Entropy doi: 10.3390/e28070822

Authors: Xinyu Tang Mingzhu Tang Na Li Shumei Zhang

Carbon market prices are jointly shaped by policy interventions, energy market fluctuations, and macroeconomic dynamics, and thus exhibit pronounced nonlinearity, non-stationarity, localized abrupt changes, and time-varying uncertainty. From an information-theoretic perspective, carbon price forecasting can be viewed as the extraction and fusion of effective information from a complex market system driven by heterogeneous endogenous and exogenous signals. To address the challenges of accurately characterizing local high-frequency fluctuations in carbon price series, effectively modeling the interactions between endogenous and exogenous variables, and mitigating the structural noise introduced by conventional serial forecasting frameworks, this study proposes ConvTimeXer, a hybrid model combining bidirectional temporal convolution and TimeXer for carbon market price forecasting. Specifically, the model first employs front-end bidirectional temporal convolutions to extract local multi-scale fluctuation features from the endogenous carbon price series. It then leverages the global token and cross-attention mechanism in TimeXer to achieve dynamic decoupling and deep interaction between endogenous and exogenous variables. Finally, residual fusion of shallow and deep features is introduced to enhance the preservation of local details. Experimental results based on data from China’s carbon market over the past three years demonstrate that the proposed framework delivers high predictive accuracy and strong robustness, effectively balancing responsiveness to local abrupt changes with global trend modeling. This study not only provides an effective approach for carbon price forecasting in complex and uncertain market environments, but also offers valuable insights into non-stationary time-series forecasting driven by multi-source heterogeneous information.

Entropy doi: 10.3390/e28070821

Authors: Bastian Castorene Francisco J. Peña Martin HvE Groves Patricio Vargas

This work presents an exact microcanonical combinatorial analysis of the one-dimensional antiferromagnetic Ising model. At the primary ground-state level crossing B/J=2, degeneracies follow the Fibonacci and Lucas sequences for open chains and periodic rings, respectively. We extend this framework to the complete excitation spectrum, demonstrating that the density of states is constructed from topological defects governed by linear Diophantine equations and p-fold Fibonacci convolutions. Open boundaries act as fractional defects, densifying the chain spectrum into energy steps of 2J, whereas the closed ring remains quantized in units of 4J. Notably, this exact topological counting exposes non-trivial spectral gaps near the fully polarized limit, strictly forbidding the penultimate macroscopic energy levels in both topologies. Using the transfer-matrix formalism, we derive exact closed-form expressions for the critical degeneracies at all energy levels. These results provide a rigorous analytical foundation for extracting exact residual entropies and exposing the intrinsic number-theoretic architecture of quantum critical manifolds.

Entropy doi: 10.3390/e28070820

Authors: Xiaohan Zhang Hailun Dai Chong Zhou Qi Shen

Rapid and accurate multi-class fault diagnosis is essential for high-end equipment because different fault categories require different maintenance responses. This study aims to develop a lightweight and discriminative diagnostic framework that can identify multiple fault categories from non-stationary vibration signals while reducing redundant computation. High-frequency vibration signals provide direct condition information, but long sequences, noise, nonlinear dynamics, and non-stationary behavior make raw-signal classification unreliable. From an entropy-based information-processing perspective, the key issue is to separate informative fault modes from redundant fluctuations and enlarge inter-class distinctions in the probabilistic decision space. This study proposes Fastformer, an integrated framework for vibration-based fault identification. Empirical Mode Decomposition first converts each signal into Intrinsic Mode Functions to reduce modal mixing and preserve fault-related oscillatory components. The resulting components are processed by an encoder-oriented Q/K/V dot-product scoring mechanism, which constructs compact spatiotemporal embeddings without adopting a complete Transformer architecture. Validation-guided pruning removes low-contribution attention responses, while a Margin-Enhanced Fault Softmax classifier optimized with a cross-entropy-based objective strengthens category separation. By combining stable decomposition, lightweight attention scoring, pruning, and probabilistic margin learning, Fastformer achieves faster and more stable convergence. On the XJTU-SpurGear dataset, Fastformer obtains precision, recall, F1-score, and AUC values of 1.000. Additional validation on the HUST bearing dataset further shows that Fastformer achieves the best overall performance among the compared methods, with an AUC value of 0.9596.

Entropy doi: 10.3390/e28070819

Authors: Shuaiqi Wang Zinan Lin Giulia Fanti

We introduce a privacy measure called statistic maximal leakage that quantifies how much a privacy mechanism leaks about a specific secret random variable, relative to the adversary’s prior information about that secret, in the worst case over all possible priors. Statistic maximal leakage is an extension of the well-known maximal leakage framework. Unlike maximal leakage, which protects an arbitrary, unknown secret random variable, statistic maximal leakage is designed to protect a known function of a public random variable. We show that statistic maximal leakage satisfies composition and post-processing properties. Additionally, we show how to efficiently compute it in the special case of deterministic data release mechanisms. We analyze two important mechanisms under statistic maximal leakage: the quantization mechanism and randomized response. We show theoretically and empirically that the quantization mechanism achieves better privacy–utility tradeoffs in the settings we study. This framework may benefit data holders and privacy practitioners who release data containing known secrets by enabling them to assess leakage without specifying an exact prior and to better preserve data utility while protecting those secrets.

Entropy doi: 10.3390/e28070818

Authors: Quan Tang Yazhi Yang Jing Liu

Complex multimodal optimization problems are widespread in machine learning, engineering design, and data science, where multiple local optima often trap conventional algorithms. Balancing global exploration and local exploitation remains a fundamental challenge for population-based optimization algorithms when solving such problems. This paper proposes a dynamic search framework optimization algorithm based on an energy level collaboration mechanism, termed DSF-ELC. The algorithm introduces two synergistic strategies. First, a population dynamic reorganization strategy adaptively adjusts particle migration between two fitness-stratified subpopulations based on real-time diversity measurements, effectively balancing exploration and exploitation. Second, a comprehensive learning strategy enables each dimension of inferior solutions to learn from the corresponding dimension of superior solutions in a randomized manner, thereby enhancing search capability on complex multimodal functions. The two strategies work synergistically to achieve an adaptive exploration-exploitation balance. Experimental validation on the CEC 2017 benchmark suite demonstrates that DSF-ELC achieves superior solution accuracy and stability compared to six representative algorithms on the vast majority of functions. Wilcoxon signed-rank tests, box plot visualization, and convergence curve analysis further validate the effectiveness of the proposed strategies. The results indicate that DSF-ELC has significant advantages and broad application prospects for complex multimodal optimization problems.

Entropy doi: 10.3390/e28070816

Authors: Wenchao Li Haitao Liu

Quantitative phylogenetics in historical linguistics has relied almost entirely on lexical cognate data. This study asks a different question: how much genealogical signal can be recovered from structural features extracted from annotated corpora, and whether it survives at deep time depths. We compute 29 structural features—including Shannon entropies of dependency direction and of dependency-relation distributions, relation-specific directionality ratios, dependency-distance measures, and constructional ratios—across 25 Transeurasian languages from the five proposed groups (Turkic, Mongolic, Tungusic, Japonic, and Koreanic) and three outgroups (Chinese, Vietnamese, and Hindi), 28 languages in all. Most of the Tungusic and Mongolic languages have no running-text corpus, so we built new Universal Dependencies treebanks for them by glossing example sentences from reference grammars; thirteen are used here. Each feature was tested for phylogenetic signal (Pagel’s λ and Blomberg’s K, with FDR correction) under four competing reference topologies, and the features that passed were used for tree inference (Bayesian inference in MrBayes, with Neighbor-Joining as a check). The same pipeline was first run on Indo-European in a companion study, where it recovers only individual subgroups and does not resolve a stable tree. At the depth proposed for the Transeurasian family (a Proto-Transeurasian root of about 9000 years before present), the structural signal was not enough to reconstruct the family’s internal relationships. The signal tests favoured a flat three-way division of the major branches (7 strict/20 relaxed features) over any nested hypothesis (≤2 strict features each), and the strongest signal lay in core word-order parameters (e.g., object direction, λ = 1.00, K = 6.06). But both Bayesian and distance-based inference returned near-complete polytomies: although the chains converged (ASDSF < 0.01), no branch reached a posterior probability above 0.75, and none of the three multi-language branches (Turkic, Mongolic, or Tungusic) was recovered. The outgroup test made the reason clear: Hindi, which is Indo-European but SOV, grouped with the head-final Transeurasian languages rather than with the other two (head-initial) outgroups, so the features are tracking typological similarity, not shared descent, at this depth. The study contributes 13 new treebanks for poorly documented languages, a reproducible framework for testing how much genealogical signal structural features carry, and direct evidence that, at Transeurasian time depths, this signal reflects typology rather than genealogy.

Entropy doi: 10.3390/e28070817

Authors: Mengqi Zhang Guangqiang Teng Xiaoyu Lei

Lei proposed an algorithm Algorithm A3 in 2023 to generate an exact discrete uniform distribution from an unknown biased Bernoulli source. The present paper does not claim a new extraction algorithm. Its contributions are analytical: first, we provide a Fourier-analytic proof of the uniformity mechanism based on roots of unity and coefficient extraction; second, we derive explicit acceptance-probability and expected-runtime bounds, with a rigorous treatment of composite moduli; third, we show how independent accepted A3 digits yield a continuous Uniform(0,1) limit through a base-n expansion and quantify the cost of finite-digit simulation.

Entropy doi: 10.3390/e28070815

Authors: Wenbo Shi Wenlong Xie Jiashen Hu Lishan Liu

Future wireless networks require channel coding schemes that can provide high reliability, low latency, and strong adaptability under finite-blocklength and structurally heterogeneous transmission scenarios. Globally coupled low-density parity-check (GC-LDPC) codes are promising for such systems because their coupled structure can enhance error-correction capability, but the additional global constraints also increase decoding complexity and make conventional fixed-parameter decoders less effective. This paper proposes a prior-assisted hierarchical alternating direction method of multipliers (ADMMs) decoding framework for GC-LDPC codes. The proposed decoder first partitions the GC-LDPC parity-check structure into two local subgraphs and performs tuned ADMM decoding on the local blocks in parallel. The local decoding outputs are then merged and verified by the full GC-LDPC parity-check matrix. If the merged local decision satisfies all global constraints, it is directly accepted, thereby avoiding unnecessary full-graph decoding. Otherwise, a global fallback ADMM decoder is activated. In this stage, the channel log-likelihood ratios are fused with soft priors extracted from the local ADMM outputs, where prior clipping and conflict scaling are introduced to control unreliable or contradictory local information. The resulting fused reliability information is used to guide full-matrix ADMM decoding. This local-to-global strategy reduces unnecessary global iterations while preserving the ability to enforce global consistency when local decoding is insufficient. Simulation-oriented metrics, including bit error rate, frame error rate, local pass rate, global fallback rate, global fallback success rate, and average iteration count, are used to evaluate reliability and decoding efficiency. The proposed framework provides an average-complexity-aware and reliability-aware decoding approach for advanced channel coding in future wireless networks.

Entropy doi: 10.3390/e28070814

Authors: Qiuyang Xue Xiu Jin Jinming Yu Yueli Liu

In light of growing concerns about sustainable development and green innovation, the green market has progressively taken center stage in the financial markets. From the nonlinear information transmission angle, we look into the interconnectedness between the global banking sectors and the green markets using transfer entropy causal networks, containing the Dow Jones Green Bond Index (SPGB), Dow Jones Sustainability Index (DJSI), The S&P Global Clean Energy Index (SPCL), and MSCI World ESG Leaders Index (ESGL). We observe significant bidirectional causal relationships between two markets. The banking industries of developed nations and emerging economies like South Korea, Indonesia, and India are the most important, while four green markets are vital. Furthermore, using the CEEMDAN-SE-KM approach, this study also investigates the two markets’ heterogeneous performance at various time scales. The causal relationships between two markets exhibit heterogeneity at time scales, and that is most noticeable at the short-term scale. Additionally, after the COVID-19 pandemic and the conflict between Russia and Ukraine, there is an increase in the causal relationships between the two markets and a higher efficiency of information transmission. These results help regulatory bodies and green market players have a more thorough understanding of and dynamic regulation of the green market.

Entropy doi: 10.3390/e28070813

Authors: Lloyd Watts Carver Mead

In 2000, Carver Mead introduced a time-symmetrical theory of energy exchange between two atoms, building on the Transactional Interpretation of Quantum Mechanics by John Cramer in 1986. In 2020, Cramer and Mead developed the theory further, proposing a conceptual path integral formulation by which energy could be completely transferred over long distances, and showing that this theory can explain the Einstein–Podolsky–Rosen paradox, the Hanbury-Brown–Twiss effect, and the Freedman–Clauser entanglement experiment. In this paper, we develop the theory further, proposing a specific formulation of the interaction between Emitter and Absorber Atoms, in which the energy density is proportional to the root-mean-square of the product of retarded and advanced four-vector potential waves, and show how this interaction efficiently and completely transfers energy from the Emitter Atom to the Absorber Atom over arbitrary distances. We use Mach’s Principle and conservation of energy to find the proportionality constant by matching the mean transition time constant for all possible Absorbers in the universe to the mean transition lifetime computed from Fermi’s Golden Rule, leading to a complete solution with no adjustable parameters. The solution represents the exchange of energy between two atoms, valid over 26 orders of magnitude in Emitter–Absorber distance, from about 0.52 m to the radius of the Hubble Sphere 1.27 × 1026 m. We define this Wave–Particle Model as the product of a retarded Emitter vector potential wave and an advanced Absorber vector potential wave, which exhibits the particle-like properties of losslessly carrying energy at the speed of light in a straight line from Emitter Atom to Absorber Atom in a vacuum in the absence of gravity.

Entropy doi: 10.3390/e28070812

Authors: Boumediene Hamzi Marianne Clausel Kamal Dingle Marcus Hutter Mohammed Terry Jack

Spurious correlations between time series are a persistent problem: simple, low-complexity patterns are abundant, so unrelated series can easily exhibit high Pearson correlation. We argue that Kolmogorov complexity—a series’ resistance to compression—provides a principled diagnostic for flagging such cases. We prove an algorithmic trilemma: a pair of binary sequences cannot simultaneously be algorithmically independent, highly correlated, and highly complex. This gives a deterministic complexity ceiling for independent correlated pairs and a probabilistic bound under which spurious correlations among independent high-complexity pairs are exponentially rare; we further bridge these results to an effective Hausdorff dimension obstruction. These guarantees hold for binary sequences under Hamming correlation; their extension to real-valued series via serialisation and LZ compression is empirically validated rather than proved, so the joint indicator JLZ=min{C˜LZ(x),C˜LZ(y)} is a calibrated diagnostic, not a causal test. On two toy models—coupled logistic maps and multivariate fractional Brownian motion (dimH=2−H)—false positives are far more common among low-complexity series. Because noise inflates complexity and non-stationary processes can be both complex and spuriously correlated, we recommend a two-stage workflow: establish stationarity, then report JLZ alongside ρ.

Entropy doi: 10.3390/e28070811

Authors: Giorgio Taricco

Entropy regularization is a recurring mechanism in reinforcement learning (RL), but its meaning changes across algorithmic settings. In classical online RL, entropy encourages exploration and smooths policy improvement; in inverse RL and imitation learning, maximum-entropy resolves ambiguity among expert-consistent behaviors; in offline RL, entropy must be balanced against data support; in generative policies, entropy becomes a tractability problem; and in reinforcement learning with verifiable rewards (RLVR) for large language models (LLMs), token entropy is tied to reasoning diversity, calibration, and collapse. This review organizes these developments into a unified taxonomy. We first summarize the mathematical foundations of maximum-entropy RL, soft Bellman equations, policy-gradient entropy dynamics, and Kullback–Leibler (KL)-constrained mirror descent. We then review entropy in imitation learning, offline RL, intrinsic motivation, diffusion and flow-based policy classes, and RLVR. Particular attention is given to recent work on entropy collapse in reasoning LLMs, entropy-based advantage shaping, covariance-based control, positive-advantage reweighting, and ordinary differential equation (ODE)-based flow-matching policies with tractable entropy. The review emphasizes that entropy is not universally beneficial: useful exploration, support preservation, multimodality, calibration, and reasoning diversity require different entropy objects and different control mechanisms.

