlations. Our results show that cross-paradigm correlations are weak and decrease systematically as model resolution decreases; for the lowest-resolution model, correlations are essentially null. In long strings, per-length correlations vanish, while global correlations appear high but are largely explained by the control estimator, indicating that they are driven primarily by trivial length effects rather than shared sensitivity to algorithmic structure. Crucially, for low-resolution models, BDMId outperforms BDM itself, indicating that the inclusion of CTM information does not improve—and may even reduce—agreement with compression-based estimators. These findings suggest that compression-based and program-execution-based estimators capture fundamentally different aspects of structure. Rather than invalidating either approach, this work provides a systematic methodology for assessing cross-paradigm correspondence and highlights the importance of explicit controls in empirical comparisons of algorithmic complexity.

Entropy doi: 10.3390/e28060600

Authors: Gengran Hu

Random integer lattices are fundamental to lattice-based cryptography and algorithmic number theory. A new random integer lattice model, free of any restrictions on the Hermite Normal Form (HNF), was introduced by in 2016. It was also observed that the probability of such a lattice being in a simple HNF form is approximately 44%. In this paper, the gap between general random integer lattices and those in a simple HNF is bridged by introducing the concept of the (n,k)-simple random integer lattice, where the first k diagonal entries of the HNF are fixed to 1. We derive the asymptotic counting formula for such lattices and compute their density among all integer lattices. Furthermore, a generation algorithm for the (n,k)-simple random integer lattice based on rejection sampling and inverse sampling methods are proposed, with the analysis showing that it achieves O(n2) expected running time. This work provides a theoretical foundation and practical toolkit for constructing structured random lattices with controlled HNF forms.

Entropy doi: 10.3390/e28060599

Authors: Oleg Gorshkov Hernando Ombao

This study examines large-scale synchronization dynamics during epileptic seizures using scalp EEG recordings, with the aim of characterizing reproducible network-level patterns across patients. Functional connectivity was estimated from the CHB-MIT database using phase-lag-based measures robust to volume conduction, specifically Imaginary Coherence and the debiased weighted phase lag index, across standard frequency bands. Synchronization features were used to train a neural network classifier evaluated under a Leave-One-Patient-Out (LOPO) validation framework to ensure patient-independent assessment. To quantify seizure-related network alterations, we introduce Relative Pathological Synchronization (RPS), defined as the median area under the ROC curve across patients. The results demonstrate that synchronization patterns deviate systematically from baseline activity in a time-dependent manner. Interhemispheric connectivity shows earlier and higher peak RPS values compared to intrahemispheric connectivity, while intrahemispheric changes develop more gradually and persist over a longer interval. Theta-band features provide the most consistent contribution, although interhemispheric synchronization involves multiple frequency bands. In addition, longer seizures are associated with higher peak RPS values. These findings indicate that large-scale synchronization patterns contain stable, patient-independent information about seizure dynamics. Specifically, interhemispheric connectivity achieved a peak RPS of 0.749 (0.609–0.891) at TAS=10 s, while intrahemispheric connectivity reached 0.640 (0.563–0.843) at TAS=30 s under strict Leave-One-Patient-Out validation.

Entropy doi: 10.3390/e28060598

Authors: Siao Liu Yongjiu Li Chunxiao Sun Yi Wang Shuxian Ji

This paper addresses issues such as nonlinearity, model uncertainty, and multiple policy constraints within the dynamic evolution of new quality productive forces systems. It proposes a research framework integrating data-driven modelling with adaptive tracking control. By merging control theory with economic dynamics, a closed-loop analytical system of ‘theory-data-control’ is constructed, providing a methodologically rigorous yet operationally feasible pathway for the precise regulation of complex economic systems. First, utilising provincial panel data, a discrete-time system model integrating linear inertia, policy effects, and nonlinear compensation is established. System parameter identification is achieved through a dual machine learning approach employing partial linear regression. Subsequently, a tracking controller integrating data-driven initial identification with online parameter adaptation is designed, incorporating a projection mechanism to strictly ensure policy variables remain within feasible adjustment ranges. Based on Lyapunov stability theory, we demonstrate that the tracking error of the closed-loop system exhibits ultimate convergence with boundedness. Simulation experiments confirm that the proposed method significantly enhances the system’s tracking performance towards the target trajectory, reducing the mean absolute error by approximately 30.8% while producing smoother control signals. Comparative studies indicate that the parameter adaptation mechanism and nonlinear compensation module play crucial roles in improving control effectiveness. This research not only expands the theoretical toolkit for analysing the dynamics of new quality productive forces but also provides an interdisciplinary methodological reference for the closed-loop management of complex socioeconomic systems under data-driven conditions.

Entropy doi: 10.3390/e28060597

Authors: Nasrin Dabirian Mansoureh Samadi Amir Babaniamansour Yameng Li Manuel E. Hernandez Yih-Kuen Jan

Cupping therapy has been demonstrated to improve hemodynamic regulation. Existing studies have reported mean changes of oxyhemoglobin (OxyHb) and deoxyhemoglobin (DeoxyHb), which do not capture the multi-scale regulatory dynamics of the microvasculature. It is therefore unclear whether cupping therapy modulates the complexity and fractal property of hemodynamic signals. The objective of this study was to examine complexity of hemodynamic response to cupping therapy. A 2 by 2 factorial design with repeated measures was used to examine the main effect of pressure (−225 and −300 mmHg) and duration (5 and 10 min) and their interaction. A near infrared spectroscopy (NIRS) was used to measure OxyHb and DeoxyHb concentrations before and after cupping therapy. A total of 18 healthy participants were enrolled in this study. The wavelet analysis, sample entropy and detrended fluctuation analysis (DFA) were used to quantify the oscillatory, complexity, and fractal scaling properties of OxyHb and DeoxyHb signals. A two-way ANOVA with Bonferroni correction was used to examine the main and interaction effects. The results demonstrated that the combined effects of pressure and duration, rather than either factor independently, were the primary determinants of the dynamic hemodynamic response to cupping therapy, with significant Pressure × Duration interactions observed in DeoxyHb myogenic wavelet power (F = 4.636, p = 0.046, η2p = 0.214), OxyHb (F = 5.704, p = 0.029, η2p = 0.251) and DeoxyHb (F = 6.600, p = 0.020, η2p = 0.280) sample entropy, and DeoxyHb DFA scaling exponent (F = 5.598, p = 0.030, η2p = 0.248). In addition, cupping pressure selectively modulated neurogenic DeoxyHb oscillatory power (F = 5.001, p = 0.039, η2p = 0.227), and cupping duration significantly altered the fractal scaling properties of DeoxyHb signals (F = 7.775, p = 0.013, η2p = 0.314). The findings indicate that the interaction of pressure and duration of cupping therapy could effectively modulate hemodynamic responses. To the best of our knowledge, this is the first study investigating the complexity of hemodynamic responses after cupping therapy.

Entropy doi: 10.3390/e28060596

Authors: Tim Genewein Jordi Grau-Moya Li Kevin Wenliang Laurent Orseau Marcus Hutter

The success of large neural networks trained for sequential prediction via log-loss minimization over massive and diverse datasets has sparked debate regarding the fundamental limits of this paradigm. While these models are not explicitly programmed to perform planning and search, their behavior increasingly resembles complex reasoning and adaptive problem-solving. This paper reviews a series of theoretical and empirical works, aiming to bridge the gap between the practical success of LLMs and formal theories of computation and intelligence—that is, algorithmic information theory and Universal Artificial Intelligence. Grounded in the framework of memory-based meta-learning, the main argument is that training sequence models to predict the next token across diverse tasks implicitly meta-trains them to perform algorithmic compression, thereby performing (amortized) Bayesian inference over the task in-context. Consequently, when pretrained on a sufficiently rich data distribution, the resulting neural networks behave as if compressing by inferring the generative algorithm producing the observed data. We discuss recent theoretical and empirical evidence demonstrating that this approach can approximate Solomonoff induction in the theoretical limit, match exact Bayesian inference on complex sources in practice, achieve strong compression on out-of-distribution data, and synthesize complex in-context algorithms like chessboard evaluations. As models become more capable and general, the theoretical understanding through the lens of algorithmic information theory, including hard theoretical limits and how far practical models are from them, becomes increasingly relevant. We thus conclude our paper by outlining a number of open research questions to further bridge the gap from well-understood theory to modern machine learning practice.

Entropy doi: 10.3390/e28060595

Authors: Zhouheng Wu Tao Zhou Jianyong Fan Ruimin Zhang Zhigang Li Kang Hu

Feature selection (FS) plays a crucial role in high-dimensional data analysis by improving model performance and reducing computational complexity. However, existing metaheuristic-based FS methods often suffer from insufficient population diversity, premature convergence, and limited capability to escape local optima, which substantially constrains their effectiveness in complex search spaces. To address these challenges, this paper proposes a novel Improved Sand Cat Swarm Optimization algorithm with multi-strategy fusion (ISCSO) for feature selection. The proposed method introduces a hybrid initialization mechanism based on the Hénon chaotic map and lens imaging reverse learning to enhance population diversity. A golden sine-based phase adjustment strategy is further incorporated to achieve a more effective balance between global exploration and local exploitation. In addition, a nonlinear adaptive weight mechanism is designed to dynamically regulate the search process, while a simulated annealing-based acceptance criterion is integrated to improve the ability to escape local optima. Comprehensive experiments are conducted on the CEC2017 benchmark suite and 18 real-world datasets from the UCI repository. The results demonstrate that ISCSO achieves superior performance over state-of-the-art algorithms, obtaining the optimal results on 82.76% of benchmark functions. In feature selection tasks, ISCSO achieves the optimal average fitness on 94.44% of datasets, reduces feature dimensionality significantly, and consistently improves classification accuracy. These findings indicate that ISCSO provides a competitive and reliable solution for high-dimensional feature selection and complex optimization problems.

Entropy doi: 10.3390/e28060594

Authors: Leonardo Ermann Alexei D. Chepelianskii Dima L. Shepelyansky

The Boltzmann–Loschmidt dispute of 1876 questioned the possibility of a statistical irreversible description by time-reversible classical equations of motion of atoms. Here we show analytically and numerically that the quantum chaos diffusion of cold atoms, or ions, in a harmonic trap and pulsed optical lattice can be inverted back in time with up to 100% efficiency. This is in sharp contrast to classical evolution, where exponentially small errors break time reversibility. We argue that the existing experimental skills allow highlighting the Boltzmann–Loschmidt dispute from a quantum perspective.

Entropy doi: 10.3390/e28060593

Authors: Tao Guo Zhangyao Song Huihui Wu Yang Li

This paper analyzes the semantic rate–distortion problem motivated by task-oriented data compression with side information. The semantic information related to a task is not directly accessible to the encoder but implicitly impacts the observations through a joint probability distribution. The decoder aims to simultaneously recover the observation and infer the semantic information under certain distortion constraints. Notably, this paper advances the related research by involving side information and the observation of two semantic segments at both the encoder and decoder, which significantly complicates the theoretic analysis. We establish the information-theoretic limits for the tradeoff between compression rates and distortions by fully characterizing the rate–distortion function. Additionally, we explicitly derive the corresponding rate–distortion functions under specific Markov conditions for two scenarios: (i) the task is a binary classification of an integer observation as even and odd; and (ii) Gaussian-correlated task and observation. Furthermore, we validate the information-theoretic analysis by conducting a classification-oriented lossy image compression based on deep learning. The results are consistent with theoretical expectations, demonstrating the effectiveness of side information on both distortion and classification accuracy and the rationality of semantic segmentation.

Entropy doi: 10.3390/e28060592

Authors: Jiangnan Zhou Sheng Zhang Bing Wu Qiuming Wang Chennan Wu Ziqiang Luo Ka Sun Hongmei Mao

Existing methods for hypergraph node classification usually rely on local message passing and use a unified strategy for topological modeling across hyperedges of different sizes. However, they have two limitations in semi-supervised settings. First, representation learning mainly depends on local neighborhoods, making it difficult to incorporate global topological information. Second, a unified structural modeling strategy cannot effectively handle both small and large hyperedges. Small hyperedges require modeling fine-grained local relations, while large hyperedges need sparse group-level structure. To address these issues, we propose S2-HGNN, a scale-aware hypergraph node classification framework with spectral inductive bias for semi-supervised learning. S2-HGNN first injects global topological information into the input features using complementary hypergraph spectral operators. It then constructs different auxiliary topologies based on hyperedge size. For small hyperedges, it uses Top-k constrained clique expansion to preserve representative local relations. For large hyperedges, it uses star expansion to reduce redundant connections while preserving sparse group-level structure. Finally, node representations are jointly learned from the original hypergraph backbone and the two auxiliary branches, and final predictions are obtained through node-level adaptive fusion. Experiments on multiple public datasets show that the proposed method consistently outperforms strong baselines and exhibits superior robustness under feature perturbations.