Entropy doi: 10.3390/e28070810

Authors: Min Wang Xiao-Fei Zhang Guo-Jun Qin Ming Liu

To effectively extract fault characteristics from complex vibration signals and improve the diagnostic performance of deep learning networks, this paper introduces a wind turbine blade fault diagnosis method that combines Multi-scale Enhanced Hierarchical Fuzzy Entropy (MEHFE), Isolation Forest, and the Grey Wolf Optimization (GWO) algorithm for optimizing the Gated Recurrent Unit (GRU). Initially, the MEHFE algorithm is applied to decompose and reconstruct three-directional vibration signals at the blade root, thereby extracting “scale-frequency” dual-dimensional features that represent the evolution of fault frequency structure and complexity across multiple scales. Subsequently, Isolation Forest is employed to assess and filter feature importance, constructing an optimal feature subset to mitigate redundancy and noise interference. Finally, the optimal features are fed into the GRU network for fault pattern recognition, and the GWO algorithm is utilized to adaptively optimize network hyperparameters, thereby enhancing classification accuracy and noise resilience. Simulation experiments on typical wind turbine blade faults reveal that when GRU serves as the classifier, the diagnostic accuracy of MEHFE exceeds 76%. After feature optimization with Isolation Forest and network parameter optimization with GWO, the diagnostic accuracy surpasses 93%, demonstrating notable advantages in both classification capability and stability. Even under conditions of noise interference, the accuracy remains above 90%. The research substantiates that the proposed method can effectively extract pattern information indicative of blade structural damage from vibration data, achieving high fault recognition accuracy and robustness.

Entropy doi: 10.3390/e28070809

Authors: Zihan Wang Yue Zhang Hengpeng Xu Zhenglu Yang Jun Wang

Feature selection is a promising dimension reduction technology that focuses on a reduced subspace by selecting excellent features. Most existing approaches tend to emphasize the discriminative ability of features based on either a global or a local evaluation criterion alone, and a few holistic approaches explore their selection granularity beyond the instance level. This study presents a novel Semantic-aware Instance-wise Feature selection model, dubbed SIF, to address the weakness of existing methods, which assess the significance of features from an individual view. Furthermore, SIF proposes to specify feature representations at the instance level, which is rarely touched by existing methods given the considerable learning complexity. In particular, SIF is designed as a sequential pipeline framework. First, it explicitly models semantic correlations and employs this information to select semantic-aware features. Then, inconsistent instances are captured and guide the instance-wise feature selection. Both types of features constitute the final optimal feature subset, which can represent semantics at a global level as well as describe instance characteristics at a local level. An extensive experimental evaluation illustrates the superiority of SIF under various metrics.

Entropy doi: 10.3390/e28070808

Authors: Jeroen Schoenmaker

This manuscript gives a solution to the black hole information paradox by bringing to the debate a fundamental aspect of information science: the process of measurement by a receiver. Bekenstein and Hawking established the foundations of black hole thermodynamics based on previous works of Brillouin and Szilard on information physics. In this work, we demonstrate that the relation between energy and information established in communication technology by Shannon and Landauer has not been adequately applied to black hole physics. As Landauer states, a computation process is closely akin to a measurement. Our argument is grounded on the physical concepts of measurement, signal-to-noise ratio, energy dissipation during the switching process in computation, and hysteresis loops. We give special attention to the role of noise and energy dissipation in the process of information transmission. We demonstrate that Szilard’s work fails to establish a connection between information and entropy in agreement with the works of Landauer and Shannon. We also demonstrate that a quantum state cannot be directly equivalent to a unit of information. The entropy and temperature attributed to black holes are questioned, and a solution to the black hole information paradox is provided. Similarly to what happens with Maxwell’s demon, the black hole information paradox is “exorcised” once we account for the process of measurement and information processing.

Entropy doi: 10.3390/e28070807

Authors: Rosa Maria Sava Annalisa Greco Alessandro Pluchino Andrea Rapisarda

The analysis of earthquake-induced damage scenarios at the urban scale is a fundamental tool for seismic risk assessment and mitigation and the management of urbanized areas exposed to seismic hazards. This paper presents a methodology for simulating earthquake damage scenarios over large urban territories that explicitly accounts for the cumulative effects of seismic sequences. The proposed approach models the progressive accumulation of structural damage and the resulting evolution of building vulnerability under repeated seismic loading. From a complex systems perspective, the methodology describes urban areas as collections of buildings whose vulnerability evolves through memory-dependent processes. Under this framework, the final damage scenario emerges from the cumulative effects of the entire seismic history rather than from the contribution of individual earthquakes considered in isolation. The study extends previous work by the authors, in which instrumentally derived macroseismic intensity maps were integrated with observed building damage data from the 2009 L’Aquila seismic sequence. The results demonstrated that the methodology could successfully reproduce the spatial distribution of observed damage when considering not only the mainshock but also all seismic events exceeding a selected magnitude threshold. In this contribution, new developments of the calibration procedure are presented, together with applications to the 2013 Garfagnana-Lunigiana and the 2016–2017 Central Italy seismic sequences. Through a comparative analysis of these case studies, the influence of different seismic sequence characteristics and building stock features on damage evolution is investigated. The results provide further insight into the capabilities and limitations of the proposed methodology, highlighting its potential as a tool for interpreting post-earthquake damage patterns and supporting seismic risk assessment and mitigation strategies.

Entropy doi: 10.3390/e28070806

Authors: Siyao Huang Yong Deng

Conflict management is crucial in information fusion. One of the efficient algorithms to address conflicting data fusion is discounting method. However, how to determine the discounting coefficient in conflict management remains an open issue. A heuristic method to determine discounting coefficient is presented based on Deng entropy and sigmoid function. Where Deng entropy quantifies the uncertainty of evidence and the sigmoid function maps it to a reasonable coefficient range. The effectiveness of the proposed method is illustrated by numerical example and real application. Compared with existing methods to determine discounting coefficients, the proposed method shows promising performance in the analyzed examples and is simple to implement.

Entropy doi: 10.3390/e28070805

Authors: András Mihály László Bacsárdi

Quantum memories are a critical component of entanglement-based quantum networks, enabling the storage and synchronisation of quantum states across dynamic links. However, current quantum memories have significantly lower capacity than the rate at which entanglement can be generated, making memory saturation a key bottleneck that reduces network efficiency and hinders the scaling of quantum networks. This problem is especially pronounced in dynamic satellite-based quantum networks, where short visibility windows constrain link availability. In this paper, we present a support entanglement-swapping algorithm that utilises leftover entanglement in quantum memories, thereby alleviating memory saturation and increasing network connectivity. Our algorithm combines two mathematical concepts, line graphs and maximum-cardinality matching, to select independent entanglement swap pairs without sharing any entanglement between concurrent swaps. This property ensures that the resulting changes to the network remain local and mutually independent, making the algorithm easy to integrate alongside any existing routing schemes without requiring network-wide coordination. We evaluate the algorithm through simulations on both static fibre-based networks and dynamic satellite networks. Across most configurations, our algorithm increases both the mean and the total number of entanglements shared between end nodes, while also increasing the network’s long-range connectivity. The ‘SwapWithToUse’ algorithm variant consistently provides the greatest improvements, with gains increasing as entanglement-generation rate increases.

Entropy doi: 10.3390/e28070804

Authors: Jakub Zwydak Marcin Wątorek Jarosław Kwapień Stanisław Drożdż

Artificial transaction generation remains an important source of potential market manipulation on cryptocurrency exchanges, as it may distort reported liquidity and reduce market transparency. This study proposes a diagnostic framework for detecting unusual trading patterns based on complexity and statistical structure measures derived from high-frequency trade-level data. The analysis considers log-returns, trading volume, and transaction counts, using tail distributions, autocorrelation functions, multifractal characteristics, approximate entropy, and detrended cross-correlations. The methodology is applied to BTC, ETH, and XRP traded on Binance, Bitget, KuCoin, and Kraken over the period from 1 April to 30 June 2025. The results reveal a pronounced anomaly on Bitget for BTC and ETH after mid-May 2025. The number of transactions increases sharply, but there is no proportional increase in traded volume or return fluctuations. This regime is characterised by numerous low-volume trades, weaker autocorrelations, reduced multifractal organisation, higher short-pattern irregularity, and weaker cross-correlations involving the transaction-count series. These features are consistent with a noise-like component in trading activity and may indicate artificially increased transaction counts, although they do not provide direct proof of wash trading. The findings show that complexity-based indicators can be useful for detecting exchange-specific trading anomalies that remain hidden in price-based measures.

Entropy doi: 10.3390/e28070803

Authors: Jiaxin Liu Qiwang Chen Yijun Chen

Very-high-resolution (VHR) multispectral satellite imagery contains rich semantic information, yet its real-time transmission is constrained by limited satellite-to-ground bandwidth and dynamic channel impairments. Conventional communication schemes prioritize pixel-level reconstruction, resulting in large transmission overhead and poor robustness under unfavorable channel conditions. To address these challenges, an end-to-end task-oriented semantic communication framework for remote sensing downstream recognition tasks, termed Semantic Transmission Architecture for Remote Sensing (STARS), is proposed. To improve transmission efficiency for very-high-resolution remote sensing images with highly redundant background regions, a Semantic Feature Reweighting Module (SFRM) is introduced to dynamically evaluate token-level semantic importance and adaptively allocate transmission resources to task-critical features. Furthermore, vector quantization and a practical digital transmission chain are jointly integrated to achieve efficient semantic compression, while dynamic channel variations are incorporated during training to improve robustness under fading channel conditions. Experimental results on the DOTA dataset demonstrate that STARS consistently outperforms conventional schemes and existing semantic baselines under Rician fading channels, validating the effectiveness of semantic-aware feature allocation for bandwidth-efficient VHR imagery transmission.

Entropy doi: 10.3390/e28070802

Authors: Meysam Motaharfar Maxwell R. Siebersma Parampreet Singh

We study the quantum dynamics in Krylov space of a spatially flat, homogeneous, and isotropic universe sourced with a massless scalar field within Wheeler–DeWitt (WDW) quantum cosmology and loop quantum cosmology (LQC) frameworks. The availability of a physical Hilbert space and physical Hamiltonian and the presence of an internal clock enable us to construct the Krylov basis analytically by applying the Lanczos algorithm. We then evaluate both the Krylov state and operator complexity for WDW quantum cosmology and LQC on this basis. In regimes where the wave function of the universe is sharply peaked, our results indicate that the Krylov complexity grows quadratically with the scalar field clock for the state and operator complexities in both the WDW quantum cosmology and LQC. We further show that the operator complexity is exactly twice the state complexity in these regimes. We discuss the interpretation of the global behavior of these systems by calculating the Krylov entropy for both quantum cosmological frameworks. We observe that in LQC, the Krylov complexity and entropy remain finite at the bounce, whereas in the WDW quantum cosmology, they diverge at the big bang/crunch singularity. Our work provides the first example of computing Krylov complexity for a system with a totally constrained Hamiltonian and no external time, a framework to calculate a purely quantum-mechanical entropy in quantum cosmology, and, to our knowledge, the first direct bridge between Krylov complexity and canonical quantum cosmology, as a first step toward understanding how polymerized quantum geometry modifies complexity and entropy.

Entropy doi: 10.3390/e28070801

Authors: Qingsong Wang Cailing Yao Jiaxing He Bingzhe Hou

In this paper, we consider a class of constrained systems named n-tuple upper bound (m1,⋯,mn)-constrained systems (n-TUB systems briefly) for 2≤n<∞, which are subshifts of finite type. We determinate the topological entropies (Shannon capacities) C(m1,⋯,mn) of all n-TUB systems and consequently order all n-TUB systems according to the size of the topological entropies. An algorithm is also presented to compute the transition matrix and topological entropy for n-TUB systems.

Entropy doi: 10.3390/e28070800

Authors: Cemile İnce

Image encryption schemes based on chaotic maps offer strong statistical properties but are vulnerable to quantum attacks, and their integration with post-quantum cryptography has not been sufficiently explored. This paper presents a post-quantum secure image encryption framework integrating ML-KEM (FIPS 203), standardized by NIST in 2024, with a two-dimensional Sinh-Logistic chaotic map, HKDF-SHA256 nonce-based key derivation, feedback diffusion, and a novel Seismic Wave Permutation (SWP). The scheme derives channel-specific encryption keys from ML-KEM shared secrets using random, channel-specific nonces via HKDF-SHA256, ensuring plaintext independence and avoiding metadata-based leakage. The proposed SWP effectively breaks spatial correlations by displacing pixels according to a chaotic SWP model. RGB images are processed with independent ML-KEM encapsulation and HKDF-derived key material per channel, enabling multi-channel encryption without cross-channel leakage. Experiments on 512 × 512 test images have demonstrated Shannon entropy exceeding 7.999 bits per pixel across all channels, NPCR of at least 99.59%, UACI between 33.41% and 33.53%, and near-zero pixel correlations, further validated across 14 standard SIPI test images. An IND-CPA game simulation using four independent distinguishers, including a learned classifier trained via chosen-plaintext oracle access, over 5000 rounds per image, showed a maximum adversary advantage of 0.0186, consistent with random prediction. ML-KEM encapsulation contributes between 3.9% (ML-KEM-512) and 8.0% (ML-KEM-1024) of total encryption latency at 512 × 512 resolution, remaining a minority cost across all security levels while keeping the total encryption time within a narrow 227–258 ms range. The proposed architecture bridges standardized post-quantum cryptography with chaos-based image security for privacy-preserving image transmission.

Entropy doi: 10.3390/e28070799

Authors: Soumyaditya Das Soumyajyoti Biswas

We analyze the effect of microscopic heterogeneity on the Lorenz curve of macroscopic observables. The Lorenz curve of a response function, being a cumulative and bounded quantity; it is often a more stable function than the corresponding probability density. We show here that by doing an exponent spectrum analysis of the complementary Lorenz curve, it is possible to obtain a reflection of the underlying heterogeneity that causes the response function to depart from a power law behavior. We demonstrate this framework first by synthetic data and then by analyzing the avalanche statistics of a two dimensional, Random Field Ising Model (RFIM) at zero temperature. This method can lead to possible use in estimating the microscopic heterogeneity of a system from the analysis of an estimated Lorenz curve, particularly in socio-economic and physical contexts where the full probability distribution function is unavailable.

Entropy doi: 10.3390/e28070798

Authors: Muhammed Yusuf Şener Gerhard Kramer Shlomo Shamai (Shitz) Ronald Böhnke Wen Xu

Dirty paper coding (DPC) is applied to multi-input multi-output (MIMO) broadcast channels with additive Gaussian noise and one message per receiver. The method decomposes each receiver MIMO channel into parallel scalar channels and applies modulo operators, amplitude-shift keying (ASK), and probabilistic shaping. The achievable rate tuples include all points inside the capacity region by choosing truncated Gaussian shaping, large ASK alphabets, and large modulo intervals. Simulations with short polar codes show significant rate and power gains from DPC compared to linear precoding, while maintaining similar encoding and decoding complexities.

Entropy doi: 10.3390/e28070797

Authors: Longlong Li Wenhao Chen Wenhui Li Yan Zhang Jiaxin Liu Runlin Chen

In the early fault stage of rolling bearings, the fault-induced impact signals in vibration data are often extremely weak and easily obscured by strong noise, making effective extraction and analysis challenging. To address this issue, this paper proposes a novel weak fault impact signal feature extraction method combining Ordered Singular Spectrum Decomposition (OSSD) and Multipoint Optimal Minimum Entropy Deconvolution Adjusted (MOMEDA). First, OSSD is employed to decompose the raw vibration signal, progressively extracting signal components across different frequency bands. The optimal signal components are adaptively selected based on mutual information criteria, effectively avoiding mode mixing issues. Subsequently, MOMEDA is applied to enhance the periodic impact features within the fault signal, improving its recognizability. To address the signal length reduction issue inherent in the MOMEDA process, a waveform extension strategy is introduced to compensate for the missing signal, ensuring signal integrity. Simulation and experimental results demonstrate that the proposed method exhibits robust noise resistance and can effectively extract early fault features of rolling bearings under strong noise conditions, validating its accuracy and effectiveness.

Entropy doi: 10.3390/e28070796

Authors: Christoph D. Dahl

Categorisation is often treated as a form of compression: a high-dimensional stimulus space is reduced to a smaller set of behaviourally or cognitively useful classes. However, compression alone does not determine whether a category map is useful. The present manuscript develops an information-theoretic framework for evaluating categorisation in terms of both category complexity and target-relevant information preservation. Here, categorisation is treated as a many-to-one mapping from stimulus instances to category labels, and category entropy quantifies the distribution of the resulting labels. Across a set of synthetic demonstrations, alternative category maps over the same stimulus space are shown to preserve different target variables, including identity, action, nuisance, and hierarchical category structure. The framework is then extended to learned visual representations by analysing layer-derived category maps from a pretrained ResNet-50 network applied to CIFAR-10 images. Clean-only, strong- and mild-nuisance controls test whether layer-derived maps preserve object or nuisance information within nuisance conditions. The results show that category maps can have substantial entropy while preserving information about a variable that is not aligned with the specified target and that the value of a categorisation depends on the target variable to be preserved. The manuscript argues that categorisation should therefore be evaluated not only by compression or separability, but by the information retained about a specified cognitive, behavioural, or computational target.

Entropy doi: 10.3390/e28070795

Authors: Xiaomeng Cui Xiaoqiang Zhang Jiaqi Ji

With the widespread adoption of digital images in network transmission and storage, the demand for image privacy protection keeps rising. We propose a robust scheme combining a four-dimensional variable-order fractional hyperchaotic system (4D-VOFHS) and a chess-game play-inspired dynamic mechanism. Firstly, we construct 4D-VOFHS, to overcome inherent limitations of constant-order systems: unlike constant-order systems that are vulnerable to deep-learning-based parameter identification attacks, this system introduces time-varying orders and high-dimensional coupling to enrich nonlinear dynamics. Secondly, inspired by the dynamic strategic interactions within chess gameplay, we design a synchronous encryption framework with a tightly coupled permutation–diffusion mechanism. This design not only significantly enhances the nonlinear complexity, confusion and diffusion performance of the algorithm, but also enables parallel synchronous processing to improve computational throughput. Finally, we propose a block-based collaborative scrambling strategy with multi-chess-piece rules, wherein traversal rules and scrambling operations are not predefined; instead, they are dynamically updated according to the real-time state evolution of the 4D-VOFHS. Through comprehensive correlation analysis and differential attack tests, the presented encryption framework achieves outstanding performance metrics: an average NPCR of 99.6%, a UACI of 33.4%, and an average information entropy of 7.9993. Overall, these results verify the strong cryptographic robustness and practical applicability of the scheme, highlighting its great potential for deployment in real-world color image encryption systems.

Entropy doi: 10.3390/e28070794

Authors: Yanqiu Zeng Xinyu Wu Shixiao Xiao

This paper investigates statistical inference for the Shannon entropy of the Transmuted Weibull Distribution under progressively Type-II censored samples. The Transmuted Weibull Distribution is obtained by applying the quadratic rank transmutation map to the cumulative distribution function of the two-parameter Weibull distribution, thereby substantially enhancing its modeling flexibility while preserving the analytical tractability of the baseline distribution. Consequently, it provides greater flexibility for modeling lifetime data exhibiting pronounced skewness and complex hazard rate behaviors. First, a closed-form expression for the Shannon entropy of the Transmuted Weibull Distribution is derived. From a frequentist perspective, the maximum likelihood estimators of the model parameters are obtained numerically using the Newton–Raphson algorithm, and the corresponding maximum likelihood estimator of Shannon entropy is derived through the invariance property of maximum likelihood estimation. To quantify estimation uncertainty, asymptotic confidence intervals are constructed using the Delta method together with the observed Fisher information matrix, while Bootstrap confidence intervals are also developed to improve finite-sample inference. From a Bayesian perspective, posterior inference is conducted using a hybrid Gibbs sampling algorithm within the Markov chain Monte Carlo framework. Bayesian point estimators of Shannon entropy are obtained under the squared error loss function, the absolute error loss function, and the 0–1 loss function, corresponding to the posterior mean, posterior median, and posterior mode, respectively. In addition, highest posterior density credible intervals are constructed for the Shannon entropy. The proposed methods are evaluated through an extensive Monte Carlo simulation study under three representative progressively Type-II censoring schemes. Estimation performance is assessed in terms of bias, mean squared error, interval coverage probability, and average interval length. The simulation results demonstrate that the Bayesian estimators consistently outperform the maximum likelihood estimator, particularly for small sample sizes and heavy censoring, while the highest posterior density credible intervals achieve more accurate coverage probabilities and shorter interval lengths. Finally, the proposed inferential procedures are illustrated using a real dataset consisting of remission times from 128 bladder cancer patients, demonstrating their practical applicability and robustness.

Entropy doi: 10.3390/e28070793

Authors: Junhui Huang Yan Guo Xiliang Chen Jianyu Wei Jiawei Yi Xinliang Chen Lifeng Chen

The Target–Attacker–Defender (TAD) pursuit–evasion game is a core challenge in multi-agent cooperative control, yet real-world settings involving dynamic team scaling and strict energy constraints remain largely unaddressed. When scalable shared-parameter policies are adopted to cope with the varying number of agents, severe policy homogeneity emerges, preventing effective division of labor. This paper proposes a Hierarchical Heterogeneous Multi-Agent Proximal Policy Optimization (HH-MAPPO) framework to resolve these challenges. Both levels employ actor–critic networks with Role-Aware Embedding (RAE). In this mechanism, each agent is assigned a unique, learnable role embedding derived from its identity. These embeddings serve as conditioning inputs to the shared policy network, enabling it to generate differentiated behaviors and effectively mitigating policy homogeneity. The upper-level policy determines the number of defenders to deploy and assigns interception targets, while the lower-level policy handles continuous control of each defender and the ground moving target (GMT). This hierarchy resolves dynamic observation spaces via a target-matching mechanism, where each defender’s observation includes only its own state and its assigned attacker’s state, keeping observation dimension constant. Experiments in a 3D TAD simulation with continuous attacker arrivals and energy-constrained defenders show the following: (1) HH-MAPPO achieves superior interception performance compared to baseline methods in both symmetric and asymmetric scenarios; (2) ablation studies confirm RAE increases policy diversity, raising Sequence-Based Action Dissimilarity (SBAD) by 15.5%; and (3) Pareto analysis demonstrates a superior performance–energy trade-off, maintaining about 70% interception rate even under an extreme energy cap (E = 30).

Entropy doi: 10.3390/e28070792

Authors: Leonid M. Martyushev

The traditional paradigm of natural science treats the laws of nature as eternal and immutable. This review examines a powerful alternative tradition that views these laws as historically evolving and constructed entities, tracing this shift from ancient roots to evolutionary epistemology, radical constructivism and physics. Specifically, it provides a chronological analysis of how ideas about the variability of laws developed from ancient Greek philosophy through Enlightenment thinkers to contemporary physicists like Ilya Prigogine and Lee Smolin. We address the resulting methodological crisis—where different branches of science optimize their own laws and isolate from one another—by proposing a strict hierarchical framework. Under this method, invariant basic concepts are strictly separated from flexible models. Crucially, the Entropic Measure of Time (EMT) is presented as the central operational tool. By defining time through entropy production, EMT enables the deductive derivation of physical laws from specific models, restoring a unified, cohesive structure to modern science and offering a robust strategy to counteract the fragmentation of scientific disciplines.

Entropy doi: 10.3390/e28070791

Authors: Florian Neukart Eike Marx Valerii Vinokur

In the original publication [...]

Entropy doi: 10.3390/e28070790

Authors: Irina Georgescu Jani Kinnunen

This paper investigates the stability of high-dimensional financial portfolios using concentration inequalities, information-theoretic measures, optimal transport metrics, and financial network analysis. Asset returns are generated under both multivariate Gaussian and multivariate Student-t distributions. Equal Weight and Regularized Minimum Variance portfolios are evaluated across alternative portfolio dimensions. The results show that increasing portfolio dimension reduces portfolio risk, tail probabilities, and risk estimation errors, indicating stronger concentration and higher stability in high-dimensional settings. Entropy and mutual information measures reveal improved diversification and weaker dependence structures as portfolio size increases. To assess distributional robustness, a novel Talagrand Stability Index (TSI), combining Wasserstein distance and Kullback–Leibler divergence, is introduced. The results show that TSI decreases with portfolio dimension. Heavy-tailed Student-t returns generate weaker concentration effects, stronger dependence structures, and lower distributional stability than Gaussian returns. Mutual information-based financial networks reveal sparse and moderately interconnected dependence structures. To illustrate the practical applicability of the proposed framework, an empirical application based on daily returns of ten large U.S. equities during 2020–2025 is conducted, showing that the Regularized Minimum Variance portfolio achieves a marginally lower TSI than the Equal Weight portfolio. Robustness checks reported further indicate that this advantage is modest and outcome-dependent rather than decisive.

Entropy doi: 10.3390/e28070789

Authors: Flavia Pennini Angelo Plastino

The onset of quantum behavior in gases is traditionally established through a criterion that is external to classical statistical mechanics. One introduces the thermal de Broglie wavelength and compares it with the mean intermolecular separation, concluding that quantum effects become relevant when nλT3∼1. This condition originates in quantum statistical mechanics and is absent from the classical ideal-gas or van der Waals partition functions. In this work, we show that a grand-canonical treatment of the van der Waals fluid naturally generates an interaction-corrected crossover temperature T3(a,b,m,n) determined by the particle mass, density, and van der Waals interaction parameters. While the thermal de Broglie wavelength provides the standard quantum crossover scale, the interaction-induced correction leading to T3 is obtained without invoking the explicit form of the Bose–Einstein or Fermi–Dirac distributions. Instead, T3 follows from a self-consistent condition within the grand-canonical van der Waals description. We demonstrate that, below this temperature, the statistical assumptions underlying the classical theory become self-inconsistent, indicating the breakdown of the classical description and the onset of the quantum-degenerate regime. The resulting temperature scale therefore provides an interaction-corrected boundary of validity of the classical van der Waals description. These findings provide a new perspective on how intermolecular interactions modify the crossover to the quantum-degenerate regime and clarify the limits of applicability of the classical van der Waals theory.

Entropy doi: 10.3390/e28070788

Authors: Longfei Wei Shouli Wang Jing Wang

This paper investigates how quantum entanglement and heterogeneous expectations jointly affect the stability, complexity, and controllability of a Cournot triopoly with isoelastic demand. Based on the Li–Du–Massar quantization scheme, we construct a discrete-time quantum Cournot triopoly in which three firms adopt different updating mechanisms: boundedly rational adjustment, naïve and adaptive expectations. The quantum boundary equilibrium and the unique interior quantum Nash equilibrium are derived explicitly. By linearizing the resulting three-dimensional nonlinear map and applying the Jury criterion, we obtain analytical local stability conditions for the interior equilibrium. The results show that increasing the entanglement level reduces the admissible range of the adjustment speed, thereby shrinking the stability domain and making the market dynamics more prone to bifurcation and chaos. Numerical simulations further reveal a typical transition from stable convergence to flip bifurcation, period-doubling cascades, chaotic attractors, and sensitive dependence on initial conditions. Finally, a control parameter is introduced to rescale the effective adjustment speed of the boundedly rational firm. This mechanism preserves the equilibrium set while restoring convergence to a stable fixed point once the control intensity exceeds a critical threshold. The findings highlight the joint role of entanglement, expectation heterogeneity, and nonlinear demand in shaping complex quantum oligopoly dynamics.

Entropy doi: 10.3390/e28070787

Authors: Ailing Zhong Chengqiang Wang

Complex nonlinear dynamical systems, often associated with high-entropy time series, have been widely employed to describe and predict intricate dynamic phenomena in real-world systems. Motivated by the need to better understand such complex dynamics in network-based epidemic processes, this paper investigates bifurcation dynamics in a fractional-order extension of the classical Susceptible–Latent–Breaking–Out model for computer virus propagation. The proposed framework incorporates two distinct transmission-related time delays and employs Caputo fractional derivatives of incommensurate orders, with the delays associated with infection rate and latent period selected as the primary bifurcation parameters. Due to the combined influence of multiple delays and incommensurate fractional exponents, the resulting system exhibits a complexity that goes beyond most existing models in the literature. By linearizing the model around its endemic equilibrium and analyzing the associated characteristic roots, we characterize how the system’s qualitative behavior depends on the magnitudes of the time delays, and establish explicit sufficient conditions for bifurcation to occur. In particular, the endemic equilibrium remains asymptotically stable as long as each delay stays below a certain critical value; once any delay exceeds its threshold, the system undergoes a Hopf bifurcation, leading to sustained periodic oscillations in virus prevalence. Numerical simulations are provided to support the analytical results, and they show strong agreement between predicted and observed system responses. These findings enhance theoretical insight into bifurcation mechanisms in fractional-order delay models of epidemic dynamics on networks, and may offer useful guidance for designing containment strategies in large-scale interconnected systems.

Entropy doi: 10.3390/e28070786

Authors: Xiao-Chen Chen Guan-Quan Dai Yang Shi Fei-Yun Wu

Aiming at the problem that sparse channel estimation in underwater acoustic communication is susceptible to complex multipath propagation, non-Gaussian impulsive noise, and channel time variations, this paper proposes a bidirectional proportionate recursive maximum correntropy criterion algorithm, referred to as Bi-PRMCC. By introducing a bidirectional filtering structure into the proportionate recursive maximum correntropy criterion (PRMCC) framework, the proposed algorithm jointly exploits the information from forward and backward data sequences, thereby improving the estimation accuracy and block-based channel variation tracking capability for sparse underwater acoustic channels. Meanwhile, the maximum correntropy criterion enhances the robustness of the algorithm against non-Gaussian impulsive noise and outlier error samples, while the proportionate update mechanism improves its identification capability for dominant taps in sparse channels. To verify the effectiveness of the proposed algorithm, short-range sparse underwater acoustic channels and long-range complex multipath underwater acoustic channels are constructed based on the Bellhop ray-tracing model. Simulation experiments are then conducted under three typical non-Gaussian noise environments, namely Cauchy noise, α-stable distribution noise, and Middleton noise. The experimental results show that, compared with recursive least squares (RLS), bidirectional recursive least squares (Bi-RLS), proportionate recursive least squares (PRLS), recursive maximum correntropy criterion (RMCC), and PRMCC, Bi-PRMCC achieves a lower steady-state normalized mean square deviation (NMSD) under different non-Gaussian noise conditions, indicating stronger robustness against impulsive noise. Under different signal-to-noise ratio conditions, the proposed algorithm still maintains superior steady-state estimation performance. In addition, in the channel abrupt-change tracking experiment, Bi-PRMCC can rapidly reconverge after channel variations occur, demonstrating favorable reconvergence capability under abrupt channel variations. The ablation study further verifies the stable performance gain brought by the bidirectional structure to PRMCC. Overall, the proposed Bi-PRMCC algorithm exhibits high estimation accuracy, robustness, and reconvergence capability under complex non-Gaussian noise and abrupt channel variation conditions.

Entropy doi: 10.3390/e28070785

Authors: Yixin Bao Wenxia Chen Chenhao Qian Titi Zhang Zidan Zhou

In the context of global climate change and China’s “Dual Carbon” target, the misallocation of carbon emission reduction responsibilities and low regulatory efficiency urgently require analysis and resolution. Based on China’s 2020 MRIO and carbon emission inventory data, this study integrates multi-regional input–output models and complex network theory to construct an embodied carbon emission (ECE) transfer network at the provincial-industrial level and analyze its structural characteristics. Drawing on complex network control theory, this paper proposes a heuristic node-ranking strategy to identify driver nodes for full controllability of the ECE transfer network and compare its regulatory effect with other topological indicators. The findings reveal: (1) At the provincial level, embodied carbon emissions show a distinct transfer pattern from central provinces to southeast coastal or economically developed regions. Jiangxi, Anhui, Shandong, etc., are net outflow provinces, while Jiangsu, Beijing, Guangdong, etc., are net inflow provinces. (2) At the industrial level, secondary industry is the main net inflow industry, and primary industry is the main net outflow industry. The secondary industries in Guangdong, Henan, etc., have high betweenness centrality, acting as “hub” nodes for carbon transmission. Community detection shows that the largest community in China is centered on the secondary and tertiary industries of Jiangsu, Henan, Guangdong, etc., and the network overall exhibits small-world characteristics. (3) Compared with other control strategies, the designed algorithm achieves the best control effect: it realizes full network controllability with the minimum number of control nodes (26), and the shortest reachable paths from the control node set to non-control nodes, meaning policy signals imposed on control nodes transmit at the fastest speed. (4) Among the control node set, 22 key control nodes are mostly secondary and tertiary industries, located at the center of the transfer network and ranking high in net outflow or inflow, belonging to the core nodes of the ECE transfer network. This study provides a scientific basis and methodological support for clarifying the attribution of carbon transfer responsibilities and formulating differentiated collaborative regulatory policies. This paper establishes a qualitative matching mechanism between network control inputs and carbon tax, emission quotas and industrial regulation to connect controllability theory and practical carbon governance.