Entropy doi: 10.3390/e28060591

Authors: Talia Costa Rodrigues David Santana Alencar Tayroni Alencar Alves Gladstone Alencar Alves Francisco Welington Lima João Antônio Plascak

The phase transition of the majority vote model with inertia has been investigated by means of extensive Monte Carlo simulations on directed Erdös–Rényi networks. Besides the usual average connectivity and local field that adds the opinion of the site itself, an additional term of inertia is considered. The relaxation time of the average opinion state of the network, together with the average opinion state fourth-order Binder cumulant and the corresponding opinion state susceptibility, have been analyzed for several different network sizes and local field and inertia parameter values, for average connectivity of 20 connections. The present results show that the phase transition of this model strongly depends on the inertia parameter, being quite different and richer than previous results of the same model on other regular networks. For inertia parameters between zero and 0.1 the system undergoes a continuous phase transition; for values in the range 0.1 and 0.2 no transition can be seen; for still larger values of inertia up to 0.5 a first-order phase transition takes place; finally, for values larger than 0.5 the dynamics is fully dominated by the inertia, and again no phase transition is observed.

Entropy doi: 10.3390/e28060590

Authors: Gaogao Dong Yili Shen Xinqi Hu Ruijin Du

The evolution of connectivity in quantum networks under decoherence and link degradation is a central problem in quantum information, calling for further understanding of the nature of its transition during structural network degradation. By diluting each link with probability 1−f, we focus on connectivity strength transitions in diluted hierarchical scale-free quantum networks, the (u,v) flowers, which are analytically tractable through two adjustable path-length parameters, u≤v. Incorporating quantum concurrence percolation and comparing it with classical percolation, we analyze the transitions of critical thresholds for various values of f and v from analytical, numerical, and simulation perspectives. The results demonstrate that quantum percolation exhibits consistently lower critical thresholds than classical percolation, even under various topologies and dilution levels. Our work implies that quantum multipath entanglement provides an intrinsic compensatory mechanism against structural degradation and that the hierarchical scale-free topology contributes to the failure resistance and robustness of quantum networks with multipath coupling.

Entropy doi: 10.3390/e28060589

Authors: Longxia Liao Long Chen

Privacy-preserving federated learning in the internet of vehicles (IoV) requires low-latency authentication, bounded privacy leakage, and robustness against malicious model updates. However, most existing studies separately design communication authentication and federated learning protection, which leads to duplicated overhead and weak resistance to cross-layer attacks. To address this issue, this paper proposes a DAG blockchain-enabled cross-layer authentication framework for trustworthy IoV federated learning (DAG-CTFL). The framework reuses authentication operations across V2X message verification and model-update delivery, incorporates trust-aware batch verification, and organizes cross-layer evidence through a two-tier DAG blockchain. In addition, differential privacy is used to reduce information leakage from uploaded model updates, while cross-layer trust evaluation improves resilience against poisoning and forged-identity attacks. Experimental results on MNIST and CIFAR-10 show that DAG-CTFL reduces single-message verification overhead by 8.2–56.1%, lowers batch-verification latency by 19.2–56.4%, and maintains model accuracy above 85% under 15% malicious nodes. These results demonstrate that DAG-CTFL achieves an effective balance among privacy preservation, authentication efficiency, and cross-layer robustness in IoV federated learning.

Entropy doi: 10.3390/e28060588

Authors: Yuxin Han Sen Wang Yaping Sun Kai Niu Nan Ma Ping Zhang

Classical information theory (CIT) characterizes the transmission limit for communication systems under syntactic accuracy, whereas semantic information theory (SIT) studies communication from the perspective of semantic fidelity induced by synonymous mapping. In this paper, we investigate the semantic channel capacity of Rayleigh fading channels under synonymous mapping of the channel gain and additive noise. We first derive the semantic capacity formula when synonymous mapping is applied to the channel fading coefficient and establish corresponding upper and lower bounds using Jensen’s inequality. To determine an optimized synonymous partition, the partition design is formulated as a constrained optimization problem and solved numerically using a neural network-based approach with the Adam optimizer. Furthermore, we extend the framework by applying synonymous mapping to both the channel fading coefficient and the additive noise and derive the corresponding semantic capacity formula together with its theoretical bounds. The numerical results illustrate the theoretical semantic channel capacity under synonymous mapping and validate the compatibility of the proposed framework with both CIT and SIT. At a 20-dB SNR with K=8 channel gain intervals and J=4 noise intervals, the semantic capacity reached 9.86 sebits/s/Hz.

Entropy doi: 10.3390/e28060587

Authors: Xiang He Yanzhu Hu Yingjian Wang Zhen Liang Binbin Xu

In recent years, multi-criteria group decision-making (MCGDM) methods have attracted widespread attention in the academic community. However, most existing MCGDM approaches suffer from limitations in decision-makers’ expressive capacity and the loss of uncertain information. To address these issues, this study proposes a novel multi-criteria group decision-making (MCGDM) framework. First, we developed an evaluation information representation method called the hybrid probabilistic information set (HPIS), which allows DMs to fully express their opinions based on individual cognition using the most suitable form of representation. Second, the criteria importance through inter-criteria correlation (CRITIC) and the combined compromise solution (CoCoSo) methods are extended into the cloud model environment, ensuring that the rich uncertainty information is fully preserved and transmitted throughout the entire evaluation process. Finally, we apply the proposed MCGDM framework to a practical case study evaluating urban flood resilience within an urban agglomeration, to identify its vulnerable components. The results indicate that Baoding, Zhangjiakou, and Chengde are identified as the most vulnerable cities, necessitating immediate and targeted measures to bolster their flood defense capabilities. At the same time, decision-makers can select both qualitative and quantitative comments simultaneously and carry uncertainty information throughout the entire calculation process. Furthermore, the sensitivity and comparative analyses demonstrate the robustness and practical utility of the proposed method under the tested scenarios.

Entropy doi: 10.3390/e28060586

Authors: Ningqiao Liu Yuxin Zhang Zhihang Liu Chao Zheng

The modern power system is characterized by large-scale networks, diverse types of sources and loads, and complex grid structures. Virtual Power Plants (VPPs) are proposed to address the operation problem after the integration of Distributed Energy Resources (DERs). Optimization problems in the VPP operation are predominantly mixed-integer programming (MIP) problems belonging to the class of NP-hard problems, motivating the application of quantum computers. Focusing on the VPP optimal dispatch problem under wind and solar uncertainty, we employ the Model Predictive Control (MPC) framework to conduct the VPP intraday rolling dispatch. The classical model and the Quadratic Unconstrained Binary Optimization (QUBO) model for the MPC-based intraday rolling dispatch problem are formulated, respectively. The QUBO formulation of the VPP dispatch problem renders it directly solvable by a specialized quantum computer based on dissipative optical systems: the Coherent Ising Machine (CIM). Compared with the benchmark classical solvers, the experimental results demonstrate the significant computational time reduction capability of CIM. Specifically, compared to Gurobi, Simulated Annealing and Tabu Search, the CIM achieves relative computational time reductions of 75.25%, 99.95% and 99.96%, respectively, while maintaining competitive solution quality. Our work demonstrates the applicability of CIM and its acceleration potential in VPP intraday rolling dispatch, paving the way for the practical application of specialized photonic quantum computers in smart grids.

Entropy doi: 10.3390/e28060585

Authors: Muen Cai Yuan Shen Xiong Luo Jian Hu

Large language models (LLMs) remain vulnerable to jailbreak attacks, especially in black-box settings where target-model gradients and internal tokenization are inaccessible. Recent information bottleneck-based defenses cast prompt protection as a compression problem, but existing methods still rely heavily on white-box optimization and the intrinsic alignment strength of the protected model. To address these limitations, we propose RIB-Guard, a safety-aware information bottleneck defense for black-box LLMs. RIB-Guard learns a token-level masking policy that extracts a minimally safety-sufficient prompt via reinforcement learning using only black-box feedback. In addition, it introduces an independent lightweight safety head to estimate residual jailbreak risk and provide model-agnostic safety guidance during training. The proposed framework jointly balances prompt compactness, benign utility preservation, and residual risk suppression within a unified objective. Experimental results on direct single-turn harmful and benign prompt settings show that RIB-Guard improves jailbreak robustness while maintaining competitive benign utility. By extending information bottleneck-based prompt protection from white-box to black-box settings, RIB-Guard provides a step toward safety-aware information-theoretic front-end defense for black-box LLMs.

Entropy doi: 10.3390/e28060584

Authors: Nathan Argaman

Bell’s theorem rules out developing a locally causal theory to describe quantum phenomena. Many take this to imply that any model of quantum entanglement must employ variables (called beables by Bell) which follow nonlocal rules, even though signaling is local. The alternative is to adopt an all-at-once (block universe) approach, with beables which may depend on both past and future inputs, even though signaling is causal. Within this lenient-causality approach (a.k.a. retrocausal), simple cases of entanglement have been successfully described by locally mediated stochastic toy models, i.e., toy models which are local in a sense which generalizes Bell’s local causality. Developing a widely applicable reformulation of quantum mechanics along these lines is a grand challenge. This work presents a general framework for such models and theories, and identifies the corresponding ontic and epistemic states. The epistemic state is closely analogous to the quantum state, yielding an explanation for the collapse of the wavefunction. In the case of the models of the framework, it is clear what the information is about. The expression for the empirically verifiable predictions of the models in terms of the ontic and epistemic states displays remarkable parallels to the Born rule. A toy-model example is discussed.

Entropy doi: 10.3390/e28060583

Authors: Randall S. Dumont

Previous 1+1-dimensional Dirac wavepacket calculations showed that the tunneling component of a relativistic electron wavepacket can generate an arrival-time distribution whose peak occurs earlier than the corresponding free-photon peak. However, adapting superluminal tunneling to signaling leads to subluminal signaling due to the low tunneling probability. In the present work we note that the barriers used in those calculations are supercritical with respect to the Sauter–Schwinger effect. Consequently, the single-electron evolution must be accompanied by spontaneous electron–positron production from the vacuum. We derive compact formulas for the electron and positron densities when one additional electron is present, showing that the evolved wavepacket contribution adds to the vacuum-produced electron density, while Pauli blocking reduces the positron density by the negative-energy component of the propagated electron. We then apply these formulas to a fourth-order super-Gaussian barrier which produces superluminal tunneling of an electron. The resulting densities are shown explicitly at several times, and are compared with a semiclassical resonance model for the pair number. The semiclassical description reproduces the numerical growth of the pair yield and clarifies the role of Klein-zone resonance energies and widths. Finally, we outline the extension from 1+1 to 1+3 dimensions by integrating over transverse momenta, using scaling properties of the 1+1-dimensional pair number.

Entropy doi: 10.3390/e28060582

Authors: Junpeng Hu Xiao Guo Jinan Shen Minghui Zheng

Automatic detection of sensitive fields in structured data is a critical prerequisite for privacy compliance and data governance. However, existing approaches face severe cross-domain generalization challenges. Hand-crafted pattern rules often fail under highly heterogeneous naming conventions, while single statistical models tend to overfit and degrade sharply under distribution shifts between training and deployment domains. These limitations stem from the weak semantic signals and distributional heterogeneity of structured data, which make it difficult to simultaneously capture explicit rules and latent, variant-sensitive attributes. To address these challenges, we propose a detection framework based on multi-view complementary features and a Hierarchical Gated Residual Network (HGRN). The framework first constructs a full-spectrum feature system that integrates explicit rules and implicit statistical fingerprints (e.g., entropy and character texture) to fill the semantic gap. It then introduces a decision mechanism combining robust priors with dynamic residual calibration: a random forest provides a stable probabilistic anchor, which is further nonlinearly corrected by a learnable gating-and-expert network. This design explicitly resolves the cognitive conflict between rule-dominated regions and complex distributional regions. Experiments on multiple real-world datasets—including DeSSI, CMS Open Payments and Home Credit—show that the proposed method achieves a Macro-F1 of 0.9408 on DeSSI and exhibits strong in-domain performance. Under strict frozen-model cross-domain transfer, HGRN mitigates the catastrophic collapse observed in pure neural baselines and maintains moderate detection capability, offering interpretable trust allocation between rule-based priors and data-driven correction in both financial and healthcare scenarios.

Entropy doi: 10.3390/e28060581

Authors: Arkady Pikovsky Andreas Deser

We perform stochastic simulations of the Hodgkin–Huxley and Morris–Lecar models with different numbers of ion channels in order to describe the effects of periodic electrical driving on spike rates and the regularity of spiking in a single neuron. For stochastic modeling, we use an efficient method that reduces the piecewise-deterministic Markov process of the membrane potential evolution to an ordinary differential equation between random opening and closing events. To characterize a regular component in the resulting voltage time series, we adopt a Wiener order parameter based on the autocorrelation function. We show that the effect of ion channel stochasticity on the spike rate is stronger at lower external force frequencies. The regular component of neural activity exhibits resonant-like behavior as a function of the driving frequency, with a maximum in the beta range.

Entropy doi: 10.3390/e28060580

Authors: Yifei Huang Siying Luo Bowen Zheng

In the traditional (K,L,MT,MU,N) partially connected linear network, a central server stores a library of N files and connects to K+L−1 transmitters, each equipped with a cache of size MT. Each user is connected to L neighboring transmitters and is equipped with a local cache of size MU. Motivated by practical scenarios in which users can access multiple cache nodes, this paper considers a (K,L,r,MT,MC,N) multiaccess cache-assisted partially connected linear network, where each user can access r neighboring cache nodes under a cyclic wrap-around topology, and each cache node has a storage capacity of MC. We propose a general construction framework based on placement delivery arrays (PDAs). The analysis shows that, when the Maddah-Ali and Niesen (MN) scheme is employed and r is sufficiently large, the achieved normalized delivery time (NDT) approaches that of existing schemes for the traditional partially connected linear network. Moreover, under the same aggregate cache size accessible to each user, numerical results demonstrate that, as the cache size ratio increases, the gap between the NDT achieved by the proposed scheme and that of the traditional partially connected linear network scheme gradually diminishes, while the proposed scheme requires a smaller subpacketization level.

Entropy doi: 10.3390/e28060579

Authors: Xibei Yu Chunyan Song

A hybrid code index modulation based on multi-carrier differential chaos shift keying (DCSK), referred to as HCIM MC-DCSK, is proposed in this paper. In the proposed system, multiple data-bearing information signals are transmitted simultaneously with one reference signal. The number of separate physical channels required for data transmission is M + 1, where M is the number of subcarriers. These data-bearing information signals are separated by different Walsh codes. The chaotic signal and its Hilbert transform are utilized to complete the hybrid index modulation. In addition, analytical bit-error-rate expressions are derived for the proposed HCIM MC-DCSK system operating over AWGN and multipath Rayleigh fading channels. The spectral efficiency and data rate of the proposed system are analyzed. The validity of the analytical results and the superiority of the proposed system are confirmed through relevant simulations.

Entropy doi: 10.3390/e28050578

Authors: Xingzhi Liu Haolan Huang Yingmei Li Zida Xia Shutong Zhao

In the analysis of complex engineering systems, managing uncertainty and optimizing information processing structures are critical for reliable state prediction. The Belief Rule Base (BRB) provides a powerful machine learning approach for integrating expert knowledge with uncertain information. However, mitigating the combinatorial complexity of BRBs through conventional structure simplification often causes unintended information loss, introducing systematic prediction biases that undermine reliability. To address the trade-off between system complexity and modeling accuracy, this study proposes an adaptive belief rule base framework integrating sensitivity analysis with posterior consistency calibration (BRB-ARR). First, an information-driven rule screening mechanism is developed to dynamically determine the pruning threshold based on optimized Mean Square Error (MSE) fluctuations. This method effectively filters redundant rules while avoiding the cognitive biases associated with fixed empirical values. Second, a low-dimensional optimization process is employed to readjust the parameter vector, significantly enhancing computational efficiency. Finally, a posterior calibration module is introduced to compensate for the systematic biases caused by dimensionality reduction, strictly preserving the interpretability of the core inference architecture. To validate the effectiveness of the proposed framework, experimental evaluations are conducted on petroleum pipeline networks and liquid propellant launch vehicles. In the petroleum pipeline scenario, the rule base scale is reduced by over 60 percent from 56 to approximately 20 rules, while the parameter dimensionality decreases from 338 to 122. Compared to the conventional model, the mean squared error is reduced from 0.5291 to 0.3619. Furthermore, in the liquid propellant launch vehicle case, the model achieves a prediction accuracy of 98.57 percent with a mean squared error of 0.00029 while reducing the rule scale from 441 to 109. These results demonstrate that the BRB-ARR model effectively balances structural compactness with high precision prediction, offering a novel approach to uncertainty modeling in intelligent systems.

Entropy doi: 10.3390/e28050577

Authors: Héctor Avilés Ingridh Gracia Rafael Kiesel Verónica Rodríguez Rubén Machucho Alberto Reyes Marco Negrete Gabriel Ramírez Nicolás Luévano Myriam Pequeño Jesús Medrano Felix Weitkämper

We investigate how causal DAG learning algorithms and structural assumptions influence counterfactual decision safety. Four structure learning regimes are compared: expert-guided edge-constrained HC+BIC, unconstrained HC+BIC, MMPC+HC+BIC, and the PC-Stable algorithm. Evaluation is conducted using a leave-one-state-out protocol over a fully enumerated state–action space in a controlled offline autonomous driving setting. The environment is characterized by seven Boolean state variables and six actions, allowing us to disentangle the effects of learning strategies on counterfactual decisions. All models are implemented as probabilistic logic twin networks (PLTNs), with additional sensitivity analysis across parameter configurations. The learning regimes produce markedly different counterfactual decisions. Edge-constrained HC+BIC recommends a diverse set of safe actions, while unconstrained HC+BIC yields fewer but consistently safe alternatives. MMPC+HC+BIC frequently fails to identify safe actions, often associated with weak connectivity of the outcome variable. PC-Stable produces varied recommendations but may include unsafe actions, which is linked to incorrect edge orientations between actions and outcomes. These findings show that structure learning choices and prior knowledge influence counterfactual decisions through the learned structure, affecting the identification of safe alternatives in safety-critical applications.

Entropy doi: 10.3390/e28050576

Authors: Xiaoqiang Zhang Pengfei Chen Xueheng Zhang

Digital images are easily transmitted in Internet, but there is also a great risk of information leakage. To meet the requirements of secure image transmission and real-time communication, an image encryption algorithm based on a novel chaotic map and a three-dimensional histogram is proposed. Firstly, a novel two-dimensional chaotic map is designed. Compared with traditional chaotic systems, it exhibits superior chaotic performance and a wider parameter range; secondly, the proposed algorithm is designed to extend the original image to three dimensions, followed by 3D simultaneous scrambling–diffusion; thirdly, the 2D exclusive OR (XOR) operation is performed for further diffusion; finally, the 3D matrix is merged to obtain the encrypted image. The encrypted images have uniform histograms and pass the Chi-square test. Information entropy is greater than 7.9992, and the average values of Number of Pixels Change Rate (NPCR) and Uniform Average Change Intensity (UACI), being 99.6137 and 33.4783, respectively, show that this algorithm can effectively resist differential attacks. On average, a 512 × 512 image can be encrypted in 0.7 s using the proposed algorithm. Thus, the proposed algorithm is applicable to image transmission over network platforms due to its high security, excellent encryption performance, and high efficiency.

Entropy doi: 10.3390/e28050575

Authors: Shanshan Li Zaizai Yan

This paper proposes a modeling framework for nonlinear degradation under constant-stress accelerated degradation testing (CSADT) to predict failure lifetime. The proposed employs a generalized Wiener process to characterize degradation, wherein the drift coefficient is stress-dependent and the heterogeneity in the diffusion coefficient is explicitly modeled. Random effects are introduced to capture volatility variability across degradation trajectories, and model parameters are estimated via the expectation–maximization (EM) algorithm. Using the law of total probability, the probability density function (PDF) and reliability function of failure lifetime under normal operating conditions are derived. The proposed model is validated using crack propagation simulation data and experimental wear scar width data from an alloy product. The results demonstrate that the proposed model improves prediction accuracy for failure lifetime and reliability, highlighting its potential utility in engineering applications.

Entropy doi: 10.3390/e28050574

Authors: Kazuya Hayata

Revealing correlations for styles in texts has been an interesting topic, providing an example of trans-disciplinarity between physics and linguistics. Typical cases can be seen for sound correlations in verses as well as for word correlations in prose. Of these, long-range correlations are of particular interest because of their connection to the Markovian nature in human cognition. For a famous novel written in an archaic style of Japanese, we conduct an analysis of a series of kanji compounds in the text. Here, kanji is a Japanese name for Chinese characters. Specifically, we focus on the number (equivalent to the length) of a compound. Subsequently, the sequence of numbers is expanded into 6-bit binary codes (equivalent to 64 modes). Replacing each compound in the text with an oscillator in a string, for the chain of the kanji compounds, one can find an analogy with the so-called Fermi-Pasta-Ulam-Tsingou (FPUT) model. Comparative analyses for 16 modern translations by humans and machines show, without exception, a strong dominance for a particular mode and its stability bearing a remote resemblance to a soliton, and at the same time, reproduce a statistical property far from a sort of ergodicity. Furthermore, comparison between the human and machine translations shows that the entropy of the latter is higher than that of the former because machines are subjected to neither a psychological bias nor an inspection by editors. Lastly, in addition to the above translated texts, 6 codices of the classic are also analyzed, and their results are compared with those of the modern translations. Note that the original of the classic has not been found yet.

Entropy doi: 10.3390/e28050573

Authors: Bekir Dogan

Entropy generation analysis provides a thermodynamic framework for quantifying irreversibility in thermal systems. However, most existing second-law studies rely on simplified boundary conditions and do not consider fully coupled conjugate heat transfer involving fluid convection, wall conduction, and external heat exchange. Consequently, thermodynamic assessments under realistic conditions remain limited. This study presents an entropy generation-based assessment of turbulent conjugate heat transfer in circular pipes by considering the combined effects of wall thickness ratio (0.02–0.08), wall thermal conductivity (0.2–400 W/m·K), and external convection (5–100 W/m2·K). A three-dimensional steady RANS-based conjugate heat transfer model is employed, and entropy generation is evaluated to quantify irreversibility within fluid and solid domains. The results indicate that wall-related thermal resistances significantly affect thermodynamic performance. Variations in wall conductivity lead to approximately 15–20% changes in total irreversibility, while increasing external convection from 5 to 20 W/m2·K results in up to 25–30% variation. Increasing wall thickness enhances conductive entropy generation, whereas higher Reynolds numbers increase overall irreversibility. These findings demonstrate that the Biot number is a key parameter governing irreversibility distribution. The results provide energy-efficient design insights for optimizing thermally coupled engineering systems under realistic operating conditions.

Entropy doi: 10.3390/e28050572

Authors: Junjie Wang Zhenyan Wang Xie Ma Xuexi Yi

A remarkable feature of non-Hermitian topological systems with skin effects is that their spectra and eigenstates are strongly dependent on the choice of boundary conditions. Here, we investigate a system where the impurity couples to a nonreciprocal Su–Schrieffer–Heeger (SSH) chain at two points with nonreciprocal coupling. We first study the spectrum of the system and demonstrate that nonreciprocal couplings between the impurity and the chain alter its spectral structure. Particularly, this effect becomes particularly prominent in the limit of unidirectional coupling, inducing a shift in the parameter regime for the zero mode. Meanwhile, the impurity–chain couplings give rise to two effective boundary conditions and determine the spatial distribution of the zero mode. In addition, the localization of bulk states is significantly altered by tuning the nonreciprocity of the impurity–chain coupling. Notably, in the unidirectional coupling regime, two distinct types of bulk states coexist near the same boundary, one differing from the other in both spatial distribution and degree of localization. We also find that the bulk states undergo significant skin phase transitions as the coupling strength varies, characterized by a transition from conventional skin states to bipolar skin states. Our findings establish the feasibility of controlling non-Hermitian topological systems by coupling an impurity.

Entropy doi: 10.3390/e28050571

Authors: Le-Bin Wang Jian Chai Ying Yang

This paper studies a dynamic price adjustment system in platform markets, where sellers continuously revise prices, and examines its implications for market stability. We develop a platform-led discrete-time Stackelberg game model to describe the evolution of sellers’ prices and price adjustment speeds under bounded rationality. Unlike previous studies that treat adjustment speed as exogenous, we model it as an endogenous state variable shaped by profit incentives, behavioral inertia, and price fluctuations. We derive the interior symmetric equilibrium and show that profit-driven acceleration increases sellers’ adjustment speed. When this speed exceeds the stability threshold, the system may leave the stable region, causing bifurcations and complex dynamics. We then introduce a platform-imposed upper bound on adjustment speeds and demonstrate that appropriate regulation can restore stability while balancing market responsiveness and efficiency. Numerical simulations illustrate that moderate acceleration improves profitability, whereas excessive acceleration can lead to low-profit regimes. Entropy-based metrics are used to quantify system complexity, and an entropy-triggered feedback-control mechanism is proposed to mitigate excessive volatility while maintaining flexibility. Overall, the study highlights the importance of governing adjustment dynamics rather than solely focusing on price levels.

Entropy doi: 10.3390/e28050570

Authors: Vasyl Martsenyuk Tomasz Gancarczyk

Understanding how information spreads in complex networks is essential for analyzing social influence, opinion formation, and the emergence of collective behavior. In many real-world systems, interactions are not instantaneous but involve delays due to communication, cognition, and response times. Motivated by this observation, the present paper investigates a delayed network model of information spreading, focusing on how time delay and interaction strength shape the system’s dynamical behavior. The novelty of the proposed approach lies in the formulation of a discrete-time network model that explicitly incorporates delayed interactions within a nonlinear dynamical framework. Using delay difference equations, the model captures both local coupling effects and memory-driven feedback, allowing for a systematic study of their combined impact on stability and complexity. Analytical results establish the existence of steady states and provide conditions for their local stability, revealing critical thresholds at which the system undergoes qualitative transitions. These findings are complemented by extensive numerical simulations. In particular, bifurcation analysis and the computation of the largest Lyapunov exponent demonstrate a progression from stable equilibria to oscillatory behavior, and further to chaotic dynamics as the delay and coupling strength increase. Our results highlight the fundamental role of delay as a mechanism that enhances nonlinear complexity and promotes unpredictable dynamics in networked systems. These insights contribute to a deeper understanding of information propagation processes, and may inform the design and control of spreading phenomena in social and technological networks.