Entropy doi: 10.3390/e28070784

Authors: Luigi Rosati Nicola Toschi Andrea Duggento

Recurrent neural networks are studied along two largely parallel tracks: as machine-learning models evaluated by task performance and as computational-neuroscience models of cortical circuits evaluated by dynamical realism. Reservoir computing offers a meeting point, yet the link between dynamical regime and computational performance has not been systematically mapped in biologically constrained spiking architectures. We treat the Brunel balanced excitatory–inhibitory network as a reservoir and characterize separation capacity (kernel quality) and transient memory (corrected linear memory capacity, validated by non-parametric mutual information) across the full phase diagram. The analysis uses a four-state Markov source whose Shannon entropy rate is set in closed form by a single parameter at fixed marginal entropy. Both capabilities increase monotonically with the inhibitory ratio g, remaining jointly highest in the asynchronous irregular regime, with diminishing increments consistent with eventual saturation; the synchronous irregular regime, despite a network timescale three orders of magnitude longer, supports neither. Memory further requires sparse input coupling: dense coupling collapses the driven timescale and erases memory in every regime. Inhibitory balance thus emerges as a unified architectural control parameter, providing a quantitative design criterion for cortical-circuit modeling and reservoir computing applications.

Entropy doi: 10.3390/e28070783

Authors: Wenbin Zhang Zikang Cao Haijian Wu Dewei Guo Yasong Pu

Long-tailed data are very common in industrial scenarios because equipment failures occur with a low probability, resulting in far fewer faulty samples than normal ones. However, when facing long-tailed data distributions, existing deep learning methods suffer from a significant degradation in performance and exhibit high bias. To overcome this limitation, this paper proposes a network that combines balanced adaptive logit-compensated cross-entropy loss with quadratic convolution (BALQNet) to improve diagnostic performance under long-tailed data conditions. The proposed method mainly consists of a balanced adaptive logit-compensated cross-entropy loss (BAL) and a quadratic convolution backbone. By jointly incorporating logit compensation, label smoothing, and class reweighting, BAL enhances the optimization of minority-class samples, thereby improving the classifier’s ability to distinguish different categories without introducing additional architectural complexity. Meanwhile, quadratic convolution further improves the effectiveness of feature representation learning. Finally, experiments are conducted on self-built bearing, gear, and motor datasets. The results show that BALQNet maintains strong diagnostic performance when handling long-tailed data. In addition, the ablation results provide further evidence for the effectiveness of the proposed approach.

Entropy doi: 10.3390/e28070782

Authors: Osei K. Tweneboah Maria C. Mariani

Forecasting stock market returns in emerging economies remains challenging due to market volatility, structural irregularities, and limited data availability. This study investigates whether discrete wavelet transformation can enhance the predictive performance of deep learning models when applied to financial time series from emerging markets. Using daily returns of the Ghana Stock Exchange Composite Index (GSE-CI) spanning 2011 to 2022, we evaluate three widely used deep learning architectures—Multilayer Perceptron (MLP), Long Short-Term Memory (LSTM), and Convolutional Neural Network (CNN)—in both their standard form and with preprocessing based on the Daubechies-4 (db4) discrete wavelet transform. The empirical results indicate that wavelet preprocessing consistently reduced forecasting errors across all three deep learning architectures, highlighting its effectiveness as a multiscale feature extraction and noise reduction technique for financial time series. Among the models considered, the Wavelet-LSTM achieved the lowest forecasting error, while the wavelet-enhanced variants consistently outperformed their corresponding baseline models. These findings suggest that the benefits of wavelet decomposition extend beyond a specific neural network architecture by providing a richer representation of nonlinear temporal dynamics in volatile and data-constrained financial environments. As one of the first studies to systematically evaluate wavelet-augmented deep learning models for stock market forecasting in an African equity market, this work contributes to the growing literature on hybrid forecasting frameworks and provides practical insights for researchers, analysts, and investors interested in forecasting emerging financial markets.

Entropy doi: 10.3390/e28070781

Authors: Ömer Akgüller Mehmet Ali Balcı Ceren Uçmakoğlu Lucian Gaban

We develop a unified information-theoretic framework for the analysis of cancer transcriptomic dysregulation across measurement modalities. Three functionals capture distributional, network-aware, and joint-dependence aspects of expression: the Shannon entropy with a Miller–Madow correction, the signalling entropy rate over the protein interaction graph, and the Gaussian total correlation on a principal-component projection. A closed-form algebraic expression yields a linear-time algorithm for the signalling entropy rate. A Compositional Dilution Lemma decomposes bulk entropy into intrinsic and compositional contributions, and a Cross-Modality Invariance Proposition provides an empirically falsifiable null hypothesis. Validation uses 700,202 single cells and 3942 bulk samples across five cancer types. Pan-cancer tumour elevation is significant at p<10−7, and cross-modality testing on 4230 observations does not reject the interaction null at p>0.5. The invariance conclusion is corroborated by cancer-level paired sign-flip permutation, cancer-block bootstrap, and empirical distribution function tests, and the prognostic Cox regressions satisfy proportional-hazards diagnostics with cross-validation concordance of 0.696±0.018. Immune deconvolution against the LM22 signature validates cell-type-specific predictions, partitioning cancers into myeloid-driven and lymphoid-driven classes. Breast cancer Cox regressions instantiate the predicted orthogonality of distributional and network-aware functionals after immune adjustment.

Entropy doi: 10.3390/e28070780

Authors: Shixiang Sun Xinjiang Wei Lewei Dong Huifeng Zhang Xin Hu

This study addresses the limitations of existing key node identification methods for multiplex networks—defined as multi-layer networks where all layers share the same node set. To mitigate the loss of critical inter-layer information and improve the reliability of influencer detection, we propose a grey correlation model-based algorithm. The method integrates three essential attributes of multiplex networks: the importance of each individual layer, the intra-layer importance of each node, and node centrality derived from compressing the multiplex network into a single weighted layer. Grey relational analysis is then employed to fuse these heterogeneous attributes and compute a final node significance score. The proposed algorithm is validated on both synthetic and real-world networks using the SIR epidemic model. Experimental results demonstrate that our approach achieves higher node ranking accuracy than six comparison algorithms, confirming its effectiveness in identifying influential spreaders in multiplex networks. Unlike conventional algorithms that typically rely on single-dimensional information, our method systematically combines multiple attributes of multiplex networks, thereby effectively preserving cross-layer topological information and enhancing the reliability of key node identification.

Entropy doi: 10.3390/e28070779

Authors: Ana Kerma Araujo Gomes de Sousa Laércio Leal dos Santos Fernando Henrique Antunes de Araujo

Hydrological reservoirs are complex systems in which storage variations integrate climate forcing, catchment response, releases, withdrawals, evaporation, and monitoring procedures. This study presents an information-theoretic characterization of storage dynamics in five semiarid reservoirs in Paraíba, Brazil. The main analytical layer is the complexity–entropy causality plane (CECP), computed from daily storage increments by permutation entropy and Martín-Plastino-Rosso statistical complexity. CECP was estimated for fixed periods and for sliding windows of 120 observations, with sensitivity tests for embedding dimension, delay, and window length. The workflow also benchmarks CECP distance against conventional descriptors, quantifies ties and zero increments, and tests window overlap. As physical context, monotonic elevation-storage curves were reconstructed for 2009–2014, 2015–2019, and 2020–2026, and storage differences at equivalent water levels were quantified by bootstrap confidence intervals. The reservoirs occupied a high-entropy, low-to-moderate-complexity region of the CECP, but their distances from the maximum-entropy/minimum-complexity vertex differed across reservoirs and periods. Sliding windows revealed temporal mobility that was hidden by fixed-period summaries, especially in Engenheiro Arcoverde, Jatobá I, and Mãe d’Água. Rankings remained strongly concordant when window overlap decreased from 94.2% to 0% (Spearman ρ=0.943), although absolute coordinates were sensitive to the treatment of reported plateaus. Elevation-storage shifts provided an independent structural context: negative shifts were compatible with possible useful-capacity reduction, although not uniquely attributable to sedimentation. The results show that CECP descriptors can reveal ordinal organization and regime mobility in reservoir storage increments, while V(H) curves supply the physically interpretable storage-capacity context. The combined evidence prioritizes Engenheiro Arcoverde and Mãe d’Água for bathymetric, curve history, and operational verification. The proposed workflow is therefore an exploratory information-theoretic screening tool for data-limited reservoir monitoring, not a substitute for bathymetric validation.

Entropy doi: 10.3390/e28070777

Authors: Ruth E. Kastner

It is often asserted that quantum states for same-type particles must be symmetrized due to “label redundancy,” i.e., the assumption that the permutations of labels in direct-product states do not reflect any real physical distinction and thus their permutations constitute an “exchange degeneracy”. This assumption is directly challenged by the case of scattering of same-type particles such as electrons, which involves two physically distinct scattering channels effectively corresponding to permutation of the labels. I discuss this counterexample with critical attention to an extant portrayal in the literature that omits pertinent physical content. I further note ways in which the assumption that symmetrization must be universally imposed is not supported by actual calculations of particle interactions, nor by seemingly viable particle states based on preparations and outcomes.

Entropy doi: 10.3390/e28070778

Authors: Zitong Diao Jie Tang Zhaoqi Lei Huicun Yu Jiahao Li Lei Shi Jiahua Wei

Quantum dense coding could be used to transmit two classical bits with one qubit when a maximally entangled state is shared. In realistic channels, entanglement degradation reduces the channel capacity, while bit-flip noise increases decoding errors. To address these issues, we propose a novel probabilistic controlled dense coding protocol that employs the three-qubit repetition code for error correction and an auxiliary qubit for probabilistic decoding. Moreover, this proposed scheme includes a third party to supervise the communication based on a three-qubit entanglement state. The implementation steps of our protocol are presented in detail, and numerical simulations show that it achieves higher average information than dense coding without error correction. The scheme provides a robust solution for quantum communication under noisy conditions and non-maximally entangled state.

Entropy doi: 10.3390/e28070776

Authors: Yu Zhang Zhiqi Chen

This paper focuses on a heteroscedastic partially linear regression model in which the errors are asymptotically almost negatively associated (AANA) random variables with a stochastically dominated and zero mean. Under some suitable conditions, the p-th p>0 mean consistency of least squares estimators and weighted least squares estimators for the unknown parameter is established, and the p-th p>0 mean consistency of the estimators for non-parametric components is also obtained. In addition, the moment convergence rate of the estimators is also investigated. Some results derived in this paper extend and improve the corresponding ones of negatively associated (NA) random errors and independent random errors. Finally, a simulation is carried out to study the numerical performance of the results that we have established.

Entropy doi: 10.3390/e28070775

Authors: Atef Gharbi Ahmad Alshammari Nadhir Ben Halima

Cyber–Physical Systems (CPS), including smart grids and industrial control networks, must maintain secure and stable operations despite increasingly adaptive cyber threats. Existing moving target defense (MTD) approaches often rely on fixed reconfiguration strategies or flat learning architectures that fail to scale and do not explicitly ensure operational resilience under real-time constraints. This study proposes a resilience-oriented hierarchical multi-agent reinforcement learning (MARL) framework for adaptive MTD in CPS environments. The attacker–defender interaction is modeled as a partially observable stochastic game, enabling defenders to learn adaptive strategies with incomplete information. The proposed architecture consists of three layers: a strategic MARL layer that optimizes high-level defense parameters, a distributed k-winner-take-all coordination layer for low-latency defender selection, and a robust execution layer based on sliding-mode control to preserve physical system stability during reconfiguration. By decoupling strategic adaptation from real-time control, the framework improves scalability and supports resource-aware defense through selective agent activation. Extensive simulations with up to 50 defender agents demonstrate that the proposed approach achieves a defense success rate of 92.4%, reduces the response time by 15% compared with the random MTD, and lowers the energy consumption by 34% on average (up to 52% at N = 50) relative to the flat MARL. These results indicate that hierarchical MARL can significantly enhance CPS resilience by enabling adaptive, efficient, and operationally safe defenses against dynamic cyber-attacks. The proposed framework is particularly suitable for edge-enabled CPS environments with strict, real-time, and safety constraints.

Entropy doi: 10.3390/e28070774

Authors: Esteban-Alejandro Durán-Yáñez Mario-Alberto Rodríguez-Díaz Ricardo Mendoza-González Francisco-Javier Luna-Rosas Julio-César Martínez-Romo

Voxel representations provide a simple way to represent three-dimensional objects as binary occupancy signals, but dense voxel grids and direct sparse encodings remain costly at medium and high resolutions. This paper addresses the gap between conventional dense-grid, octree, and point-cloud-codec representations and deterministic contour-first source serialization for exact binary voxel occupancy. We propose a contour-based chain-code serialization that decomposes a voxel grid into two-dimensional slices, extracts foreground components and holes, encodes their contours using F4, 3OT, and F8 variants, and separates contour symbols from positional metadata before applying general-purpose lossless compression. The method is evaluated on 3983 ModelNet40-derived voxelized objects across 40 classes and resolutions N = 8, 16, 32, 64, 128, 256, and 512, using the X-axis for the main evaluation. It is compared against OCC1, BINVOX, breadth-first octree masks, and geometry-only G-PCC. The proposed streams are not competitive at N = 8, where zstd-compressed octree masks achieve the best mean bpv. From N = 16 onward, however, the best proposed stream outperforms the strongest evaluated baseline, with gains increasing from 20.93% at N = 16 to 84.37% at N = 512. The best proposed configuration is zstd + 3OT at N = 8 and N = 16, while zstd + F8 dominates from N = 32 through N = 512. Entropy, ablation, timing, memory, and validation analyses further show that the advantage comes from the interaction between contour-aware source serialization and backend compression, rather than from the backend compressor alone.

Entropy doi: 10.3390/e28070773

Authors: Lanxiang Jiang Xuanya Zhang Qun Chen Xin Wan Fei Yang Gang Yang

Multi-antenna-based covert communication techniques exploit spatial degrees of freedom to improve transmission efficiency under covertness constraints, but this generally comes at the cost of increased hardware complexity and power consumption. To this end, time-modulated arrays (TMA) enable multi-user covert communication with a single radio-frequency (RF) chain, providing a promising solution for low-complexity and energy-efficient covert communication. However, the infinite-order harmonics generated by time modulation spread signal energy over the entire spectrum, allowing the warden to enhance detection capability via cross-band observations, which aggravates signal leakage toward unintended directions. This paper develops a binary hypothesis testing model from the perspective of the warden based on infinite-order harmonic characteristics, to characterize the statistical properties and power distribution of harmonic-induced leakage. Furthermore, since the Kullback–Leibler (KL) divergence is intractable under infinite-order harmonic conditions, a computable upper bound is derived to enable covert constraint analysis. Considering the strong coupling among system parameters, an optimization problem is formulated to maximize the minimum covert transmission rate, and a genetic algorithm (GA) is employed for the joint design of time modulation, power allocation, and spatial phase. Simulation results demonstrate that the proposed scheme effectively suppresses signal leakage and improves covert transmission performance.

Entropy doi: 10.3390/e28070772

Authors: Gian Paolo Beretta Rohit Kishan Ray Michael R. von Spakovsky

A formal development of the HypoEquilibrium (HE) state concept within the Steepest-Entropy-Ascent Quantum Thermodynamics (SEAQT) framework is presented, emphasizing its rigorous mathematical formulation. Using a general decomposition of the Hilbert space, HE states are defined in operator language and the reduced evolution of the associated intensive parameters for the regime where the dissipative dynamics commutes with the Hamiltonian is derived. It is proved that the M-th-order HE family (where M is the number of spectral sectors) constitutes an invariant manifold under the SEAQT equation of motion, ensuring that states initially representing a “mixture of canonicals” maintain this structure throughout their evolution. Furthermore, a formal connection is established between the HE ansatz and the rate-controlled constrained equilibrium (RCCE) method, identifying HE variables as constraint potentials. Finally, the model is extended to Non-Hamiltonian SEAQT (NH-SEAQT) interactions to describe thermodynamically consistent energy and entropy exchanges between subsystems and heat baths. This work provides the formal foundation for reduced-order modeling of far-from-equilibrium relaxation and transport processes, and supports a methodology previously applied across various physical and chemical systems.