Entropy doi: 10.3390/e28050569

Authors: Franck Florin

This paper introduces a geometric framework for analyzing finite-energy signals observed with additive noise by representing them as points on statistical manifolds equipped with the Fisher–Rao metric. Each signal is associated with a parameter vector θ, which defines a unique probability distribution p(x|θ) on a statistical manifold. We propose a unified approach based on the normal multivariate model to describe a raw signal mixed with additive stationary noise. In the approach considered, the background noise is typically assumed to be stationary, whereas the unknown signal is regarded as deterministic. Leveraging tools from information geometry, we compute geodesic equations for the statistical manifolds. We re-derive known results regarding the multivariate normal models and extend them to the signal processing domain. We show that in some cases, the geodesic equations can be solved to obtain a closed-form expression of the Fisher–Rao distance. This expression corresponds to a minimum bound when the sub-manifold is not geodesic, revealing a fundamental geometric constraint in signal parameter estimation. We introduce the spectral distance function, which characterizes the influence of each spectral component of the signals on the Fisher–Rao distance. Our findings provide theoretical insights for signal clustering and machine learning applications, where geometric distances can characterize classification and estimation tasks.

Entropy doi: 10.3390/e28050568

Authors: Xinyu Li Zhaofa Zhou Zhili Zhang Zhe Liang Zhenjun Chang Yiyi Li

Dynamic gravity measurement, crucial for engineering application, suffers accuracy degradation due to errors induced by carrier maneuvers. To address the limitations of conventional swarm optimization algorithms in precision, stability, and generalizability, this study proposes a Swarm Cooperation Evolution Strategy (SCES) that efficiently integrates multiple algorithms. The proposed SCES is extensively evaluated on the CEC2022 benchmark suite in comparison with several cooperative fusion-related algorithms and representative single optimization algorithms. The experimental results demonstrate that SCES achieves an overall effectiveness score of 0.034 and an optimal accessibility rate exceeding 95%. Compared to the best-performing fusion-based algorithm, these metrics represent improvements of 54.67% and 31.11%, respectively. Moreover, relative to the best-performing single optimization algorithm, the improvements amount to 37.73% and 32.69%, respectively. These findings robustly validate the superior performance of the proposed algorithm. Moreover, an in-depth investigation based on SCES into dynamic error compensation methodologies is conducted. Firstly, a polynomial compensation model is established through error mechanism analysis, with parameters identified via SCES. Secondly, a data-driven compensation model employing a multi-layer long short-term memory (LSTM) network optimized via neural architecture search (NAS) guided by SCES is proposed, circumventing the performance limitations inherent in manually designed networks. Furthermore, an innovative two-stage hybrid strategy is introduced. Systematic trend errors are compensated using the polynomial model, followed by the NAS-LSTM model addressing complex residual nonlinear errors, effectively combining mechanism-based and data-driven approaches. Validation on three lines exhibiting varying maneuverability shows all methods significantly improve accuracy. The hybrid strategy delivers optimal performance, achieving 0.58 mGal internal coincidence accuracy on stable lines and up to 91.58% improvement in external coincidence accuracy under high maneuverability. This research provides an effective high-precision dynamic gravity measurement and compensation solution, advancing engineering applications.

Entropy doi: 10.3390/e28050567

Authors: Dayu Qin Yuzhe Chen Ercan E. Kuruoglu

Designing informative descriptors for time-varying complex networks is important for characterizing structural reconfiguration in evolving graph data. This paper introduces a thermodynamics-inspired framework for dynamic graph analysis, centered on Spectral Core Entropy (SCE), node energy, internal energy, and a temperature-like reconfiguration index. These quantities provide a compact representation of how graph organization changes over time. We apply this framework to resting-state fMRI data from autism spectrum disorder (ASD) and control subjects. At the event level, the temperature index shows a statistically significant but modest association with low-SSIM reconfiguration events, indicating that it serves as a weak yet reproducible marker of rapid network change. On controlled synthetic dynamic graphs, the framework exhibits regime-dependent sensitivity: spectral-core change is more informative under rewiring, whereas the temperature index is more informative under gain modulation. At the node level, node energy highlights regional differences between ASD and control groups, providing interpretable neuroscientific context for dynamic brain connectivity. Overall, the proposed framework provides a promising and computationally tractable approach for characterizing reconfiguration patterns in dynamic brain networks and other evolving complex systems.

Entropy doi: 10.3390/e28050566

Authors: Topi Halme Visa Koivunen

This paper provides an overview of recent developments in quickest change detection (QCD) for high-dimensional multi-sensor systems, with an emphasis on settings involving structural constraints and limited sensing resources. Classical QCD methodologies, while well understood in low-dimensional and fully observed regimes, face significant challenges when extended to modern applications characterized by large-scale data, constrained sampling or communication, and heterogeneous signal structures. We review key approaches for handling high dimensionality, including methods that exploit sparsity, and other forms of signal heterogeneity. Additionally, we discuss sampling constraints, where observations must be selected or acquired sequentially under resource limitations. Multi-stream applications can require making multiple detections, for example when detecting changes separately in different streams. The underlying assumptions on probability models, the types of changes taking place, commonly used decision-making criteria, performance indices, and error types are described. We also briefly discuss the application of machine learning in cases where the underlying probability models are not known, or there is a need to select which sensors should monitor the phenomena because of the large scale of the system.

Entropy doi: 10.3390/e28050565

Authors: Yiming Geng Tianshuo Yu Yan Liu Jiayin Zhao

Accurate Remaining Useful Life (RUL) prediction is pivotal for the intelligent operation and maintenance of high-precision equipment. However, existing deep learning-based prognostic methods predominantly focus on point estimations and often overlook the non-Markovian characteristics and stochastic uncertainties inherent in complex mechanical degradation. To bridge this gap, this study proposes a novel uncertainty-aware hybrid prognostic framework by synergizing TCN–Transformer architectures with fractional Brownian motion (FBM). Specifically, a TCN–Transformer hybrid network is developed to adaptively learn a multi-scale drift function, effectively capturing both localized causal features and global long-range temporal dependencies. Concurrently, the FBM component is employed to model the diffusion process, explicitly accounting for the long-range dependence and inherent stochasticity of degradation. By leveraging the first hitting time (FHT) principle, an approximate analytical expression for the RUL probability density function (PDF) is derived based on an established approximation treatment for FBM-driven degradation processes, enabling robust uncertainty quantification. Experimental results on both the XJTU-SY bearing dataset and the servo tool holder power head system dataset demonstrate that the proposed method achieves superior predictive accuracy and reliable uncertainty quantification, thereby providing effective support for condition-based maintenance and intelligent decision-making.

Entropy doi: 10.3390/e28050564

Authors: Melvin M. Vopson

In this paper we investigate the relationship between Shannon information entropy and symmetry in closed Euclidean polygons within the framework of the second law of information dynamics. Using Lagrange multiplier formalism, we derive the condition for minimum entropy in a system of fixed size, showing that it occurs when all elements have equal multiplicity. Applying this result to two-dimensional polygons, we demonstrate that zero-symmetry configurations maximize entropy, while maximally symmetric shapes correspond to minimum entropy states. We show that although entropy increases with geometric descriptor complexity for asymmetric shapes, it remains invariant for maximally symmetric configurations. These results provide a quantitative basis for the association between symmetry and low information entropy, within the broader framework of information dynamics and entropy minimization principles.

Entropy doi: 10.3390/e28050563

Authors: Qian Jun Beh Moritz Straeter Zeen Sun Leong Chuan Kwek Yuancheng Zhan

Quantum cloning machines are essential in quantum information processing, finding applications in areas such as quantum communication and cryptographic protocols. However, the fidelity of universal quantum cloning machines diminishes as the dimension of the Hilbert space increases, resulting in significantly lower efficiency when cloning high-dimensional quantum states compared to qubits. In this study, we introduce a Hybrid Quantum Autocloning Machine (HQAM) that combines quantum autoencoding with universal quantum cloning. The core concept involves compressing a high-dimensional quantum state into a lower-dimensional effective subspace through a quantum autoencoder, conducting the cloning process within this reduced subspace, and then reconstructing the state in the original Hilbert space. Our results show that, for input states with a strong overlap with the effective qubit subspace, the HQAM achieves cloning fidelities exceeding the benchmark fidelity of direct qutrit universal cloning and approaching the optimal qubit cloning limit, while maintaining robustness under noise. These findings demonstrate that compression-assisted cloning provides a practical strategy for improving cloning performance in high-dimensional quantum systems and may enable more efficient quantum information processing protocols.

Entropy doi: 10.3390/e28050562

Authors: Sovon Chakraborty Shakib Mahmud Dipto Kevin R. Pilkiewicz Michael L. Mayo Pratip Rana

Evaluating the trustworthiness of black-box machine learning models remains a significant methodological challenge. Their lack of transparency and interpretability limits applicability, because stakeholders often seek transparency before trusting the results of black-box machine learning models. Explainable AI (XAI) methods provide for human-understandable justifications and informed decision-making of these black-box architectures. Therefore, it is imperative to select the proper XAI model tailored to specific tasks. In this research, we focus on examining four XAI techniques: PEEK, LRP, GRAD-CAM, and LIME to understand how they perform against each other for image classification tasks. We evaluate the performance, robustness, generalizability, noise stability, and computational efficiency of these methods using a globally recognized dataset. With 7390 images, the Oxford IIT pet dataset provides a comprehensive resource for training a custom Convolutional Neural Network (CNN) and VGG16, enabling a consistent evaluation of each XAI method. First, we analyze the saliency maps of the input images and observe the regions predicted by these XAI methods, and then leverage a noise analysis approach to evaluate their performance in terms of accuracy. We further explore the robustness, run-time, and “faithfulness” metrics of each XAI method. In general, we find that these methods can identify a set of input-data features that are critical for accurate classification but also intuitive, such as the outline, face, and eyes of subjects. However, our analysis reveals only marginal consensus among XAI methods in identifying those critical features. Grad-CAM demonstrates strong robustness and stability in VGG16, but the performance on the shallow CNN model remained inconsistent.

Entropy doi: 10.3390/e28050561

Authors: Xiaonan Ma Hua Yang Yanli Xu Naoki Wakamiya

Maritime emergency response requires broadband and reliable communications in sea areas where shore coverage is limited or emergency connectivity is temporarily unavailable, making rapid on-demand aerial networking essential. Unmanned aerial vehicles (UAVs) acting as aerial base stations can be rapidly deployed to provide on-demand coverage; however, ship mobility, heterogeneous emergency priorities, and UAV endurance limitations make the joint optimization of user association and multi-UAV deployment a challenging mixed-integer, long-horizon decision problem. This paper considers a multi-UAV maritime emergency communication system where ships are categorized into multiple priority classes and served links must satisfy a minimum signal-to-noise ratio (SNR) constraint. We formulate a long-term system-utility maximization problem that jointly determines (i) per-slot association between UAVs and ships under capacity, priority, and SNR constraints, and (ii) dynamic UAV deployment under mobility, geofencing, and battery constraints. To obtain tractable and high-quality solutions, we decompose the problem into two coupled subproblems. For user association, we propose a Priority-Aware Branch-and-Cut (PA-BAC) algorithm that integrates linear programming relaxation, cutting-plane tightening, and priority-guided branching, with a priority-greedy feasible initialization to accelerate incumbent improvement. For dynamic deployment, we develop an Enhanced Multi-Agent Proximal Policy Optimization (E-MAPPO) method featuring a global value network, entropy regularization, and sequential actor updates to enhance learning stability and exploration. Importantly, the PA-BAC association is embedded into the learning loop to provide reliable, constraint-satisfying per-slot rewards and reduce the burden of end-to-end learning over hybrid-action spaces. Simulation results demonstrate that PA-BAC consistently improves normalized priority-weighted throughput over heuristic association baselines. Moreover, by mathematically enforcing priority and QoS feasibility at every slot and delegating only continuous mobility to MARL, the integrated E-MAPPO-PA-BAC framework achieves higher long-term system utility, improved energy efficiency, and strong robustness across varying ship densities—properties that are vital for time-sensitive maritime emergency communications. Additional runtime, sensitivity, and AIS-driven trace evaluations further verify the computational practicality of PA-BAC and the applicability of the proposed framework under realistic ship mobility patterns.

Entropy doi: 10.3390/e28050560

Authors: Siyu Xie Gang Zhou Haizhou Wang

In recent years, cyberspace governance has become a critical component of national security strategies worldwide. Although social network platforms provide users with convenient channels for expression and information acquisition, unregulated, harmful euphemisms have become increasingly prevalent. These euphemisms disrupt the order of the digital space and trigger secondary harms such as cyberbullying and regional discrimination. Currently, researches on Chinese harmful euphemism detection face three key challenges: the lack of large-scale annotated datasets, the cognitive reasoning deficit in lightweight models, and the latency constraints of Large Language Models (LLMs), which collectively constrain detection performance and real-world generalization. To address these issues, this study first collected a large corpus from social networking platforms and constructed a fine-grained annotated harmful euphemism dataset. Then, a representation learning framework was designed by integrating deep prompt-based chain-of-thought reasoning with multi-head contrastive learning. This framework introduces external knowledge from LLMs to enhance the diversity and precision of semantic representations. Finally, a multi-dimensional semantic perception fusion framework was proposed. It incorporates multiple semantic perception channels and a cross-channel dynamic fusion mechanism, enabling the model to better capture implicit semantics and integrate external contextual knowledge. Experimental results show that our approach significantly outperforms state-of-the-art lightweight models. While large-scale LLMs exhibit superior zero-shot transferability in cross-domain tasks, our proposed model maintains highly competitive performance with substantially lower inference latency and computational overhead. This research provides a novel methodological and technical foundation for detecting harmful euphemisms in social networks.

Entropy doi: 10.3390/e28050559

Authors: Ntebogang Dinah Moroke

The attribution of systemic financial stress to specific market sectors requires metrics that are faithful to the model’s computations, statistically consistent, and connected to a physically meaningful measure of directed information flow. This paper addresses all three requirements through information geometry, contributing to SDGs 7, 8, 9, and 17 through an entropic causal chain linking energy infrastructure failure to financial market stress. We conjecture and empirically verify the Entropy–Saliency Equivalence: Metabolic Saliency is an asymptotically unbiased estimator of the local Kullback–Leibler divergence between stressed and resting sector return distributions, with bias decaying at a parametric rate under Gaussian regularity conditions. The finite-sample bias–variance decomposition of the Kraskov–Stögbauer–Grassberger transfer entropy estimator is derived, establishing a minimax-optimal convergence rate. A novel metric, the Spatio-Temporal Information Flux (STIF), quantifies directed inter-sector stress transmission in bits per trading day, providing a bootstrap-calibrated audit trail aligned with the South African Financial Sector Regulation Act and MiFID II. Empirical validation on the JSE canonical panel (87 securities, 2857 trading days, 2015–2026) with Eskom load-shedding stages as exogenous stress injectors confirms the equivalence (R2=0.810, ρ^=0.90), with walk-forward R2=0.789 and placebo R2=0.081 ruling out estimation artefacts. The energy sector is identified as the primary stress transmitter during Stage 4+ Eskom events (STIF rising from 0.14 to 0.43 bits/day, directional asymmetry ratio 4.7). Robustness checks confirm stability across non-Gaussian securities and rolling transfer entropy windows.

Entropy doi: 10.3390/e28050558

Authors: Fatimah A. Almulhim Mohammed B. Alamari Ali Laksaci

We propose a new nonparametric estimator of the conditional mode in a regression framework where the covariates are functional in nature. The estimator is constructed through a quantile regression approach, which provides a robust alternative to classical density-based procedures. It is well documented that employing the L1-structure in quantile regression, the estimation procedure improves robustness properties, particularly resistance to outliers and heavy-tailed error distributions. This feature makes the L1 estimation of the conditional mode more stable and reliable in complex and high-variability functional data settings. The main objective of this paper is to establish strong consistency, with explicit convergence rates, for the associated kernel estimators, uniformly over a range of bandwidth parameters. The latter is developed under general regularity conditions involving the concentration distribution of the functional regressor, smoothness assumptions on the structural components of the model, and entropy conditions ensuring adequate control of the functional class complexity. Uniformity in bandwidth is essential both from a theoretical and practical issues, as it guarantees stability of the estimator under data-driven smoothing parameter selection. Beyond its theoretical contribution, this paper has direct implications for applied statistics. Specifically, it provides mathematical support for the automatic bandwidth selection procedures in the high-dimensional data context. Furthermore, the main theoretical novelty is highlighted through simulation experiments and applications to real data.

Entropy doi: 10.3390/e28050557

Authors: Cristián Alberto Antiba

This work proposes that fundamental physics remains unchanged, but its local classical representability is limited by the finite entropic capacity of causal horizons. Geometric entanglement entropy is treated as a thermodynamic potential, and horizons as finite-capacity information systems. When this capacity is saturated, local semiclassical descriptions break down without affecting underlying unitary dynamics. This defines a representational, not dynamical, transition, where configurations persist but cannot be locally encoded. This work formalizes this limit of representation as an intrinsic entropic bound.

Entropy doi: 10.3390/e28050556

Authors: Amaury Dechaux Aliza T. Sloan J. A. Scott Kelso

Many of life’s biggest dilemmas can be summed up as a tension between holding on and letting go. The very language evokes a notion of intentionality which, for the most part, has evaded scientific understanding. How might we even get a window into it? Important insights have come from a seemingly simple task: wiggling one’s fingers to and fro to the beat of a metronome. As the metronome pace increases to some critical frequency, one coordinative pattern becomes unstable and switches spontaneously to another. Such transitions are typically preceded by critical fluctuations, a predicted feature of self-organization in complex, dynamical systems. Here we address the nature and source of these fluctuations, usually assumed to be: (1) random; (2) of external origin; and (3) of fixed magnitude. We performed an experiment in which participants were instructed to oscillate their fingers in either an in-phase or anti-phase pattern in time with a metronome and instructed them to either “hold-on” or “let-go” should they feel the pattern begin to change, yielding a 2 by 2 within-subjects design. We observed that as the metronome frequency was increased from 1.00 to 3.00 Hz, fluctuations in the relative phase between the fingers were significantly altered both by the starting coordinative pattern as well as the participant’s intention to “hold it on” or “let it go”. Specifically, the intention to hold on to the anti-phase pattern delayed the spontaneous transition to in-phase, an effect that was paired with increased fluctuations beyond the critical frequency. These observations were analyzed under the extended Haken–Kelso–Bunz (HKB) model which describes the non-linear stochastic dynamics of the order parameter (relative phase) as a gradient descent on a certain potential. Our analysis, in line with experimental results, suggests that intention transforms the HKB potential not only by stabilizing unstable coordination states but also (paradoxically) by increasing fluctuations around them. Such findings may offer new interpretative light on the relation between intention and fluctuations in the coordination dynamics of living things.

Entropy doi: 10.3390/e28050555

Authors: Paulina Trybek Jerzy Dajka

A quantum model describing ion channels from an information-theoretic perspective is considered. The information χ-capacity of an ion channel, treated as an information channel whose properties are modified by continuous quantum measurements, is investigated. The behavior of the χ-capacity is analyzed as a function of the measurement parameters, in particular the type of measured observable, the measurement duration, and the measurement strength. It is shown that the information χ-capacity exhibits qualitatively different behaviors depending on the measurement conditions, including regimes of rapid decay as well as regimes where it remains finite for long observation times. These results indicate that, within the considered model, continuous observation may significantly influence the information-theoretic properties of the effective ion-channel dynamics.

Entropy doi: 10.3390/e28050554

Authors: Jiatong Wu Sen Wang Kai Niu Yifei She Ping Zhang

Classical Algorithmic Information Theory (AIT) provides a rigorous foundation for information-based similarity measurement, but classical formulations and their compression-based approximations largely operate at the syntactic level, making them sensitive to surface-level variation and insufficient for semantic equivalence. To address this limitation, this paper introduces Semantic Algorithmic Information Theory. The contributions are organized around three core aspects. First, regarding algorithmic extension, we formalize the Semantic Turing Machine System (STMS) to decouple abstract concepts from their diverse syntactic realizations. Within this framework, Semantic Complexity is defined as the minimum program length required to generate some realization in a synonymous set, thereby characterizing compact meaning representation. Second, to enable approximate computation, we move from the ideal, uncomputable semantic information distance to a model-based direct estimator of the Normalized Semantic Information Distance (NSID), which uses neural autoregressive models as conditional probability estimators. Finally, through experimental validation and comparative analysis, we show that the NSID estimator suppresses syntactic variance while preserving semantic structure. Empirical results indicate that NSID provides a practical, computable surrogate for semantic distance and improves upon classical syntactic metrics in evaluating cross-representational equivalence.

Entropy doi: 10.3390/e28050553

Authors: Valerian V. Popkov

Bayesian inference is predominantly formulated in a continuous framework, in which posterior beliefs are represented by smooth probability densities. However, an alternative discrete representation—already implicit in Bayes’s original construction—remains conceptually distinct and structurally informative. This paper develops a representation-level analysis of Bayesian updating in the binomial setting and shows that discrete and continuous posteriors may exhibit qualitatively distinct behavior under finite parameter resolution. In particular, coarse discretization can induce regime-dependent divergence from the continuous posterior, even when the algebraic form of the likelihood is identical. The analysis further demonstrates that divergence is not determined solely by grid resolution but also by the balance between prior strength and sample size. By introducing a scale-dependent perspective in which representational resolution and prior magnitude jointly define distinct regimes of inference, the paper clarifies how structural and analytic descriptions interact under finite conditions.

Entropy doi: 10.3390/e28050552

Authors: Oleksii Zhokh Peter Strizhak

The overall rate of a heterogeneous catalytic process may be limited by the rate of a chemical transformation on the catalyst surface, reactant transport to either external or internal catalyst surface, or by the interplay between these factors. In this paper, we consider each case concerning the influence of diffusion limitations on the overall process rate. The internal, external, and overall effectiveness factors are obtained for various Langmuir–Hinshelwood kinetic rate equations and catalyst shapes via numerical simulations. It is shown that different kinetic rate equations provide an equally good description of the experimental data obtained under reaction-rate control. In contrast, the internal, external, and overall effectiveness factors may obey dissimilar trends for various kinetic rate equations. The obtained findings are of practical interest since the external and internal diffusion limitations can be achieved by simply changing the feed flow rate in a chemical reactor, catalyst particle size, or temperature increase. Therefore, the presented simulations deliver an easy and comprehensive tool for kinetic model discrimination based on the comparison of the overall effectiveness factors derived in the frame of various rate equations with the experimental one. This result represents a new utilization of an old concept, which is the effectiveness factor, for the selection between the plausible kinetic models.

Entropy doi: 10.3390/e28050551

Authors: Chandrajit Bajaj

Real-world autonomous agents learn under nonstationarity, safety constraints, and finite energetic budgets. We develop a framework for perennial learning—agents that continuously refine their models while provably controlling the cost of forgetting—by unifying three classical pillars: Kolmogorov complexity, which equates scientific discovery with algorithmic compression; Landauer’s principle, which assigns a minimal thermodynamic cost of kBTln2 per erased bit to every irreversible model update; and port-Hamiltonian (PH) dynamics, whose (J−R)∇H decomposition separates zero-cost reversible inference from costly irreversible forgetting by construction. The Maxwell demon analogy is formalized: each learning episode is a Szilard cycle in which information acquisition, belief transport, and memory erasure must balance thermodynamically. The information-distance framework, comprising the normalized information distance (NID) and normalized compression distance (NCD), provides a computable geometry for measuring learning progress and guiding curriculum design. We separate theideal uncomputable regularizer based on prefix complexity from the practical compressor/MDL (minimum description length) surrogate that appears in optimization and prove a calibration lemma linking the two under a mild uniform-accuracy assumption. Under explicit regularity, compact-sublevel, and non-energy-extracting assumptions, we prove a passivity speed limit for curriculum-induced contractions of the effective feasible set. Under local asymptotic normality, we reprove that Fisher information is a local posterior codelength proxy rather than an exact theorem about algorithmic entropy. A conditional sequential information-budget proposition shows that the per-stage sample requirement scales as O˜(Δkt/λ⋆), where Δkt is the number of materially changed model coordinates (not the total model complexity kt); the k3→Δk improvement is conditional on a warm-start assumption and a chosen cold-start baseline. A double-integrator running example with a moving obstacle illustrates the architecture.

Entropy doi: 10.3390/e28050550

Authors: Sara Aminpour Yaser M. Banad Sarah S. Sharif

Quantum machine learning integrates quantum computing with classical machine learning techniques to enhance computational power and efficiency. A major challenge in quantum machine learning is developing robust quantum classifiers capable of accurately processing and classifying complex datasets. In this work, we present an advanced approach leveraging data re-uploading, a strategy that cyclically encodes classical data into quantum states to improve classifier performance. We examine two cost functions, fidelity and trace distance, across various quantum classifier configurations, including single-qubit, two-qubit, and entangled two-qubit systems. Additionally, we evaluate four optimization techniques (L-BFGS-B, COBYLA, Nelder–Mead, and SLSQP) to determine their effectiveness in optimizing quantum circuits for both linear and non-linear classification tasks. Our results show that the choice of optimization method significantly impacts classifier performance, with L-BFGS-B and COBYLA often yielding superior accuracy. The two-qubit entangled classifier shows improved accuracy over its non-entangled counterpart, albeit with increased computational cost. Also, the two-qubit entangled classifier is the best option for real-world random datasets in terms of accuracy and computational cost. Linear classification tasks generally exhibit more stable performance across optimization techniques compared to non-linear tasks. Our findings highlight the potential of data re-uploading in quantum machine learning, outperforming existing quantum classifier models in terms of accuracy and robustness. This work contributes to the growing field of quantum machine learning by providing a comprehensive comparison of classification strategies and optimization techniques in quantum computing environments, offering a foundation for developing more efficient and accurate quantum classifiers.