Entropy doi: 10.3390/e28070771

Authors: Malte Henkel

The exact time–space correlation function of the 1D Glauber–Ising model, quenched to temperature T=0 and on a semi-open lattice of finite size N, is obtained. This also enables deducing the exact empty-interval probability of the dual 1D coagulation–diffusion process on a periodic finite ring and reproducing the long-time decay of the particle concentration. These results are consistent with the generic expectations of dynamical finite-size scaling theory.

Entropy doi: 10.3390/e28070770

Authors: Pete Herzog Michael Sletten Šarūnas Grigaliūnas Rasa Brūzgienė

This paper proposes a quantum-inspired hypothesis that cybersecurity can be modeled as information persistence: the maintenance of separation between protected and adverse system states under entropy, latency, and control cost. The objective is to provide a time- and energy-aware framework for comparing security architectures without claiming that cybersecurity is literally quantum or that a universal law has been proven. We define a dimensionless Security Persistence Index, P=Δ/(E+L+S), and map controls across three temporal phases—Intent, React, and Resolve—within a 5×3 Control Lattice. The resulting Principle of Energetic Asymmetry predicts that React-dominated architectures should require greater energy, latency, and residual-entropy cost than architectures that shift control weight toward Intent and Resolve. We evaluate this prediction through a simulation of four architectures—Intent-heavy, Balanced, Misaligned, and React-heavy—using 1000 trials per condition. The expected pattern was observed: Intent-heavy achieved the highest simulated persistence, Psim=5.93, vs. 3.45 for React-heavy, and lower normalized energy cost, CPU load, false positives, latency, and residual entropy. These results provide simulation-based internal-consistency evidence only; the framework remains a hypothesis requiring hardware-level measurement, independent replication, and field validation.

Entropy doi: 10.3390/e28070769

Authors: Roxana Irina Iancu Călin Gheorghe Buzea Florin Nedeff Diana Mirilă Valentin Nedeff Mirela Panainte-Lehaduș Claudia Manuela Tomozei Maricel Agop Alina Ștefania Doboș Dragoş Petru Teodor Iancu Lăcrămioara Ochiuz Decebal Vasincu

Complex diseases often involve distributed interactions among biological regions, physiological systems, imaging phenotypes, and clinical variables that are not fully captured by anatomical proximity, isolated biomarkers, or conventional feature-based representations. In oncology, neuroimaging, critical care, and systems medicine, distant or apparently separate biomedical sectors may show strong statistical or functional coupling associated with multimodal imaging signatures, inflammatory responses, metabolic constraints, treatment-induced changes, or shared disease-state organization. In this work, we introduce a proof-of-concept relational graph framework for representing such candidate hidden connectivity in terms of correlation-induced accessibility bridges. The novelty of the framework is that it does not treat biomedical correlation, graph distance, and network connectivity as separate descriptors but explicitly couples non-factorizable inter-sector correlation to localized accessibility compression in an emergent disease-state geometry. The proposed framework represents a biomedical system as a weighted relational graph in which nodes correspond to clinically relevant entities, such as tissue regions, imaging-derived features, biomarker modules, physiological variables, or disease states, while weighted edges encode constraints on functional, statistical, or pathological accessibility. Within this structure, coarse-grained biomedical sectors are defined as organized subsystems, and non-factorizable coupling between sectors is quantified using mutual-information-type measures. Candidate biomedical bridges are then defined operationally as localized, high-gain reductions in effective inter-sector accessibility distance. We introduce explicit coupling rules linking sector-level correlation to bridge-specific accessibility compression, including an effective distance-compression model and an ensemble-based formulation. Numerical proof-of-concept simulations on randomized modular graph ensembles show that increasing correlation strength systematically reduces effective inter-sector distance and increases bridge gain. The strongest compression occurs when correlation modulates a designated bridge architecture, exceeding the effects observed under random non-bridge or generic inter-sector modulation. These simulations are not intended to validate a disease-specific biological mechanism but to test whether the proposed correlation–compression rule produces bridge-specific effects distinguishable from null graph perturbations. The resulting structures should not be interpreted as physical anatomical tunnels or direct causal pathways unless supported by additional biological evidence. Rather, they represent correlation-induced accessibility bridges: localized, high-gain routes in a patient- or disease-specific relational geometry. The framework may therefore provide a theoretical and computational basis for prioritizing candidate hidden connectivity patterns in radiomics, multimodal prognosis, physiological deterioration, recurrence modeling, and systems-level disease networks.

Entropy doi: 10.3390/e28070768

Authors: Kazuo Ishii Bishnu Prasad Gautam Jieling Wu Javaid Saher

We characterize observability in representation learning through Platonic Projection Structures (PPS), an operator-theoretic framework for analyzing representation accessibility under partial observation. Rather than treating observable outputs as direct reflections of latent representations, PPS models observation as a geometry induced by a self-adjoint positive semidefinite operator acting on a latent Hilbert space. A system is represented as a triple (H,Π,O), where H denotes a latent representation space, Π⪰0 is an observation operator, and O(v)=⟨v,Πv⟩ defines an induced scalar observable. The framework characterizes observability through the quotient geometry H/ker(Π), which represents equivalence classes of latent states that are indistinguishable under observation. From this perspective, observable behavior is governed not by latent representations themselves, but by the geometry induced through the observation operator. We show that both quantum measurement and representation inference under linear observation models can be formulated within this common operator-theoretic structure while differing in the algebraic properties of their observation operators. Within this perspective, quantum measurement serves primarily as a mathematically canonical example of projection-mediated observability. The correspondence developed in PPS is therefore structural rather than physical. Within the same framework, representation transfer and knowledge distillation can be interpreted as approximate preservation of observable geometry through the intertwining condition ΦΠT≈ΠSΦ. PPS further reveals a structural limitation of output-based interpretability: latent components contained in ker(Π) are fundamentally inaccessible from observables generated through the induced observation process. Accordingly, attribution and explanation methods inherit intrinsic constraints imposed by the observation geometry itself. We provide controlled empirical validations demonstrating kernel-invariant observability, projection-induced attribution gaps, and rank-controlled observable geometry in latent representation spaces. Overall, PPS provides a mathematically explicit characterization of observability through operator-induced quotient geometry, offering a unified perspective on representation accessibility, interpretability, and representation transfer.

Entropy doi: 10.3390/e28070766

Authors: Yun Jeong Kang Sung Kyu Seo Kyungsik Kim

In this paper, we apply the double Fourier transform method to the two-dimensional Vlasov equations for a charged particle subjected to white noise, exponentially correlated Gaussian forces, trap forces and thermal and active noises in a magnetic field. By deriving the corresponding Fokker–Planck equation, analytical solutions for the joint probability density are obtained in different time domains. The mean squared displacement and velocity of a charged particle driven by white noise exhibits a super-diffusive behavior, scaling as ~t2 in the short-time regime, while it grows linearly with time (~t) in the long-time regime, in agreement with numerical simulations of the mean squared displacement. When thermal noise is included together with harmonic trap and viscous forces, the characteristic time scale increases as ~t2h+1 in the corresponding time domains, whereas the mean squared velocity scales as ~t2h+3. The moments of the joint probability density under thermal noise scale as ~t2h+5. Furthermore, when the persistent Hurst exponent h→1/2, the entropy of the joint probability density associated with thermal noise coincides with that obtained for active noise in both the short-time (t≪τ) and long-time (t≫τ) limits.

Entropy doi: 10.3390/e28070767

Authors: Marius Vila Qiaohong Hao Miquel Feixas Micaela Y. Martin Mateu Sbert

The gaze information channel paradigm models fixation sequences as a first-order Markov chain and quantifies gaze behaviour through Shannon entropy and mutual information (MI), where I(X;Y) measures the reduction in uncertainty about the next fixation state given the current one. This paper extends the framework by introducing two new channels: the saccade amplitude channel, which discretises saccade angular distance into three categories (short, medium, long) with a four-category variant also analysed, and the pupil diameter channel, which discretises fixation-period pupil size into three categories. Both are applied to 10 observers viewing 12 Van Gogh paintings. The amplitude channel shows that observer-driven variation exceeds stimulus-driven variation. The pupil channel yields the highest I(X;Y) among the two new channels (0.489±0.209 bits per participant), consistent with the slow dynamics of pupil responses. Goodness-of-fit tests confirm significantly non-random sequential structure in both channels (p<0.01) for all pooled matrices. A simultaneous cross-channel association analysis across all five channels finds that 19 of 20 pairwise Spearman correlations are non-significant; the single nominally significant result (pupil–duration, ρ=+0.697, p=0.025) does not survive Bonferroni correction and is not robust to outlier removal. Two theoretical observations are presented: an upper bound on conditional entropy in terms of transition persistence (Proposition 1), and a refinement monotonicity result showing that finer discretisation cannot decrease channel MI (Remark 2). An exploratory comparison with five computational aesthetics measures finds a nominally significant negative correlation between pupil I(X;Y) and Bense’s palette redundancy (ρ=−0.692, p=0.013, uncorrected), suggesting that diverse colour palettes are associated with stronger sequential pupil dynamics; permutation entropy and statistical complexity show no association with any channel.

Entropy doi: 10.3390/e28070765

Authors: Xiangping Huang Thi-Kien Dao Trong-The Nguyen

The rapid growth of heterogeneous IoT devices in 5G environments has created stringent requirements for low-latency edge-based protocol processing. Existing static parsing frameworks lack adaptability to dynamic multi-protocol traffic, resulting in increased processing delays and quality-of-service (QoS) violations under bursty workloads. This paper presents HeteroEdge, a latency-aware adaptive protocol parsing framework for 5G Multi-access Edge Computing (MEC) environments. HeteroEdge integrates four tightly coupled components: (i) a lightweight machine-learning-based Heterogeneous Protocol Parsing Layer (HPPL) built on gradient-boosted decision trees (XGBoost); (ii) a Network Digital Twin (NDT) that maintains a compressed and continuously updated representation of IoT endpoint states; (iii) a Real-Time Inference Engine (RTIE) that dynamically reallocates parsing resources at 50 ms intervals; and (iv) a What-If Simulation (WIS) module that proactively evaluates resource-allocation strategies under hypothetical traffic scenarios. Experimental evaluation on a physical 5G MEC testbed comprising four Intel Xeon Silver 4316 edge nodes and 2000 emulated IoT endpoints spanning twelve protocol classes demonstrates the effectiveness of the proposed framework. HeteroEdge reduces median edge parsing latency (including parsing, classification, and queuing delays, but excluding the 5G radio component) by up to 44.7% compared with static MEC baselines, achieves a macro-averaged protocol classification accuracy of 97.8%, and sustains sub-7 ms edge parsing latency at a line-rate NIC injection throughput of 18 Gbps. Furthermore, latency spikes under bursty traffic are reduced by 39% at the 95th percentile, while SLA violation rates decrease by a factor of 3.9 relative to static resource allocation. These results demonstrate that HeteroEdge provides an effective and scalable solution for latency-critical IoT applications, including smart manufacturing, connected vehicles, and urban sensing.

Entropy doi: 10.3390/e28070764

Authors: Feng Liu Zheng Wang

We propose a mathematical framework for domain generalization in medical image segmentation built on dual-tree complex wavelet transform (DTCWT) and variational information theory. The core premise is that, under adequate spatial normalization and acquisition-style shifts, DTCWT phase components are more closely associated with anatomical structure, whereas amplitude components are more sensitive to domain-specific intensity and style variations. We formulate this as a local phase–magnitude complementarity premise and construct an information bottleneck that operates on structured subband representations. The framework provides several key theoretical results under explicit structural assumptions: an information bound showing when DTCWT amplitude subbands better isolate domain-related information than global Fourier representations; a variational information bottleneck encoder that compresses domain-specific amplitude information into low-dimensional latent codes; a triple constraint mechanism (domain supervision, KL compression, and orthogonality) that controls domain–task information leakage; and a predictive feature modulation scheme with O(1) spatial complexity. We further analyze test-time adaptation via calibrated uncertainty, deriving a sufficient condition under which a two-pass inference strategy reduces the expected generalization gap. Finally, we include illustrative public-dataset checks on FeTS 2022 and BraTS 2023 to test the central phase–amplitude premise and the feasibility of DTCWT-front-end segmentation. All theorems are stated with their assumptions and verifiable conditions, offering a physically motivated approach to domain generalization in medical imaging.

Entropy doi: 10.3390/e28070763

Authors: Qiuming Wang Sheng Zhang Bing Wu Jiangnan Zhou Chennan Wu Yirong Zeng Ka Sun Chang Liu

Attributed graph clustering partitions nodes in an unsupervised manner by leveraging graph topology and node attributes. Existing deep methods face challenges including local structural bias, high noise in unsupervised graph editing, and insufficient discriminative ability for hard samples. To address these issues, we propose a deep graph clustering framework based on confidence-guided graph enhancement and dual-negative sample contrastive learning (CGEN). CGEN constructs a local–global dual-view representation learning module to fuse local neighborhood attributes with high-order global topological information. It then utilizes a confidence-guided conservative graph editing mechanism that integrates multiple constraints, specifically feature similarity, intra-cluster consistency, multi-view consistency, and pairwise node confidence, using a progressive update strategy for stable structural optimization. Furthermore, a dual-negative sample contrastive learning strategy dynamically adjusts the weights of attribute-confused and inter-cluster-confused negative samples to enhance discriminative ability near adjacent cluster boundaries. Extensive experiments on four benchmark datasets demonstrate that CGEN achieves highly competitive performance, outperforming the majority of state-of-the-art methods across core clustering metrics, thereby validating its effectiveness in addressing local structural bias, graph editing noise, and hard sample discriminative limitations.

Entropy doi: 10.3390/e28070762

Authors: Wilson Alexander Rojas Castillo Alexander Zamora Velandia Luis Fernando Quijano Wilchez Yaneth Beltrán Peña

We develop a geometric thermodynamic framework for the analysis of sectoral economic dynamics grounded in statistical physics principles. By constructing a Legendre-invariant thermodynamic metric within the formalism of geometrothermodynamics (GTD), we establish a minimal effective structure consistent with extensivity and entropy-based representations of macroscopic economic systems. The resulting thermodynamic curvature provides a coordinate-independent measure of structural interactions and equilibrium stability across economic sectors. Applying this framework to satellite account data, we find that the thermodynamic curvature of the equilibrium manifold remains finite and regular across the empirically relevant range, with no curvature singularity in the period studied. In particular, the 2020 contraction—the most pronounced macroeconomic disruption in the sample—is not reflected as a curvature singularity in the equilibrium geometry. We read this regularity as a diagnostic of structural stability: the sectoral system absorbs such disruptions without an abrupt reorganisation of its equilibrium geometry. The geometric invariants thus capture stability properties not directly accessible through standard entropic indicators alone, offering a complementary statistical description of economic dynamics. Our results demonstrate that thermodynamic geometry furnishes a consistent bridge between entropy-based macroeconomic modelling and coordinate-invariant measures of equilibrium stability, extending the applicability of geometric methods in statistical physics to complex economic systems.

Entropy doi: 10.3390/e28070761

Authors: Antonio J. Ibáñez-Molina Sergio Iglesias-Parro M. Carmen Gálvez-Garzón María Felipa Soriano

Cognitive neuroscience has addressed the understanding of human brain processes through numerous techniques and psychological paradigms. In general, different types of tasks have been used depending on the specific cognitive operation under study. Since these tasks are usually designed to register responses at the single-trial level, the most common methodological approach to electroencephalography (EEG) is to obtain event-related potentials (ERPs). Crucially, the linear analysis methods associated with ERPs often overlook the intrinsic non-linear and multiscale dynamics of brain activity. Hence, to better characterize brain activity, there is increasing interest in the study of the non-linearity and complexity of EEGs. Given that experiments relating cognitive processing and EEG complexity are still scarce, this work is a narrative review of studies in which non-clinical cognitive processing, such as memory, perception, or attention, is addressed using complexity measures. Here, we focus on EEG metrics derived from the concepts of fractality, information, and randomness across different temporal and spatial scales. We discuss how these measures complement more classical analyses, try to integrate the findings using a predictability–regularity framework, and finally, we point out possible future directions with which to advance current knowledge about the relationship between cognition and EEG complexity.