Entropy doi: 10.3390/e28050549

Authors: Shengli Xu Yangwang Fang Hanqiao Huang

Accurately determining the parameters of an unmodeled spacecraft is crucial. Filtering methods that are resilient to uncertainty, employing dual quaternion frameworks to ascertain orientation and position, introduce a design for an extended Kalman filter based on variational Bayesian inference and dual vector quaternions (VB-DVQEKF) to carry out parameter estimation for a non-cooperative spacecraft. The system kinematics and dynamics are modeled using dual vector quaternions, rendering the representation manifestly concise. The method achieves thoroughness by accounting for the coupled interactions between translational and rotational motions. Furthermore, to address uncertainties in the measurements, a variational Bayesian approach is employed for the dependable simultaneous estimation of state parameters and measurement noise covariance. Mathematical simulations are used to verify the proposed VB-DVQEKF, and its robust capabilities are demonstrated through comparisons with several conventional parameter estimation techniques, including the conventional DVQ-EKF and the Sage–Husa adaptive DVQ-EKF (SH-DVQEKF). Quantitative results based on root-mean-square error (RMSE), convergence time, and final estimation error confirm that the proposed VB-DVQEKF achieves the smallest steady-state error among the compared methods and remains stable under white-burst, gradient (drift), and outlier-type measurement anomalies.

Entropy doi: 10.3390/e28050548

Authors: Lei Liao Chao Wang Jun Wang Yinchao Liao Yanjie Lai

Stock price prediction is a challenging problem in quantitative investment, as financial markets generate complex, noisy, and dynamic time series containing heterogeneous signals. Short-term fluctuations usually exhibit greater uncertainty and stronger local variation, whereas long-term trends convey relatively stable and persistent information shaped by market and macroeconomic conditions. However, most existing methods struggle to distinguish these two components effectively, often leading to interference between short-term fluctuations and longer-term trends. In addition, they fail to capture dynamic temporal dependencies and cross-stock information propagation while preserving the causal structure of financial time series. To tackle these issues, we propose the Wavelet-Decoupled Spatiotemporal Network (WaveDSTN). It leverages wavelet transformation to decompose stock returns into high-frequency and low-frequency information, corresponding to short-term fluctuations and long-term trends, respectively. This decomposition enables the model to learn complementary predictive patterns more effectively. Furthermore, WaveDSTN incorporates a Dual-Path Spatiotemporal Encoder to capture complex temporal dependencies and evolving cross-stock information propagation while preserving temporal order and causal consistency. Extensive experiments demonstrate that WaveDSTN achieves significant improvements over existing methods, showing that explicitly modeling trend and fluctuation components can enhance predictive accuracy and reduce uncertainty in stock return forecasting.

Entropy doi: 10.3390/e28050547

Authors: Ernest Nieznaj

We consider a Markov chain that can be termed a discrete version of Blackwell’s example from 1958. It is constructed with the aid of a sequence of independent Markov chains with two states. It turns out its stationary distribution π and transition matrix P are in detailed balance. As a result, the transition operator associated with P is self-adjoint in ℓ2(π), the Hilbert space of all square summable sequences with respect to π. All eigenvalues of P are therefore real, and we give explicit formulae for them. Their corresponding eigenvectors form an orthogonal family in ℓ2(π). Consequently, P can be diagonalized, and we find manageable formulae for Pn, where n≥2.

Entropy doi: 10.3390/e28050546

Authors: Mrittunjoy Guha Majumdar

The standard Landauer bound W≥kBTln2 sets the fundamental thermodynamic cost for information erasure under ideal conditions: weak system–bath coupling, quasistatic operation, and equilibrium reservoirs. However, realistic quantum error correction (QEC) operates in a profoundly different regime—finite-time syndrome extraction, strong coupling to cryogenic environments, and non-equilibrium dynamics. Here, we develop a unified thermodynamic framework for fault-tolerant quantum computing that incorporates corrections beyond the ideal Landauer limit. We derive a generalized bound on the heat dissipation per QEC cycle: Qmin≥kBTln2+kBTΔISB+ℏτ, and scaling this result to large-scale quantum computers, we find that the total heat load grows polynomially with code distance but remains in the nanowatt range for million-qubit systems—well within the cooling power of modern dilution refrigerators. Applying our model to superconducting qubit architectures, we show that while strong coupling can add up to ∼20% to the ideal cost, finite-time effects contribute approximately 0.55% at 100 ns and 5.5% at 10 ns reset operations. Our results establish that the true thermodynamic cost of fault tolerance, while exceeding the naive Landauer estimate, does not pose a fundamental obstacle to scalability; the dominant engineering challenges lie in the heat load of control electronics and wiring, not in the fundamental dissipation of qubit reset.

Entropy doi: 10.3390/e28050545

Authors: Patric C. Nordbeck Christine Shi Anjali Dutt

People’s willingness to include others and their level of suspicion of those perceived to not belong are constrained by political affiliation and adherence to ideologies such as neoliberalism, Islamophobia, and ethnocentrism. In an era of heightened polarizing discourse about immigrants, the interaction between changing information and constraints can be leveraged to understand how dynamic narratives affect inclusory and exclusory behaviors. This study provides a combination of time-series generating methods and the survey approach to situate participants in a developing scenario. Eighty-two participants completed the dynamically modified survey in a scenario involving an immigrant family moving in next door and responded to two affordances: perceived invitability and reportability of the family. Participants’ responses to each of twelve new situations formed time series of changes in reported inclusion and exclusion. Shannon information was used to quantify the amount of information (or uncertainty) that any given instance in the data series represents about the set of reported behaviors. The results showed clustering around adherence to neoliberal ideology, Islamophobia, ethnocentrism, and political affiliation, along with their relation to uncertainty of the time series. We discuss potential implications for perceptions of others’ behavior and the potential of the modified survey and affordance-focus as dynamic and relational methods.

Entropy doi: 10.3390/e28050544

Authors: Michaël Vaillant Tony C. Scott

We propose a microphysical completion for the scalar sector of dilatonic gravity by identifying the dilaton with the coarse-grained stiffness mode of a constrained complex tension field defined on a discrete relational network. Under a controlled ordered-regime coarse-graining, the real projection of the tension scales as Φ(Θ)=Φ0cosΘ, so the Planck mass varies with the phase angle Θ and the Einstein-frame canonical scalar becomes φ∝ln[Φ(Θ)/Φ0]. This logarithmic structure emerges naturally from the Weyl map and provides the correct canonical variable for vacuum models inspired by the Logarithmic Schrödinger Equation (LogSE). We outline how this scalar–tensor interface can satisfy Solar-System constraints through environmental locking and discuss avenues for laboratory and astrophysical tests based on stiffness–coherence coupling. This paper does not introduce a new scalar–tensor EFT class as such; rather, it provides a controlled microphysical origin for a specific scalar stiffness law, Φ(Θ)∝cosΘ, and for the resulting logarithmic Einstein-frame canonical structure.

Entropy doi: 10.3390/e28050543

Authors: Lingfeng Lyu Xiao Guo Zongqi Liu

This paper studies transfer learning for ridge-regularized robust linear regression in the moderate–dimensional regime, where the number of predictors is of the same order as the sample size and the regression coefficients are not assumed to be sparse. We propose Trans-RR, which combines a robust ridge estimator from a source study with a robust ridge correction based on the target study. Under mild assumptions, we characterize the asymptotic estimation error of the proposed estimator and show that leveraging source data can substantially improve estimation accuracy relative to the traditional single-study ridge-regularized robust estimator. To guard against negative transfer when the source study is not sufficiently informative, we further propose an adaptive aggregation of Trans-RR with the single-task estimator that selects the mixing weight by cross-validation. Simulation studies and a real-data analysis support the theory and illustrate the transition between positive and negative transfer as the discrepancy between the source and target studies varies.

Entropy doi: 10.3390/e28050542

Authors: Zhigang Chen Xinxia Song Liqun Chen

Deploying neural networks on encrypted data requires efficient evaluation of nonlinear activations, especially the ReLU function, without decryption. While the CKKS homomorphic encryption scheme supports packed arithmetic over approximate numbers efficiently, its approximate semantics make direct nonlinear evaluation difficult, and polynomial surrogates often introduce approximation error and non-discrete outputs. In this work, we present a task-specific, non-interactive construction for discrete ReLU evaluation in CKKS by combining modulus-switch-based discretization with interpolation-driven lookup-table (LUT) evaluation. We instantiate this design in two complementary schemes. The first uses trigonometric Hermite interpolation and functional bootstrapping to compute a discrete sign indicator, which is then combined with the encrypted input through conditional multiplication to obtain the ReLU output; this variant is compact and suitable for lightweight settings. The second uses iterative most-significant-bit (MSB) bootstrapping to support larger plaintext moduli and higher-precision regimes through repeated digit extraction. A common enabler of both schemes is a discretization step that maps approximate CKKS plaintexts to a finite integer representation; exactness in our setting therefore refers to exact evaluation over this discretized representation, while the deviation from the original CKKS plaintext is governed by the discretization error analyzed in Lemma 1. Experiments on encrypted MNIST inference and the accompanying LUT/storage analysis indicate that the proposed schemes preserve competitive accuracy relative to polynomial-approximation baselines while maintaining manageable auxiliary storage under the reported parameter settings. These results suggest that interpolation-based discrete activation is a promising alternative to polynomial approximation for selected CKKS-based encrypted inference tasks.

Entropy doi: 10.3390/e28050541

Authors: Yuqi Zhou Junyao Sun

In realistic social networks, individuals are influenced not only by current interactions, but also by recent historical opinions, prior experience, and external guidance. However, historical dependence and its decaying effect remain insufficiently studied in bounded-confidence opinion dynamics. To address this issue, this paper proposes an extended Hegselmann–Krause (HK) model in which each individual updates its opinion according to four information sources: the current opinion, historical opinions, neighbors’ opinions, and a target opinion. The historical-opinion term is modeled as a weighted average of recent historical opinions, and its influence is regulated by an attenuation rate to capture memory decay over time. Simulation experiments are conducted to examine the effects of confidence thresholds, attenuation rates, weighting coefficients, and network topology on opinion evolution. The results show that low confidence thresholds tend to generate fragmented clusters, moderate thresholds facilitate opinion integration, and excessively high thresholds may lead to rapid homogenization. The attenuation rate regulates the balance between historical dependence and adaptability to new information, while different weighting configurations produce distinct evolution patterns. In addition, comparisons across ER random, WS small-world, and BA scale-free networks show that network topology significantly affects convergence speed and final opinion distributions. Finally, simulations on a real-world review-network topology derived from the Epinions dataset illustrate the applicability of the proposed model in an e-commerce-related setting. These findings extend the HK framework from a memory-aware perspective.

Entropy doi: 10.3390/e28050540

Authors: Weihan Zhang Zhangtao Liang Yun Mao Hang Zhang Ying Guo

Quantum secret sharing (QSS) has been previously demonstrated with conceivability in optical-fiber channels. However, extending this framework to the microwave frequency band presents challenges in achieving secure quantum communications over turbulent channels, as intricate turbulence can induce amplitude and phase jitter in quantum signals, leading to decoherence or even interruptions in the communication link. In this work, we propose a microwave-enabled continuous-variable quantum secret sharing (CVQSS) scheme operating over turbulent free-space channels. The protocol explicitly addresses the extreme sensitivity of microwave quantum states to environmental turbulence, which manifests as severe amplitude and phase fluctuations. It incorporates the Shamir threshold scheme to facilitate multi-user secret sharing. We suggest a flexible approach to solving problems of adaptive phase compensation and multi-aperture reception techniques when characterizing an equivalent noise channel based on the Kolmogorov turbulence model. The proposed measurement-device-independent (MDI) architecture renders the protocol immune to all detector-side attacks, provided that the state preparation at the users’ side is trusted. Numerical simulations ascertain the performance of the microwave continuous-variable measurement-device-independent quantum secret sharing (CV-MDI-QSS) system and demonstrate the feasibility of practical deployment in complicated turbulent channels. This approach offers a turbulence-resistant solution for dynamic quantum networks through harsh free-space channels implemented in microwave-propagated environments.