Entropy doi: 10.3390/e28070760

Authors: Javiera Olave Eugenio E. Vogel Pablo Díaz Denisse Pastén Gonzalo Saravia

Three entropic functions are used to characterize the seismic activity prior and after the major Mw8.8 earthquake of Maule (Chile) dated on 27 February 2010. Shannon entropy, mutability, and Tsallis entropy are calculated globally on the seisms extracted from the catalog of the National Center of Seismology (Chile). Calculations are done both globally on the whole data and also dynamically on windows of different number of seisms after filtering data with a Gutenberg–Richter analysis. The data are time series based on two observables: seism magnitudes and inter-event intervals. It is found that the two entropies and mutability give similar descriptions on the different regimes present between 2005 and 2022. However, mutability can be calculated in a direct and straightforward way. It is also found that the results for inter-event intervals produce more contrast between periods than the corresponding ones for magnitudes. The region spanning six degrees in latitude is split in five overlapping subzones of two degrees each. This allows us to find the way the rupture takes place: from south to north, for about 400 km.

Entropy doi: 10.3390/e28070759

Authors: Ewa Roszkowska Tomasz Wachowicz

This study applies the Belief Structure TOPSIS (B-TOPSIS) method to analyse cross-country attitudes toward AI-driven workplace practices across the EU27. The proposed approach preserves the full distribution of survey responses, explicitly incorporates uncertainty, and evaluates alternatives based on their distance from ideal and anti-ideal belief structures. Using data from Special Eurobarometer 554, we construct individual B-TOPSIS indexes for eight AI-related workplace applications and an aggregated B-TOPSIS index capturing overall acceptance. The results reveal systematic cross-country differentiation. Activities such as gathering applicant information, allocating work, and processing employee data generally receive moderate acceptance. Safety-focused applications are widely supported, whereas ethically sensitive practices, such as employee monitoring and automated dismissal, face low acceptance. Additionally, sensitivity analysis based on Monte Carlo simulation and stochastic dominance demonstrates that the obtained rankings remain highly stable under alternative assumptions regarding utility functions, confirming the robustness of the proposed framework. A comparison with rankings derived from total positive responses, commonly used in EU reports, shows that although the two approaches are strongly correlated, they are not interchangeable. By retaining the complete response structure, the proposed method captures differences in response intensity that are obscured by conventional summary measures. The findings highlight the multidimensional and conditional nature of workplace AI acceptance in the EU and demonstrate the value of belief-structure-based approach for analysing survey data.

Entropy doi: 10.3390/e28070758

Authors: Michel Broniatowski Justin Moutsouka

Adequacy for estimation between an inferential method and a model can be defined through two main requirements: firstly the inferential tool should define a well posed problem when applied to the model; secondly the resulting statistical procedure should produce consistent estimators. Conditions which entail these analytical and statistical issues are considered in the context when divergence based inference is applied for smooth semiparametric models under moment restrictions. A discussion is also held on the choice of the divergence, extending the classical parametric inference to the estimation of both parameters of interest and of nuisance.Classical arguments in favor of the omnibus choice of the L2 and Kullback Leibler divergences are discussed and motivation for the class of power divergences is presented in the context of the present semi parametric smooth models. A short simulation study illustrates the method.

Entropy doi: 10.3390/e28070757

Authors: Yukio-Pegio Gunji Andrei Khrennikov

We introduce a lattice-theoretic framework for composite information systems in which tensor-like composition and entanglement are defined without presupposing Hilbert spaces or quantum states. Starting from approximation operators induced by indiscernibility relations, we construct composite systems via interaction-dependent closure operators and characterize their fixed-point lattices. Entanglement is defined structurally as the impossibility of generating a fixed point of the composite system from local fixed points alone. This notion does not rely on non-distributive logic a priori and remains meaningful even when local lattices are Boolean. Non-distributive and orthomodular structures arise only under additional conditions and are treated as emergent properties rather than assumptions. The proposed framework generalizes the concept of entanglement as a property of composition and interaction, providing a unified information-theoretic perspective on non-separability beyond standard quantum-mechanical formalisms. By mapping quantum states to correlation patterns via row-set tensor products, we demonstrate that standard quantum entanglement can be understood as a stabilized structural constraint. In this context, maximally entangled states, such as Bell states, correspond to diagonal constraint sets that are non-generable from local components, confirming that the structural core of entanglement exists independently of linear or probabilistic interpretations. Beyond quantum mechanics, the framework admits a natural interpretation in terms of relational databases, where entanglement corresponds to irreducible global relations stabilized by interaction-induced fixed points.

Entropy doi: 10.3390/e28070756

Authors: Jiawei Yu Lu Wang Xinyan Sun

Forecasting multi-asset volatility spillovers and turning the forecasts into risk-aware portfolios requires methods that uncover directional information flow between assets, compress the state into a minimal sufficient representation, deliver calibrated uncertainty, and respect explicit tail-risk limits. We propose TDV (Transfer-entropy, Dynamic-graph-attention, Variational-information-bottleneck), an information-theoretic artificial intelligence framework that couples a time-varying transfer entropy network with a graph attention encoder regularised by a variational information bottleneck, and demonstrates the practical value of the calibrated predictive distribution through a downstream entropy-regulated, CVaR-constrained portfolio application. We establish three theoretical results: L2 consistency of the k-nearest-neighbour transfer entropy estimator on α-mixing returns with rate OP(n−2/(2+d)), a PAC–Bayes generalisation bound of order O((I(X;Z)+log(1/δ))/n) for the bottleneck-encoded forecaster, and asymptotic CVaR feasibility of the plug-in allocation. In simulations across sparse Granger networks, contagion DCC–GARCH ensembles, and regime-switching factor models, the framework cuts spillover forecasting errors by 24 to 42 percent against LSTM, vanilla GAT, and Transformer baselines, and it recovers 1.6 additional nats of mutual information with the realised connectedness matrix. On a 32-asset global panel covering 2014 to 2025, the model delivers an out-of-sample R2 of 0.331, an annualised Sharpe ratio of 1.46 against 0.83 for an equally weighted benchmark, a maximum drawdown of 7.8 percent, and 95 percent CVaR reductions of 28 to 36 percent across sub-periods relative to a shrinkage minimum-variance baseline.

Entropy doi: 10.3390/e28070755

Authors: Anfal A. Alqefari

This paper introduces an entropy-based scale-free L-statistic for exponentiality testing in lifetime data under decreasing mean residual life (DMRL) aging. The proposed method is derived from a mean-residual-life characterization of the DMRL class through a cumulative residual entropy-type departure functional that quantifies deviations from the constant mean residual life property of the exponential distribution. An L-functional representation of this departure measure is established, leading to an order-statistic-based nonparametric test statistic. To remove the effect of the unknown exponential scale parameter, the statistic is normalized by the sample mean, yielding a scale-invariant testing procedure under the null hypothesis. The exact finite-sample null distribution is obtained using normalized spacings, and asymptotic normality is established through standard L-statistic theory. Monte Carlo simulations under linear failure rate, gamma, and Weibull DMRL alternatives show that the proposed test maintains the nominal significance level and provides competitive power relative to existing DMRL procedures. Real-data applications further illustrate the usefulness of the method as an entropy-based statistical tool for detecting DMRL aging patterns in reliability and lifetime-data analysis.

Entropy doi: 10.3390/e28070754

Authors: Tomas Veloz Dirk Bruin Cedric De Coning

Evidence suggests that protracted conflicts persist because several forms of socio-political organization run simultaneously on the same population, resources, and territory. Reading Service’s typology of bands, tribes, chiefdoms, and states not as evolutionary stages but as coexisting and superposed social organizations, we model conflict as a reaction network where each social form is a self-maintaining set of stocks—a chemical organization—and conflicts arise where competing productive logics between organizations generate stocks with negative connotation, such as grievances and displacement. Taking the Lake Chad Basin as inspiration, we build a ladder of progressively richer models arriving a mixed chiefdom–state configuration compatible with current views on the conflict. As the model complexifies, kinetic approaches become uninformative; we therefore develop complementary stoichiometric methods that are parameter-free and thus are far easier to measure and compute. These diagnostics reveal a structural bias toward conflict: transitions into conflict regimes are systematically richer than transitions out. We show how a dual chiefdom–state form acts as a conflict attractor within a closed conflict–peace loop that transits among documented different forms of organization. Conflict management then becomes the identification of the mechanisms that redirect rather than change the state of a self-sustaining organization—here, elite-surplus redistribution—and of the timescales at which such redirection is observable, turning intervention design into a structural rather than a parameter-tuning problem.

Entropy doi: 10.3390/e28070753

Authors: Han Chen Qingye Huang Yingjie Su Lezhu Chen Baoyi Liao Linqing Huang Changwen Chen

To address the limitations of existing chaotic systems such as complex structure and potential chaotic degradation, this paper proposes a novel four-dimensional discrete chaotic system (4D-DCS) and an image encryption algorithm based on it. The 4D-DCS is constructed by integrating a feedback controller and modulo operation into a linear discrete-time system, featuring a simple structure without the need for intricate matrix reconstruction or memristor circuits. Mathematical analysis confirms its chaos in the sense of Li–Yorke and numerical simulations including Lyapunov exponent (LE) analysis, 0–1 test, and NIST SP 800-22 test demonstrate its hyperchaotic characteristics and excellent pseudorandomness. Based on the 4D-DCS, the proposed encryption algorithm employs SHA-256 to generate initial states for key uniqueness, combines row–column permutation to disrupt pixel correlation, and adopts a reversible neural network for diffusion to enhance confusion capability. Comprehensive security analysis shows that the algorithm achieves an NPCR of ∼99.61% and a UACI of ∼33.46%, a key space of 2216, information entropy close to 8, and correlation coefficients of encrypted images near 0. It also exhibits strong robustness against differential, cropping, noise, and chosen-plaintext attacks. Comparative analysis with state-of-the-art algorithms validates the 4D-DCS’s advantages in structural simplicity and stability, and the encryption algorithm’s superiority in security and practicality, making it suitable for security-critical applications such as image encryption.

Entropy doi: 10.3390/e28070752

Authors: Yanhang Chen Gaoyang Du Chenglong Yang Shuyu Dai Bo Cui

We propose a theoretical framework based on an affine-eigenstate transformation for analyzing ultracold atom–molecule conversion dynamics with particle loss. The transformation maps the mean-field dynamics to an effective two-mode representation in which fixed points, Bloch-sphere trajectories, and linear stability can be examined in a common set of variables. We give the derivation of the transformed Hamiltonian and specify the invertibility and conjugate-condition requirements under which the mapping is used. Within this representation, we distinguish ordinary, pseudo, and strict self-trapping regimes. The strict regime is associated with the balanced condition S=0 in the transformed variables; in the corresponding linearized dissipative flow, the leading attractor/repeller bifurcation term controlled by SΓ− vanishes, explaining the observed robustness against atom- and molecule-loss imbalance. We also introduce von Neumann and linear-entropy diagnostics for future mixed-state or ensemble descriptions in the transformed two-level representation, and we provide an inverse reconstruction procedure for preparing initial states that realize strict self-trapping. Finally, we discuss the limits of the mean-field and Markovian approximations and outline how finite-particle simulations and phase-modulated control protocols could connect this mechanism to decoherence-resilient quantum simulations and information-processing architectures.

Entropy doi: 10.3390/e28070751

Authors: Jiaxin Zhu Luyan Wang Qiubao Wang

This study proposes a stochastic time-delay pheromone feedback model to describe ant foraging dynamics, and investigates how response delays and environmental noise jointly induce stochastic oscillations and reorganize the system’s probabilistic structure. By employing near-Hopf center-mode projection and stochastic averaging, we derive the first-order stochastic amplitude equation and analyze the stochastic dynamical properties near the deterministic delay-induced Hopf bifurcation. Subsequently, normalized Shannon entropy and Jensen–Shannon divergence, computed relative to a pre-Hopf stochastic stationary reference distribution, are used to quantify uncertainty expansion and distributional reorganization in the stationary amplitude distribution and reconstructed state-variable distributions. The analytical results are supported by numerical simulations, which indicate that response delay primarily determines the transition from stable foraging to oscillatory behavior, while noise intensity mainly affects the dispersion and uncertainty of the amplitude distribution. Information-theoretic metrics further reveal noise-induced uncertainty growth and delay-induced probabilistic restructuring. This study elucidates the stability regulation mechanisms of ant foraging systems under stochastic conditions from a combined dynamical and information-theoretic perspective, and provides a theoretical reference for the design of delayed feedback in swarm intelligence systems.

Entropy doi: 10.3390/e28070750

Authors: Xiaoliang Feng Jiawei Zhang

This paper investigates state estimation for strongly nonlinear systems with unknown inputs under non-Gaussian heavy-tailed impulsive noise. Conventional recursive three-step filters (RTSF) based on the minimum-variance criterion are sensitive to outliers, while a single local linearization is often inadequate for strongly nonlinear dynamics. To overcome these limitations, a multi-model minimum error entropy recursive three-step filter (MMMEERTSF) is proposed. The minimum error entropy criterion is embedded into the RTSF framework to enhance robustness against abnormal disturbances, and iterative reweighted solutions are developed for unknown-input estimation and state correction by combining residual whitening with entropy-based optimization. Meanwhile, multiple local linear submodels are constructed to approximate the nonlinear system, and compatibility-based posterior fusion is employed to obtain the final estimate. The proposed method shows improved robustness and competitive estimation accuracy under non-Gaussian mixture and impulsive noise, especially in the nonlinear multi-model case.

Entropy doi: 10.3390/e28070749

Authors: Norihiro Maruyama Michael Crosscombe Shigeto Dobata Takashi Ikegami

We investigate functional behavioural differentiation in genetically homogeneous animal collectives using the ϵ-machine and ϵ-transducer frameworks from symbolic dynamics. Long-term tracking of unmarked individuals in colonies of the clonally reproducing ant Pristomyrmex punctatus reveals two distinct movement modes—clustering within the group and solitary exploration outside it. Reconstructed individual ϵ-transducers expose a sharp asymmetry in computational structure between these modes: solitary explorers are described by a deterministic machine, whereas clustering ants require stochastic machines to capture their complex patterns of micro-movement. A population-level (universal) ϵ-transducer, inferred from pooled data, captures the shared behavioural repertoire across all individuals. Individual differences are parsimoniously explained as biased and partial traversals of a common state space rather than as distinct generative programs. We compare three predictive models: the ϵ-machine, which relies solely on an ant’s own output history; a memoryful ϵ-transducer, which additionally conditions on changes in the local neighbour count as social input; and a memoryless ϵ-transducer, which uses this social input alone. The memoryful transducer matches the ϵ-machine in prediction accuracy despite requiring ten times as many states, while the memoryless transducer performs substantially worse. This shows that an ant’s own behavioural history is the essential predictor of its future movement at the temporal resolution examined here. We argue, however, that this predictive redundancy does not entail the causal irrelevance of social input: the behavioural history itself accumulates the trace of past social encounters so that any role differentiation established through prior interactions is already inscribed in the output sequence that the ϵ-machine reads, and mode transitions—the moments at which social input most plausibly exerts causal influence—are rare events that contribute negligibly to aggregate one-step accuracy. Agent-based simulations driven by the universal ϵ-transducer reproduce basic motion statistics and transient aggregations but fail to generate the stable macroscopic clusters observed experimentally, pointing to the role of additional mechanisms such as longer-term memory or stigmergic coupling. Nevertheless, ants do respond to their social environment: an explorer encountering an increase in neighbours is absorbed into the cluster and ceases directed movement. Together, our results suggest a two-level organisation: within each behavioural mode, individual dynamics are self-sufficient for one-step prediction, while transitions between modes are environmentally triggered and represent switches between fundamentally different classes of dynamical organisation.