Entropy doi: 10.3390/e28050539

Authors: Tuğçe Ekiz Yılmaz Cemal Zehir

Brent crude oil prices are strategically important due to their sensitivity to geopolitical developments, financial market stress, and global monetary conditions. This study examines whether strategic risk indicators improve the forecasting performance of Brent crude oil returns within an integrated econometric and machine learning framework. Monthly data from January 2001 to December 2025 are employed, using the Global Geopolitical Risk Index (GPR), the CBOE Volatility Index (VIX), and the U.S. 10-year Treasury yield (DGS10) as key explanatory variables. Methodologically, the analysis first estimates benchmark econometric models, including ARIMAX (AutoRegressive Integrated Moving Average with Explanatory Variable) and ARIMAX-gjrGARCH (Glosten-Jagannathan-Runkle Generalized Autoregressive Conditional Heteroscedasticity, and then implements machine learning models, namely XGBoost (eXtreme Gradient Boosting), LightGBM (Light Gradient Boosting Machine), and Random Forest, to capture potential nonlinear relationships. Using sMAPE (Symmetric Mean Absolute Percentage Error), forecast performance is assessed over multiple forecast horizons under a rolling-origin framework. Across several forecasting horizons and train-test split configurations, the empirical results consistently show that machine learning techniques, especially LightGBM, offer superior out-of-sample forecasting accuracy. These findings suggest that the dynamics of Brent crude oil returns are influenced by complex and nonlinear relationships between macro-financial conditions, financial uncertainty, and geopolitical risk. The study concludes that flexible data-driven forecasting frameworks offer stronger predictive performance than benchmark econometric models under strategic risk conditions and provide useful implications for energy market risk management and policy decision-making.

Entropy doi: 10.3390/e28050538

Authors: Julian Feiveson Mateu Yearian Maddie Jones Andrés Aragoneses

Feedback delays and the coexistence of multiple timescales are central features of complex dynamical systems, ranging from neural networks and ecosystems to electronic and optical devices. Interactions between fast and slow dynamics can give rise to rich emergent behaviors that are absent in single-timescale systems. Here we investigate how these coupled timescales shape the dynamics of a photonic neuron with single and dual delayed feedback. Using ordinal pattern analysis and recent ordinal-based complexity measures, we characterize the temporal correlations and symmetry properties of the fast peaks and slow spikes generated by the system. Our results show that the signatures of determinism exhibited at fast and slow timescales differ markedly, revealing a strongly multiscale organization of the dynamics. Despite these differences, when represented in the symmetry-based Φ-space, all cases, fast peaks and slow spikes under both single and dual feedback, collapse onto a common curve. This universal structure indicates the presence of underlying constraints governing the system’s dynamics across temporal scales and feedback configurations. These results highlight the power of ordinal-based approaches to uncover hidden symmetries and multiscale organization in delayed nonlinear systems.

Entropy doi: 10.3390/e28050537

Authors: Lu Qiu Yanzhe Jiao Gege Dong Guangcan Cui

Mapping directed spillover pathways in urban housing prices is essential for monitoring the contagion of housing prices across cities. However, existing studies typically rely on either spatial gravity models or time-series models in isolation to analyze intercity connections, thus failing to simultaneously capture the spatiotemporal integration characteristics of housing price contagion. To address this, we embed a finite-length sequence correlation analysis (Correlation-Dependent Balanced Estimation of Diffusion Transfer Entropy, CBEDTE) into the gravity model, yielding the CBEDTE-GM integrated model. Using housing price data from 296 Chinese cities, we construct a spatiotemporal correlation matrix and employ the directed minimum spanning tree algorithm to extract core directed spillover pathways. Results reveal that China’s urban housing price spillover network exhibits a hierarchical architecture with pronounced ripple effects, where eastern coastal cities and the national core city serve as dominant radiation hubs. The East China sub-network occupies a distinctive net spillover position. We identify heterogeneous structural evolution patterns across regional sub-networks: (1) North China evolved from a dispersed multi-centered configuration to a Beijing-dominated single-core structure; (2) East China developed a robust multi-centered architecture anchored by Shanghai; and (3) South China transitioned from a Guangzhou-centered single-core pattern to a tri-polar configuration co-driven by Guangzhou, Shenzhen, and Nanning.

Entropy doi: 10.3390/e28050536

Authors: Mark Kelbert El’mira Yu. Kalimulina

We study binary hypothesis testing for i.i.d. observations under a multiplicative context weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-II losses, we prove the logarithmic asymptotic Ln∗=exp{−nDCw(P,Q)+o(n)},n→∞, where DCw is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, φ(x1n)=∏i=1nφ(xi); this restriction is essential for the single-letter representation and should be distinguished from the weaker qualitative description “multiplicative context weight”. The proof embeds the weighted geometric mixtures φpαq1−α into a likelihood-ratio exponential family and identifies the rate through its log-normaliser. We also derive concentration bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian, Poisson, and exponential models, and extend the exponent characterisation to finitely many hypotheses.

Entropy doi: 10.3390/e28050535

Authors: Jie Zhang Wenjie Zhou Mingxu Wang Yiting Lin

Most existing low-dimensional chaotic maps suffer from a limited dynamical complexity and dynamic degradation, which restrict their effectiveness in image encryption. To address this issue, a novel three-dimensional chaotic map (3D-TCM) was constructed to improve dynamical complexity and stability, and its superiority was verified through a dynamical analysis. Based on these advantages, a plaintext-related image encryption scheme was designed by combining bit-level permutation and S-box-based diffusion. The experimental results show that the proposed scheme achieved high information entropy, a low pixel correlation, and desirable NPCR and UACI values, and successfully passed NIST SP800-22 statistical tests, demonstrating a strong resistance to differential attacks and overall robustness.

Entropy doi: 10.3390/e28050534

Authors: Christoffer Lundbak Olesen Nace Mikuš Mads Hansen Nicolas Legrand Peter Thestrup Waade Christoph Mathys

Biological cognition depends on learning-structured representations in ambiguous environments. Computational models of structure learning typically frame this as an inference problem, but often overlook the temporally extended dynamics that shape learning trajectories under ambiguity. In this paper, we reframe structure learning as an emergent consequence of constraint-based dynamics. Informed by the literature on the role of constraints in complex biological systems, we develop a constraint-based approach to computational cognitive modelling and provide a proof-of-concept model. The model consists of an ensemble of components, each comprising an individual learning process, whose internal updates are locally constrained by both external observations and system-level relational constraints. This is formalised using Bayesian probability as a description of constraint satisfaction rather than epistemic inference. Representational structure is not encoded directly in the model equations, but emerges over time through the interaction, stabilisation, and elimination of components under these constraints. Through a series of simulations in environments with varying degrees of ambiguity, we demonstrate that the model reliably differentiates the observation space into stable representational categories. We further analyse how global parameters controlling internal constraint and initial component precision shape learning trajectories and long-term behavioural alignment with the environment. We discuss the formal relationship between the present approach and Bayesian inference accounts, and argue that a constraint-based approach offers a conceptually distinct foundation for relating computational models to biological systems.

Entropy doi: 10.3390/e28050532

Authors: Xiao Li Xun Wang Zhong-Ming Liu Jin-Biao Liu Xin Huang

Accurate matching between molecular structures and NMR spectra is an important task in automated structure elucidation. However, existing methods still face difficulties in jointly modeling multi-scale molecular topology and effectively exploiting the complementary information provided by paired 1H and 13C NMR spectra. To address these limitations, we propose SpecMol-MatchNet, a multimodal matching framework that integrates a hybrid molecular graph encoder, branch-specific spectral feature learning, and residual multimodal fusion. In the molecular branch, attention-based graph interaction is combined with multi-scale neighborhood aggregation to capture structural cues at different receptive fields. In the spectral branch, branch-specific attention enhancement and joint gating are introduced to better exploit the complementary characteristics of paired 1H and 13C spectra. The resulting molecular and spectral representations are integrated through a residual fusion module for final matching prediction. Experimental results on benchmark datasets demonstrate that SpecMol-MatchNet achieves consistently better overall performance than representative baseline methods.

Entropy doi: 10.3390/e28050533

Authors: Wenbo Li Zhichao Feng Yijie Sun Xinyi Zhang

Health-state assessment is a critical component of prognostics and health management (PHM) for complex equipment. Previous studies on assessing the health state of complex equipment have overlooked the statistical dependence arising from causal coupling relationships between subsystems, which is defined as causality-informed correlation in this study. This correlation introduces redundancy in health information, leading to assessment bias. To address these limitations, this study proposes a health-state assessment model based on the evidential reasoning rule considering causality-informed correlation (ERr-CIC). First, the causal coupling relationships in dynamics and their effects on health-assessment results are analyzed. Based on this analysis, the convergent cross-mapping (CCM) method is employed to examine causal coupling between subsystems. Subsequently, a health-assessment model based on ERr-CIC is developed. This model incorporates a discount factor to quantify the causality-informed correlation among indicators, realized using a conditionally hybrid correlation coefficient (CHCC), and a fusion order derived from signaling sequences. Furthermore, a sensitivity and robustness analysis of the model output to the CHCC is conducted to identify the key parameters governing system behavior and to assess the reliability of the model results under parameter perturbations. Finally, experiments are performed on the PAMD simulation device for validation, and the proposed model is compared with three other typical health-state assessment models. The results show that the ERr-CIC model proposed in this paper achieves relatively balanced performance in terms of stability and interpretability while maintaining competitive model accuracy.

Entropy doi: 10.3390/e28050531

Authors: Adrian-Mihail Stoica Isaac Yaesh

This paper presents an optimal L2-induced control problem for systems with multiple sector-bounded nonlinearities. Sufficient boundedness conditions for the L2-induced norm are derived in terms of a specific system of linear matrix inequalities (LMIs). Based on these conditions, an optimal state feedback control problem is then formulated and solved for the considered class of nonlinear systems. A procedure to reduce the conservatism of the derived conditions is also provided. The proposed formulation, which explicitly considers multiple sector-bounded nonlinearities, is useful because it enables optimal L2-control problems for a much wider class of nonlinearities. Indeed, by invoking the universal approximation theorem, one may represent nonlinearities that do not satisfy sector-bounded conditions as a weighted sum of sector-bounded sigmoid functions. The theoretical and procedural developments are illustrated by a numerical example consisting of the state feedback optimal L2-induced control of a forced van der Pol oscillator.

Entropy doi: 10.3390/e28050530

Authors: Xiaoqiang Zhang Haoran Hu

With the rapid advancement of information technology, the secure transmission and storage of digital images have garnered increasing attention. To safeguard image information from theft and enhance security during network transmission, a novel image encryption algorithm based on a two-dimensional chaotic system named the two-dimensional sine-cubic modular map (2D-SCMM) and a rotating dial model is proposed. First, the 2D-SCMM is designed, and comprehensive dynamic analyses along with randomness assessments are conducted. Second, in the scrambling phase, a diagonal cyclic-shift transformation is employed to dynamically update the distribution of pixel positions. Third, during the diffusion phase, inspired by the dial phone, the rotating dial model is utilized to achieve dynamic pixel updates. Finally, extensive testing and comparative analyses reveal that image pixels are evenly distributed, the average entropy value for grayscale images is 7.9993, and the correlation coefficients approach 0. Meanwhile, the encryption algorithm is highly secure against various attacks, such as noise attacks and cropping attacks.

Entropy doi: 10.3390/e28050529

Authors: Luis Adrián Lizama-Pérez

We introduce a quantum multiplexer (GHZ MUX) architecture that enables deterministic routing of an unknown qubit from a single sender to one of 2n receivers using only local tripartite Greenberger–Horne–Zeilinger (GHZ) states arranged in a binary tree. At each level of the hierarchy, a Bell-basis measurement and classical feed-forward propagate the encoded quantum information along a selected branch while maintaining the appropriate Pauli correction frame. Unlike quantum routing architectures that rely on globally entangled multipartite states, the proposed design composes small GHZ clusters into a modular teleportation hierarchy that requires only local entanglement generation and coherence. This structure achieves full input–output connectivity while preserving deterministic routing control and experimental feasibility for near-term small-scale quantum networks. Beyond routing functionality, we show that the same GHZ-tree structure naturally supports hidden-destination communication. We formalize this extension as the Hidden-Secret GHZ-Tree Routing (HS-GTR) protocol, in which the final receiver remains unknown to external observers and the transmitted quantum state may optionally be protected by a quantum one-time pad. This construction demonstrates that hierarchical GHZ routing can serve not only as a quantum switching architecture but also as a building block for privacy-preserving communication and multi-receiver key establishment in distributed quantum networks.