Entropy doi: 10.3390/e28070748

Authors: Juexiao Chen Yinlin He Lei Shi Yihao Tao Donghuan Liu

Due to the stochastic and fluctuating nature of output from roadway high-entropy energy units, traditional evaluation metrics based on fault statistics struggle to comprehensively assess their availability. To address this, this paper proposes a novel unit availability assessment method that integrates output capacity and behavioral orderliness, defining availability as the mathematical product of an output capacity factor and a behavioral orderliness factor. By overcoming the intrinsic flaws of conventional binary status judgments, this approach yields intuitive and logical evaluation results upon case study validation. The proposed model successfully distinguishes the comprehensive performance of equipment under diverse output conditions, thereby offering a fresh perspective for the refined evaluation of roadway high-entropy energy units.

Entropy doi: 10.3390/e28070747

Authors: Jidong Li Maolin Cai Weiqing Xu Guanwei Jia

To enhance the compression efficiency and refrigerant flow capacity for low-temperature refrigeration applications, the vapor-injection strategy is innovatively synthesized with two-stage cascade refrigeration systems. Two cascade vapor-injection configurations with subcoolers and flash tanks (CSVIRS and CFVIRS) are compared with the conventional cascade refrigeration system (CCRS) through integrated thermodynamic simulations. The impacts of crucial temperature and injection parameters are comprehensively analyzed through energy and exergy methods, while the performance comparisons of various refrigerant combinations are also conducted. The coefficient of performance (COP) of the CFVIRS exceeds that of the CCRS and CSVIRS by 33.84% and 2.10% under the default condition. The cascade vapor-injection configurations exhibit a performance advantage at higher condensation temperature and lower evaporation temperature of the low-temperature cycle (LTC). The evaporation temperature of the high-temperature cycle (HTC) and injection pressures are examined with optimum solutions. Decreasing the entrainment ratio of the HTC and increasing the entrainment ratio of the LTC within appropriate ranges are beneficial for the refrigeration performance. R1270-R170 demonstrates superior energy and exergy performance, whereas R143a-R23 shows the highest improvement ratio among the compared refrigerants. The implementation of cascade vapor injection substantially reduces exergy destruction in the compression and expansion devices, while the exergy characteristics of various refrigerant pairs are extensively investigated.

Entropy doi: 10.3390/e28070746

Authors: Sadullah Çelik Cemile Zehra Köroğlu Muhammet Ali Köroğlu

Multidimensional poverty is a socioeconomic problem that results from nonlinear interdependencies of various socioeconomic indicators such as educational, health, and living standards indices. This paper considers an XAI-based approach for studying the structural topology and dependency structures of multidimensional poverty systems. It relies on a combination of machine learning approaches, network science, spectral graph theory, and entropy-based complexity measures for revealing the systemic interdependencies between different poverty indicators. The structure of interdependencies between socioeconomic indices associated with education is represented by the multilayer perceptron (MLP). The interpretability of the model is provided via the computation of SHAP values by means of the KernelSHAP method. Higher-order interactions between variables are revealed via the construction and analysis of SHAP-based interaction networks. The proposed methodology is then employed on the GEMPI 2025 dataset consisting of 109 countries. The results show significant consistency in the structural map (R2 = 0.9890) in combination with stable internal consistency in cross-validation (R2 = 0.9864, SD = 0.0074). The SHAP analysis shows that standards of living and health have a high influence on the structural mapping of education, while the contribution of income-related features is lower compared to other features. Entropy analysis points toward partially fragmented dependency networks with a moderate concentration of explanatory influences (H = 1.705). The proposed framework can be used to characterize structural dependencies, identify influencing system components, and map informational processes in multidimensional poverty systems by combining the methodology of explainable artificial intelligence with entropy–spectral network analysis.

Entropy doi: 10.3390/e28070745

Authors: Lang Yang Jichong Lei

Accurate break size assessment is critical for the safety response of small modular reactors (SMRs) during loss-of-coolant accidents (LOCAs). Traditional methods struggle with the rapid transient features, strong spatiotemporal coupling, and complex uncertainty characteristics of SMR-LOCA, leading to low accuracy and poor stability. To address these issues, this study proposes an entropy-driven intelligent diagnosis approach based on a CNN-LSTM-Attention hybrid model. The framework adopts information entropy for data uncertainty quantification, adaptive weighting, and loss constraint, so as to realize high-precision break size assessment. A time-series dataset covering break sizes from 0.05 to 10 cm2 was constructed using the PCTRAN/SMART platform. The CNN module extracts spatial coupling features of multi-sensor parameters, the LSTM module captures long-term temporal dependencies, and the attention mechanism dynamically weights key information to enhance feature representation under high uncertainty. Experimental results show that the model achieves a mean absolute error (MAE) of 0.096311, reducing errors by over 64.4% compared with baseline models; more than 90% of prediction errors are within ±5%, and the correlation coefficient reaches 0.994902. Based on the well-validated PCTRAN/SMART simulation platform, the proposed entropy-informed spatiotemporal learning framework provides a technical solution for intelligent LOCA diagnosis, uncertainty quantification, and safety assessment of SMRs.

Entropy doi: 10.3390/e28070744

Authors: Zican Jiang Yongjun Li Kai Zhang Jianguo Liu

UAV-assisted communication is widely regarded as a key component of next-generation Space-Air-Ground Integrated Networks (SAGINs), where integrated sensing and communication (ISAC) further drives the demand for accurate and reliable channel modeling. Terahertz (THz) communications are particularly attractive for UAV platforms, offering ultra-high data rates and physically secure transmission. However, the physical heterogeneity between reflection and scattering mechanisms in THz UAV channels poses significant modeling challenges, as conventional unified approaches tend to introduce energy distribution distortion and non-stationary prediction errors. To address this, we propose a 3D non-stationary geometry-based stochastic model (GBSM) based on an ellipse-sphere hierarchical geometric framework, where reflection paths are confined to ground-plane ellipses and scattering paths are distributed over spatial spheres. The model accounts for atmospheric molecular absorption, multipath fading, and non-stationarity induced by random 3D UAV trajectories. A cluster birth-death mechanism is introduced to capture the time-varying evolution of scattering clusters. Key statistical properties, including the temporal auto-correlation function (T-ACF), spatial cross-correlation function (S-CCF), and Doppler power spectral density (DPSD), are derived and analyzed. Simulation results agree well with theoretical derivations, validating the proposed model and providing practical guidance for THz UAV-ISAC system design.

Entropy doi: 10.3390/e28070743

Authors: Rafik Aliev Oleg Huseynov Aziz Nuriyev

The concept of the Z-number was introduced to formalize partially reliable information. A Z-number represents linguistic evaluations of a random variable’s value and the associated reliability degree. The latter is defined as a fuzzy restriction on the value of a probability measure since the actual probability distribution is unknown. Lotfi Zadeh formalized an extension principle for computation with Z-numbers based on fuzzy and probabilistic arithmetic and noted that the problem of computing with Z-numbers is easy to formulate but difficult to solve. Since then, a series of theoretical studies and practical applications of Z-numbers has been proposed. However, the computational complexity of Z-numbers remains a challenge. Because the actual probability distribution is unknown, a set of probability distributions is considered, which is the main source of computational complexity. In this study, we outline a new approach to computation with Z-numbers that relies on the concept of imprecise probability. Specifically, we use a lower prevision measure (the lower envelope of a set of probability measures) as the basis for computation. The reason is a one-to-one correspondence between lower previsions and convex sets of probability measures. Experimental results show that the proposed approach reduces computational complexity compared with existing methods.

Entropy doi: 10.3390/e28070742

Authors: Mengyuan Liu Xing Liu Ziheng Cheng Xin Yuan

Snapshot compressive imaging (SCI), including coded aperture compressive temporal imaging (CACTI), reconstructs high-speed video frames from compressed low-frame-rate measurements. Most deep SCI reconstruction networks are designed around a measurement-wise formulation: each compressed exposure is reconstructed independently, and the resulting frame segments are concatenated to form a continuous video. This protocol is effective for within-measurement reconstruction, but it leaves cross-measurement temporal continuity largely unmodeled. Boundary artifacts such as flickering, texture drift, or motion jumps can therefore appear between adjacent reconstructed segments, even when frame-wise reconstruction metrics remain competitive. This work identifies and empirically analyzes the underexplored problem of cross-measurement temporal continuity in continuous SCI, and it provides TCA-EfficientSCI as a lightweight, causal, and reproducible baseline. The Temporal Context Adapter uses the last m reconstructed frames from the previous measurement as causal temporal context and injects this history through a gated residual feature pathway. A boundary consistency loss regularizes the predicted temporal variation across measurement boundaries without forcing adjacent frames to be identical. In a controlled three-seed comparison, Full TCA with boundary loss reduces mean Boundary Difference Error (BDE) by 2.23% relative to the matched-epoch EfficientSCI control while maintaining similar PSNR and SSIM. Correct-history inference gives BDE 0.01615, while zero and shuffled history give 0.01725 and 0.01810, respectively. The adapter adds 1,019,905 parameters, or 11.56% relative to the EfficientSCI baseline parameters, and it changes 256×256 mean latency from 54.35 ms to 68.58 ms per measurement in the profiling protocol. Rather than claiming broad reconstruction-quality improvement, this study highlights cross-measurement continuity as an important evaluation and design dimension for continuous SCI deployment.

Entropy doi: 10.3390/e28070739

Authors: Xuecheng Yang

A color image encryption scheme is developed by integrating a two-dimensional nonlinear exponential chaotic system (2D-NECS), Queen-driven permutation, and indexed row-column diffusion. The proposed 2D-NECS generates highly sensitive pseudo-random sequences for constructing dynamic permutation indices and diffusion parameters. A Queen-driven traversal mechanism achieves multi-directional pixel scrambling and enhanced cross-channel coupling, while indexed row–column diffusion propagates local changes throughout the entire image. Experimental results show that the encrypted images exhibit uniform histogram distributions, low pixel correlations, and information entropy values close to the theoretical ideal. Moreover, differential, chosen-plaintext, and known-plaintext attack analyses verify the strong security of the proposed scheme. These results demonstrate that the proposed method provides effective resistance against various cryptographic attacks while ensuring accurate image reconstruction.

Entropy doi: 10.3390/e28070741

Authors: Wen Liang Chunyu Yang Qiwei Liu Wenbo Zhang Hongliang Wang

Accurate prediction of missing or potential links is crucial for understanding complex network dynamics and supporting applications such as social recommendation and infrastructure planning. To effectively exploit both global and local structural information, this study proposes a spectral-adaptive fusion (SAF) algorithm. SAF first constructs a spectral embedding matrix by retaining a subset of spectral components, from which a row-column normalized matrix and a Gaussian kernel matrix are derived. These matrices are then adaptively fused to produce link scores, using a common-neighbor-based mechanism that dynamically balances their contributions, capturing both local and global network features while mitigating the influence of highly central nodes. Energy retention and spectral gap analyses set the truncated ratio to 5%, resulting in an average runtime reduction of 71.0% across eight datasets. Under the AUC index, SAF achieves an average relative improvement of 2.22% over advanced graph neural network methods and 10.65% over matrix factorization approaches. Importantly, even at low training ratios, SAF maintains AUPR values above 0.91 on four networks and exhibits stable performance on recall, confirming its robustness and effectiveness for link prediction.

Entropy doi: 10.3390/e28070740

Authors: Ruikang Liu Zixiao Wang Cheng Zha Kaijing Song Lu Hu

Pavement crack segmentation is crucial for enhancing traffic safety, improving maintenance efficiency, extending road lifespan, and supporting smart city development. Utilising computer vision technology to automate crack detection can significantly reduce time and labour costs, improving both accuracy and efficiency. However, pavement crack images exhibit complex visual features, irregular distributions, and diverse shapes and textures, posing challenges for accurate segmentation. To address these issues, a pavement crack segmentation network (PCSNet) based on a dynamic dual-branch encoder and deformable spatial focusing is proposed. The dual-branch encoder employs pre-trained and self-trained branches to extract general and specific crack features, respectively. Dynamic feature fusion optimises the contribution of each branch, enhancing model generalisation. The deformable spatial focusing module refines crack morphological features, improving the model’s ability to identify and localise cracks of varying shapes. Extensive experiments on the DeepCrack dataset show that PCSNet achieves precision, recall, F1 score, and Mean Intersection over Union of 85.34%, 86.16%, 85.75% and 75.23%, respectively, outperforming all comparative methods, thereby validating its superiority.

Entropy doi: 10.3390/e28070738

Authors: Arda Bayer Zhiyao Zhang Ahmet Emre Ipek Rose Khavari Behnaam Aazhang

Functional magnetic resonance imaging (fMRI) signals exhibit complex temporal structure arising from multivariate neural dynamics, physiological variability, and measurement uncertainty. In this work, we formulate region-of-interest-level fMRI analysis as a probabilistic multi-step forecasting problem and investigate the predictability of blood-oxygen-level-dependent (BOLD) activity from an information-theoretic perspective. Using the Natural Scenes Dataset, we model multiregional BOLD activity as a stochastic process with finite memory and train multiple forecasting architectures, including linear regression, exponential smoothing, recurrent neural networks, and transformer-based models, to predict future BOLD samples from preceding temporal observations. Forecasting performance is analyzed together with entropy-based quantities, including marginal entropy, conditional entropy, and normalized predictive information measures estimated directly from model-derived predictive distributions without imposing restrictive Gaussian assumptions on the underlying BOLD dynamics. The transformer model achieved significant improvement over a naive persistence baseline (p=0.001) while yielding a high predictive information fraction (η=75.49%). Post hoc directed information analysis revealed that short-horizon prediction was dominated primarily by autoregressive, within-ROI, temporal structure. Overall, the proposed framework demonstrates how probabilistic forecasting and information-theoretic analysis can be integrated to characterize the predictability, uncertainty structure, and directional organization of large-scale fMRI dynamics and may support future downstream neuroengineering and neural-state inference applications.

Entropy doi: 10.3390/e28070737

Authors: Paolo Marcandelli Stefano Mariani Martina Siena Stefano Markidis

Fourier Neural Operators have become a central tool for learning solution operators of partial differential equations, but their spectral layers remain entirely classical and rely on digital Fourier processing. In this work, we introduce the Continuous-Variable Quantum Fourier Neural Operator (CV-QFNO), a Gaussian photonic formulation of the FNO spectral layer. The proposed architecture maps the essential operations of Fourier-domain operator learning, Fourier transformation, mode selection, and channel mixing, onto native continuous-variable optical primitives. In this way, the CV-QFNO provides a photonic quantum analogue of the truncated spectral mechanism underlying the classical FNO, while avoiding the compilation overhead and spectral mismatch that arise in qubit-based Quantum FNO constructions. We extended the framework to both one- and two-dimensional operator learning and validated it on standard PDE benchmarks, including Burgers’ equation, heat equation, Navier–Stokes dynamics, and Darcy flow. The results show that the proposed model preserves the predictive accuracy, resolution generalisation, and spectral inductive bias of Fourier neural operators while using structurally constrained photonic parameterisation. Since all the experiments were performed as classical simulations, the contribution should be understood as an architectural and algorithmic blueprint for photonic neural operators rather than as a demonstration of quantum computational advantage.

Entropy doi: 10.3390/e28070736

Authors: Chanpyo Kim Jeongsoo Kang Younghun Kwon

Superconducting transmon processors represent a leading platform for large-scale quantum computing due to their high gate fidelities and scalability. However, conventional qubit–coupler–qubit (QCQ) architectures face critical physical and structural bottlenecks, notably frequency crowding [spectator qubit collisions] during system scaling and inefficient mapping onto the standard surface code. To overcome these limitations, we propose a novel lattice-patch architecture that couples four fixed-frequency transmons to a single fixed-frequency coupler. This design enhances qubit connectivity and maps directly onto the surface-code lattice unit [plaquette], thereby minimizing the compilation overhead associated with logical qubit implementation. Furthermore, utilizing an entirely fixed-frequency design intrinsically eliminates susceptibility to external flux noise, ensuring robust operational stability. Multi-level numerical simulations demonstrate CNOT gate fidelities exceeding 0.98 across all six connectivity directions within the patch. Nevertheless, the complex interaction network of the four-qubit architecture induces unintended residual phase accumulation during cross-resonance driving. This parasitic effect necessitates precise calibration, achievable via virtual Rz gates [software phase updates]. Ultimately, our results establish the lattice-patch architecture as an efficient, robust building block for future fault-tolerant quantum computers.

Entropy doi: 10.3390/e28070735

Authors: Antony Mizzi David M. Walker Michael Small

We derive a penalty strength criterion for ridge regression using stochastic complexity, which is a refined variant of the minimum description length principle. Since stochastic complexity does not typically account for the effect of regularization on complexity, despite its ability to simplify models, we are required to make a slight modification to the underlying coding scheme. Our scheme makes use of a weighted ensemble of regularized model fits rather than a mixture of maximum likelihood estimates. Under this modification, regularization is interpreted as reducing model complexity by constraining flexibility. In the case of ridge regression, the complexity penalty term that we derive can be expressed analytically as the log determinant of the residual operator. We demonstrate the effect of this complexity penalty by fitting a linear readout to a reservoir computer, and by performing benchmark testing on publicly available datasets.

Entropy doi: 10.3390/e28070734

Authors: Ken Chen Wenyao Zhu Tiansong Li Hongkui Wang

Outlier detection serves as an effective technique for identifying anomalous samples in complex data. Existing methods are often disturbed by noise and boundary samples, which degrade the quality of sample relationships. Moreover, traditional random walk approaches are vulnerable to weak and spurious connections that can mislead the walking process. To address these issues, this paper proposes a robust random walk based on natural neighbors for outlier detection (RWNOD) method. First, an adaptive smoothing mechanism is proposed to leverage natural neighbors to actively adjust sample positions, reducing local noise while preserving structural information. Then, a robust random walk strategy is developed to incorporate shadowed sets into the transition matrix, preserving reliable connections while suppressing unreliable ones. At the same time, a corresponding outlier detection algorithm is proposed. Experiments on datasets are conducted to compare the proposed algorithm with seven other algorithms. The experimental results demonstrate that the proposed algorithm achieves superior performance and strong robustness.

Entropy doi: 10.3390/e28070733

Authors: Chao Zhang Yufei Zhang Feng Wang Xiaoxu Su Zhijie Dong Linlin Zuo

The multi-valued dependency matrix (MVD matrix) is an important testability modeling approach, which can deliver more comprehensive testability information than the traditional dependency matrix (D-matrix). However, existing testability strategy optimization algorithms perform poorly in handling the MVD matrix, and the high-dimensional MVD matrix further aggravates these limitations as system complexity increases. To address these problems, a novel testability strategy optimization method under multi-valued dependency conditions based on deep reinforcement learning (DRL) is proposed. Firstly, the sets of elements and two reward functions to minimize test sequence length and test cost are established from the MVD matrix. Subsequently, the algorithm for selecting test points based on Deep Q-Network (DQN) is proposed. The DQN parameters are updated to fit the Q-value of test points. Thirdly, Double DQN (DDQN) and the prioritized experience replay (PER) mechanism are introduced to address the overestimation problem and sample redundancy problem, respectively, in high-dimensional matrix environments. The experimental results show that the testability strategy generated by this method can isolate all faults with fewer steps or at a lower cost. In a high-dimensional matrix environment, it can reduce test costs compared with the other heuristic algorithms while maintaining a good level of stability.

Entropy doi: 10.3390/e28070732

Authors: Brian Keith-Norambuena

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric C=A·T, where A is the angular similarity of document embeddings and T=1−dJS is the topic proximity through the Jensen–Shannon distance of soft cluster memberships, and we provide an information-geometric reading of this metric together with an axiomatic characterization of the geometric-mean combinator. On the product manifold Sd−1×Δ+K−1, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen–Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher–Rao metric singled out by Chentsov’s theorem. Within a parametric family of combinators (the compensability spectrum), the geometric mean is the unique combinator consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction also motivates a proper product metric d× that we use as a reference distance. Experiments on four corpora spanning news and academic domains (40 to 6000 documents), three general-purpose embedding families (GPT-4/ada-002, MPNet, MiniLM-L6) plus citation-aware SPECTER2, and three alternative topic models (LDA, soft k-means, GMM) are consistent with the framework: the Fisher identity holds with R≥0.99, the geometric mean tracks d× closely (ρ=0.999), and a downstream LLM-as-judge consistency check shows that the geometric mean is not empirically dominated by any alternative combinator or single-channel baseline. Sweeping the compensability spectrum, the bottleneck-coherence gap between extracted storylines and random sequences splits into a symmetric component—maximized at the geometric mean on the four corpora above and a fifth, human-navigation corpus—and a displacement term; a cross-modal case study on a human-curated image narrative reproduces the same effect in a second modality. Together, these results provide an information-geometric justification for the composite coherence metric and articulate the conditions under which the geometric mean is the natural choice.

Entropy doi: 10.3390/e28070731

Authors: Antonio Chella Salvatore Gaglio Giovanni Pilato Filippo Vella

A reactive agent operating in a complex environment must classify its perceived state and select an action under uncertainty. This uncertainty may arise from sensor noise, ambiguous perceptual configurations, or the limited separability of the action regions induced by the agent’s policy. We propose a hybrid classical–quantum architecture for a reactive agent in which the perceived state, represented as a classical sensor vector, is mapped onto a quantum feature space. In this space, learned conceptualizations or rule-defined perceptual regions are represented as reference states, and similarities between the current perception and such references are used to support action selection. The architecture is evaluated on a public wall-following robot dataset. Two implementations are considered: (i) a quantum-kernel classifier based on ZZ feature maps and (ii) an illustrative quantum circuit that explicitly encodes sensor conditions into qubits and performs measurement-based action selection. The experimental evaluation is intended as an offline proxy for reactive decision-making, not as a demonstration of a complete closed-loop robotic controller or of quantum advantage. The results show that the proposed framework can represent perceptual ambiguity and connect quantum-state measurement to the selection of discrete reactive actions.

Entropy doi: 10.3390/e28070730

Authors: Marc A. Scott Fulvia Pennoni Ignacio Bórquez

State sequence analysis of longitudinal categorical data seeks to synthesize pathways through different dimensions of the life course for descriptive, associative and predictive purposes. Given the number and variety of patterns in such data, measures of the dynamic features of sequences are used to characterize them. One, based on the information-theoretic notion of entropy, measures the uncertainty in the state that will be active at a given time. We customize its use to establish the extent to which we are ignorant, or unsure, of what happens next in a dynamic process, conditional on its past. Relying on different Markov chain models for nominal state sequences, we establish multiple measures of uncertainty that allow us to adjust expectations to reflect individual-specific differences and historical information. We establish complementary measures to assess the predictive power of the models in the context of this uncertainty. In so doing, we can summarize and contrast the change in uncertainty associated with different models. As is common in this field, we consider ways in which data can be stratified through demographics and clustering, and how this additional level of partitioning builds a more complete narrative of the social process.

Entropy doi: 10.3390/e28070729

Authors: Meilin Liu Wenping Zheng Shuxia Yuan

Hypergraph neural networks have shown strong potential for node classification due to their ability to capture high-order relationships and multi-granularity structural patterns. However, real-world hypergraphs are often sparse, which limits interaction modeling through node–hyperedge incidence and, in turn, weakens reliable attribute propagation and global dependency capture. To address this issue, we propose DDEC, a Dual Dependency-Enhanced Contrastive learning framework for sparse hypergraph node classification. To compensate for relational information lost under sparse structures, DDEC introduces an attribute view to complement the structural view. Since attribute information can be noisy and unreliable, we first design an entropy-guided feature recalibration mechanism to estimate node uncertainty and emphasize trustworthy attribute interactions. Building upon this, DDEC performs dual dependency enhancement from both structural and attribute perspectives. Specifically, we exploit the duality between a hypergraph and its line graph to perform line-graph transformation in both views, thereby constructing a shared dual relational space for interaction enhancement under sparse topologies. Within this dual space, we perform attention-based dependency enhancement in both views, so that the structural view captures explicit topological dependencies among hyperedges, while the attribute view uncovers latent semantic correlations beyond sparse incidence relations. The resulting representations from the two views are then adaptively fused, and collaborative contrastive learning is further performed at both the node and hyperedge levels to enforce multi-granularity semantic consistency. Experiments on eight public datasets demonstrate that DDEC consistently outperforms competitive baselines, validating its effectiveness and robustness.

Entropy doi: 10.3390/e28070728

Authors: Yalong Xie Yuming Liu Xianhui Nie Jiaao Guo Chengfeng Huang

Against the backdrop of the advancing Transportation Power Strategy, long and large tunnels face critical challenges in ensuring the safety and efficiency of transportation scheduling due to their harsh environment, complex traffic network, and the need for coordination among multiple types of vehicles. Addressing the shortcomings of existing research—such as the disconnection between path planning and dynamic environments, insufficient coordination between timetables and paths, and incomplete conflict management—this paper constructs a comprehensive optimization model for the scheduling of construction vehicles in tunnel traffic networks. Firstly, integrating the improved social force model with the BPR function, an adaptive social force-BPR path planning model with a collision compensation mechanism is proposed, and the weights of sub-items are optimized using the improved AHP algorithm. Secondly, a constraint system covering paths, spatio-temporal logic, and three types of conflicts (crossing conflicts, head-on conflicts, and congestion conflicts) is established, and a bi-objective function of “minimum total scheduling time” and “minimum number of conflicts” is designed. Combined with the improved NSGA-II algorithm, the collaborative optimization of departure intervals and paths is realized. In particular, a conflict entropy repair operator is introduced to quantify the conflict chaos through node conflict entropy and vehicle conflict entropy, and the scheduling strategy is accurately adjusted based on the logic of “priority ranking-dynamic delay” to balance conflict resolution and efficiency loss. Finally, a case verification is carried out relying on a tunnel topological network with 30 nodes and 41 edges. The experimental results show that the optimal repulsion coefficient kf of the social force model is 20, and the maximum departure interval of 8 min is the best configuration after introducing the repair operator. At this time, the total scheduling time is 136 min, and the total number of conflicts is only 2, completely avoiding high-risk head-on conflicts and congestion conflicts. The research outputs a vehicle scheduling scheme, enriches the theory of tunnel traffic scheduling, and provides scientific and feasible technical support for the coordinated scheduling of construction vehicles in long and large tunnels.

Entropy doi: 10.3390/e28070727

Authors: Katarína Karl’ová Jozef Strečka

The transfer-matrix method is employed to investigate a spin-1/2 Ising chain under open and periodic boundary conditions. It is demonstrated that finite-size Ising chains with antiferromagnetic coupling may exhibit significantly distinct magnetic behavior under open and periodic boundary conditions. While the open Ising chains display intriguing magnetic features regardless of the system size, mainly due to a specific contribution of boundary spins, the magnetic behavior of closed Ising chains depends basically on the number of spins. The closed Ising chains with an odd number of spins are subject to a geometric spin frustration leading to an additional plateau in the magnetization curve, which is naturally absent in the closed Ising chains with an even number of spins. Despite different microscopic origins, the magnetization curves of open and closed Ising chains with an odd number of spins exhibit an identical intermediate plateau, with only small quantitative differences appearing at moderate temperatures, which means that a geometric spin frustration of odd-membered rings is somewhat similar to the effect of open boundaries. The magnetization curves of the open Ising chains with an even number of spins differ drastically from those of the closed Ising chains due to the presence of an additional intermediate magnetization plateau. Furthermore, the initial susceptibility, inverse initial susceptibility, and susceptibility–temperature product are examined in detail as functions of temperature. These magnetic response functions demonstrate that the Curie constant and Weiss temperature represent fundamental characteristics of the magnetic system that are independent of the choice of boundary conditions.

Entropy doi: 10.3390/e28070726

Authors: Jiaqi Tang Mu-Jiang-Shan Wang

The severe susceptibility of qubits to environmental noise remains the primary obstacle to practical quantum computing. To overcome this, we introduce a purification-assisted quantum error-correction (QEC) framework that embeds a symmetric subspace projection module between the encoding and physical layers. Acting as an information-theoretic noise-entropy filter, it compresses von Neumann entropy before encoding. Under depolarizing noise, a three-copy scheme elevates the surface-code threshold from 1.1% to a 2.0% noiseless bound (~1.6% at circuit level). Our iterative purification-assisted error-correction (IPEC) algorithm dynamically modulates purification depth via syndrome feedback, delivering a 46-fold logical error reduction for surface codes (d = 7) at a 1.0% physical error rate.

Entropy doi: 10.3390/e28070725

Authors: Weijian Pang Jun Zhou Jingwen Xu Xinian Zhi

Early fault detection in microsatellite thermal control systems (TCS) faces fundamental challenges: high-dimensional redundant telemetry channels, overlapping multi-scale periodicities that obscure anomaly signatures, and severely limited daily data downlink (1–2 passes per day) that restricts the temporal window for diagnosis. Existing data-driven approaches either rely on supervised learning, requiring labeled fault data that are scarce in practice, or employ univariate analysis that fails to capture inter-sensor spatial correlations. To address these limitations, this paper introduces a hybrid framework integrating information-theoretic feature selection and spatiotemporal deep learning. The Generalized Maximum Information Coefficient (GMIC) quantifies nonlinear dependencies between temperature channels for key channel selection, reducing dimensionality by 82% while preserving diagnostic information. A dual-level Seasonal Trend Decomposition (STL) method disentangles orbital-periodic dynamics from diurnal cycles, effectively isolating distinct thermal characteristics at multiple timescales. Each decomposed component is modeled using Convolutional Neural Network–Long Short-Term Memory (CNN-LSTM) networks to capture spatiotemporal dependencies for accurate temperature prediction. An adaptive threshold-based weighted error fusion mechanism enables early fault detection within a single day of telemetry data. Experimental validation on real satellite telemetry data demonstrates that the proposed framework achieves high-precision fault detection across multiple fault types using a minimal set of temperature channels, significantly outperforming existing benchmarks in both prediction accuracy and detection reliability.

Entropy doi: 10.3390/e28070724

Authors: Marcin Kacprowicz

Outlier detection in time series is traditionally formulated as the identification of rare or extreme observations with respect to global statistical properties. While effective for stationary processes, this perspective becomes insufficient in complex and non-stationary systems, where atypical behavior may manifest as disruptions of stable relationships rather than numerical extremeness. This paper proposes a rule-based framework for detecting structural outliers in non-stationary time series. Regular system behavior is represented by an interpretable set of deterministic IF–THEN rules describing stable relational patterns between features. Each rule defines a logical context and an admissible range of a diagnostic quantity, estimated nonparametrically from historical observations satisfying the rule condition. For a given observation, the set of active rules is identified and a structural inconsistency score is computed as the fraction of violated rule consequences. Additionally, observations lacking support from high-frequency contexts are treated as candidates for structural atypicality. The method is deterministic and avoids the need for explicit probabilistic modeling or iterative parameter learning, which simplifies interpretation and implementation. The framework is illustrated on daily EUR/USD data (2010–2022) using technical indicators (EMA, RSI) and absolute log-returns as the diagnostic measure. Results provide evidence that structurally atypical events can be identified even when global statistical thresholds remain unviolated, suggesting the practical relevance of relational analysis for non-stationary time series monitoring contexts.

Entropy doi: 10.3390/e28070723

Authors: Huanyu Ren Xiaozhen Lian Qiong Chen Ziheng Lin Zonghui Jiang Zhenglong Li

Global container liner shipping networks (GCLSNs) underpin world trade, yet their organization is increasingly reshaped by geopolitical fragmentation. Existing studies often model GCLSNs as single-layer networks, overlooking how carriers’ geopolitical affiliations structure both connectivity and disruption risk. This study constructs a weighted carrier–geopolitical multiplex network in which layers are defined by carriers’ geopolitical affiliations and coupled through shared port calls. Structural analysis reveals pronounced asymmetry in layer size, cohesion, and inter-layer dependence, with overlap concentrated in a limited set of shared hubs. Using the Red Sea crisis as an empirical stress-test scenario, we develop a load–capacity propagation model, incorporating intra-layer load redistribution, rerouting to substitute shared hubs, and inter-layer resource squeeze at same-port layer copies. Results show that direct losses concentrate in corridor-exposed layers, while indirect losses propagate selectively through bridge hubs, especially Singapore, Shanghai, Shenzhen, and Port Klang. Sensitivity analysis indicates nonlinear amplification when low tolerance, strong inter-layer squeeze, and elevated rerouting pressure coincide. These findings show that multiplexity does not imply resilience by itself; cross-layer connectivity buffers disruption only when spare capacity is distributed but amplifies vulnerability when it converges on a narrow set of shared hubs. The paper contributes a carrier–geopolitical perspective to shipping network analysis and a dynamic framework for studying disruption propagation in complex logistics systems.