Entropy doi: 10.3390/e28050528

Authors: Joanna Olbryś Dawid Toczydłowski

The aim of this study is to thoroughly assess the informational content of the CBOE Volatility Index® (VIX® Index) in the context of various turbulent periods. The VIX Index is especially important from an investor perspective. It is often referred to as the “investor fear gauge”, because its level tends to spike during periods of market turmoil and other extreme events. Therefore, this index significantly differs from other market indices and financial instruments. Information theory and normalized Shannon entropy, combined with a rolling-window dynamic approach, are used to explore the evolution of the VIX Index over time. The research hypothesis states that the informational content of the VIX Index varies substantially across periods affected by crucial events. To verify this hypothesis, three important periods of the twenty-first century are analyzed: (1) the Global Financial Crisis, (2) the COVID-19 pandemic outbreak, and (3) the period covering the sub-periods before and after the Donald Trump’s Presidential Inauguration. The results provide no reason to reject the research hypothesis. The empirical findings show that the entropy values appear to be quite sensitive to the choice of discretizaton procedure. However, this evidence is consistent with the existing literature.

Entropy doi: 10.3390/e28050527

Authors: Tomasz Górecki

Expected goals (xG) is widely used to quantify offensive performance in football by summarizing the expected number of goals from shot-level scoring probabilities. However, xG reflects only the first moment of the underlying Bernoulli system and does not capture how scoring probability is distributed across shots. As a result, teams with identical total xG may nevertheless have different probabilities of scoring at least once. In this paper, we study the quantity xG+=−logP(G=0), which is a monotone transform of the exact no-goal probability and, equivalently, of the probability of scoring at least once. We interpret xG+ as an additive, event-specific summary of aggregated Bernoulli risk and analyze its main structural properties. In particular, we show that xG+≥xG, with equality only in the degenerate case pi=0 for all i, and we derive a second-order approximation linking xG+−xG to the second moment of shot probabilities, the effective number of shots, and Rényi-2 entropy. Empirical illustrations on football data show how concentrated shot profiles can increase scoring certainty relative to total xG and how exact Bernoulli aggregation differs from a Poisson approximation based only on the mean. While xG remains an appropriate measure of expected scoring volume, xG+ provides a complementary summary targeted at the probability of scoring at least once.

Entropy doi: 10.3390/e28050524

Authors: Natalia Buitrago-Ricaurte José Javier Reyes-Lagos Karsten Berg Rafael González Niño Thomas Penzel Niels Wessel

Heart failure (HF) is accompanied by autonomic dysregulation and disrupted physiological rhythms, yet how cardiorespiratory coupling (CRC) reorganizes across the circadian cycle during everyday life remains incompletely characterized. We studied 24 h ambulatory ECG recordings from 88 healthy controls and 75 patients with HF. Cardiac autonomic dynamics were quantified from RR intervals using standard HRV indices and symbolic/entropy descriptors, and circadian organization was assessed with 24 h cosinor modeling. Respiratory timing was derived from ECG-derived respiration (EDR) to obtain breath-to-breath (BB) intervals and the pulse–respiration quotient (PRQ). System-level coupling was evaluated primarily as event-timing coordination using coordigram-based coordination percentages computed at two timing tolerances (ε = 0.1 s and ε = 0.2 s) over 24 h and hour-by-hour, and complemented by entropy-based timing irregularity for RR, BB, PRQ, and RR–BB cross-entropy. Patients with HF exhibited lower global HRV and reduced information content in RR dynamics, together with circadian chronodisruption characterized mainly by weaker rhythmic expression and increased inter-individual phase dispersion. CRC differences depended on both tolerance and time of day. In entropy-based profiles, RR–BB cross-entropy and RR entropy did not show hour-specific differences; instead, group separation was localized to higher early-night ApEn in BB in HF and to consistently higher daytime/early-evening FuzzEn in PRQ in controls. Together, these findings indicate a time-structured remodeling of cardiac autonomic dynamics and CRC in HF, in which autonomic function is compromised, and coupling alterations become most evident when examined at appropriate timing tolerances and with circadian (hour-resolved) resolution.

Entropy doi: 10.3390/e28050526

Authors: Xiyu Fang Yuan Xue

The Anninghe Fault Zone is an active, deep–large fault in southwestern China, with a history of multiple strong earthquakes. To reveal the temporal patterns of seismicity and improve medium- to short-term earthquake frequency prediction, this study constructs a quarterly seismic frequency sequence (M ≥ 3.0) from May 1972 to September 2025 and applies the SARIMA (seasonal autoregressive integrated moving average) model for modeling and prediction. The hypothesis is that the frequency sequence exhibits modelable seasonality, trends, and nested periodic structures. The ADF test and Ljung–Box test confirm that the sequence is stationary and non-white noise, satisfying the prerequisites for SARIMA modeling. The centered moving average method is used to extract short-term (1 year), medium-term (5 years), and long-term (10 years) periodic components, and corresponding SARIMA models are constructed. Results show that the medium-period model ARIMA(2,0,1) × (1,0,0)20 achieves the best prediction accuracy (RMSE = 0.6868, MAE = 0.6143), followed by the short-period model, while the long-period model yields slightly higher errors. All selected models pass residual white noise tests and parameter significance tests, and exhibit good robustness under different training–test splits. The main innovations are: (1) the first systematic application of SARIMA to earthquake frequency prediction in the Anninghe Fault Zone, and (2) a preliminary physical interpretation of multi-scale periodic components (e.g., seasonal loading, strain accumulation fluctuations). This method offers significant application value in regions with sparse seismic networks or limited precursory data, providing a new statistical tool for regional seismic hazard assessment and disaster mitigation planning.

Entropy doi: 10.3390/e28050525

Authors: Chao Li Yanfei Liu Jieling Wang Zhong Wang Kewei Lu Chengjin Wang

Multi-agent reinforcement learning (MARL) relies on trial-and-error interactions to update policies. However, trial-and-error learning typically requires extensive interactions to achieve satisfactory performance, resulting in low sample efficiency, which limits its application in the real world. To reduce the trial-and-error costs of MARL and accelerate the convergence of multi-agent collaborative policies, we propose a MARL policy transfer method named LoLM-MARL, based on fine-tuning large language models (LLMs). First, leveraging the general world knowledge and reasoning capabilities of LLMs, low-rank adaptation (LoRA) is employed to fine-tune the pre-trained model on source tasks, thereby providing general decision-making knowledge for cross-scenario policy transfer. Second, a dynamic prompt construction method for LLMs is designed. By dynamically eliminating the state information of ineffective agents from the prompts, the method provides denser observation data for the large language model, thereby enhancing its policy representation capability in specific complex collaborative scenarios. Meanwhile, the dynamic prompt design concept enriches the training sub-scenarios for the algorithm, thereby laying the foundation for the model to learn more general decision-making knowledge. Finally, a Kullback–Leibler (KL) divergence regularization method based on an annealing strategy is constructed to ensure consistency between the policy distributions of the fine-tuned model and the pre-trained model, effectively mitigating the catastrophic forgetting problem during the fine-tuning process of the pre-trained model. Experimental results show that in zero-shot transfer tasks, LoLM-MARL achieves a maximum improvement of 101.4% in average win rate compared to existing state-of-the-art (SOTA) methods. In six few-shot transfer tasks, our method consistently achieves better generalization performance than traditional SOTA methods, and improves the convergence speed by 4 to 30 times compared to the training-from-scratch approach, providing a new solution paradigm for efficient policy transfer in complex dynamic environments.

Entropy doi: 10.3390/e28050523

Authors: Takashi Ikegami Hiroki Kojima Akiko Kashiwagi

Collective systems often exhibit emergent behaviors that cannot be reduced to the properties of individual components. A central question is whether individuality itself is a precondition for collective organization, or whether it arises from it. Here we develop and empirically test Community First Theory, which proposes that collective organization is the generative substrate from which individual dynamical identity emerges. To operationalize this claim, we introduce non-trivial information closure (NTIC), which quantifies whether an individual’s temporal predictability is self-determined or distributed across collective relations. Using high-resolution tracking of complete Tetrahymena populations across four generations, we show that information closure emerges transiently in the middle phase of the cell cycle, flanked by strong collective coupling. Cells in the information-closed regime show significantly greater divergence from parental phenotypes, demonstrating that community organization actively generates behavioral diversity. These results provide initial empirical support for Community First Theory in a single-model system and suggest that NTIC offers a substrate-independent tool for locating agency transitions in collective systems.

Entropy doi: 10.3390/e28050522

Authors: François Dubois Zeno Toffano

This contribution proposes a conceptual framework for quantum relations understood as operator-based, scale-dependent semantic structures. It explores the “fractaquantum” hypothesis, emphasizing that nature exhibits quantum properties at all scales, from subatomic particles to social structures. Using Pauli operators, we propose a semantic theory of quantum relations based on the “semiotic square” and on eigenlogic. The "two one-half spin" quantum composition defines the exchange operator at the basis of fundamental quantum relations. The approach is applied to macroscopic phenomena such as “social lasers” and the rhythmic “breathing” of entanglement, suggesting that individuality and social coherence are governed by scale-invariant quantum principles. This project aims to unify several quantum-like approaches under a common relational paradigm and highlights the role of fractal scaling, contextuality, non-commutativity, exchange, indistinguishability and entanglement in the emergence of semantic relations across physical, cognitive, social and artistic domains.

Entropy doi: 10.3390/e28050521

Authors: Jin Wei Xinyuan Wang Liam Xu Yunfei Li

Multimodal models such as CLIP and ALBEF essentially maximize cross-modal mutual information to align heterogeneous modalities, utilizing semantic consistency as an implicit prior. However, this alignment mechanism creates a structural vulnerability: the models rely heavily on invariant information coupling. In this work, we investigate this vulnerability and propose a symmetry-driven adversarial attack framework. Unlike standard methods that inject high-entropy unstructured noise, our approach designs collaborative perturbations by modeling semantic-consistent mappings between geometric image transformations and syntactic text variations. By explicitly exploiting the information redundancy inherent in cross-modal symmetries, our method effectively reduces the entropy of the adversarial search space. This reveals a fundamental trade-off between information invariance and robustness, achieving state-of-the-art attack success rates with imperceptible perturbations.

Entropy doi: 10.3390/e28050520

Authors: Jinhui Hu Lihong Qiao Yucheng Shu

Robust representation learning from multi-source data necessitates the effective orchestration of complementary information while preserving semantic integrity. Existing methods primarily focus on class-level or instance-level alignment, neglecting fine-grained feature consistency and hierarchical collaborative mechanisms, which consequently limits representation precision. To address these issues, we propose the Tri-Level Consistency–Diversity Calibration (TCDC) method, a hierarchical framework designed to optimize information flow across feature, instance, and class levels. Specifically, at the feature level, TCDC imposes a variance–covariance constraint to align fine-grained features, thereby decorrelating dimensions. At the instance level, semantics-guided multi-objective graph learning is integrated with contrastive learning to adaptively calibrate global topology and capture high-order category correlations. Finally, a class-level attraction–repulsion constraint leverages category prototypes as global semantic anchors to promote intra-class aggregation and enhance inter-class separability. Extensive experiments on multiple public datasets demonstrate the effectiveness of TCDC.

Entropy doi: 10.3390/e28050518

Authors: Yang-Fan Jiang Liang Huang Yu-Zhe Zhang Likang Zhang Qi-Chao Sun Zheng-Ping Li Hao Li Weijun Zhang Lixing You Feihu Xu Qiang Zhang Jian-Wei Pan

Entanglement is one of the most striking phenomena in quantum physics, playing important roles in fundamental physics and quantum information science. It enables a secure means of communication—quantum cryptography—and builds up the foundation of its unconditional security. Entanglement-based quantum cryptography has received great attention from the early demonstrations to the recent remarkable achievements. In a practical scenario, although entanglement-based quantum cryptography can provide inherent source-independent security, its detection side has been shown to be vulnerable to external probing attacks. Here we show that entanglement swapping can effectively solve this critical issue, enabling a side-channel-free quantum cryptography.Entanglement swapping allows each user’s quantum state preparation and detection in a completely private station, which is immune to any external probing side channels. We demonstrate the entanglement-swapping quantum cryptography scheme in the field based on two independent entanglement photon sources. Based on the remote entangled photon pairs, we implement the Ekert-1991 protocol under a channel attenuation equivalent to 100 km of standard optical fiber, achieving a Bell violation value of S=2.659±0.092 and a secret key rate of 0.0163 bit/s. While recent device-independent QKD demonstrations have reached 100 km using atoms or ions, our photonic ES-QKD offers a complementary, all-optical pathway that is directly compatible with existing fiber networks and quantum repeaters.

Entropy doi: 10.3390/e28050519

Authors: Guillermo Ezequiel Perna Federico Centrone Esteban Calzetta

We study the fundamental limitations of implementing time-dependent Hamiltonian protocols when “time” is provided by a quantum clock rather than an external classical parameter. For a parametric harmonic oscillator controlled through a shortcut-to-adiabaticity (STA) schedule and coupled to a minimal clock degree of freedom, tracing out the clock yields an effective reduced dynamics that is a mixture of unitary Gaussian trajectories. Within a noise-dominated regime, we compute the energetic deviation from the target STA outcome and its fluctuations, together with the fidelity to the target evolution and the purity loss of the reduced state, for vacuum and coherent initial states. Combining these observables produces a thermodynamic-uncertainty-type tradeoff that links achievable precision to an irreducible loss of purity set by the clock precision and the protocol sensitivity.