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Volume 28
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Entropy, Volume 28, Issue 7 (July 2026) – 117 articles

Cover Story (view full-size image): 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. Applied to BTC, ETH, and XRP on Binance, Bitget, KuCoin, and Kraken, the analysis reveals a pronounced anomaly on Bitget for BTC and ETH after mid-May 2025: transaction counts rose sharply without a proportional increase in traded volume or return fluctuations. Our findings show that complexity-based indicators can be useful for detecting exchange-specific trading anomalies that remain hidden in price-based measures. View this paper
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26 pages, 1831 KB  
Article
Data-Driven Quantification of Quantum k-Entanglement via Machine Learning
by Jie Guo, Jinchuan Hou, Xiaofei Qi and Kan He
Entropy 2026, 28(7), 832; https://doi.org/10.3390/e28070832 - 22 Jul 2026
Viewed by 356
Abstract
k-entanglement, including entanglement relative to full separability and genuinely multipartite entanglement, is a fundamental quantum resource in multipartite quantum systems. Its identification and quantification play essential roles in quantum information processing, quantum simulation, and quantum metrology. However, the practical computation of rigorous [...] Read more.
k-entanglement, including entanglement relative to full separability and genuinely multipartite entanglement, is a fundamental quantum resource in multipartite quantum systems. Its identification and quantification play essential roles in quantum information processing, quantum simulation, and quantum metrology. However, the practical computation of rigorous k-entanglement measures remains highly challenging due to the need for high-dimensional optimization. In this work, we propose a machine-learning-based surrogate framework for approximating the witness-based k-entanglement measure Ew(k,n). The numerical evaluation of the computationally realized quantity E˜w(k,n)(ρ) is reformulated as a supervised regression problem, where the input is the density matrix ρ and the labels are obtained from finite witness databases. The framework combines multilayer perceptrons (MLPs), convolutional neural networks (CNNs), and light gradient boosting machine (LightGBM) through a stacking ensemble. Numerical experiments are performed for 3- and 4-qubit systems as representative demonstrations of the proposed workflow. The results show that the learned models achieve high predictive accuracy in terms of MAE, MSE, and R2, while providing millisecond-level inference for single-state evaluation. Werner state tests serve as symmetric benchmark checks, and an additional four-qubit noisy circuit-generated state family, obtained from finite-depth circuit preparation followed by local amplitude-damping noise, is used as a structured physical test beyond random density matrices. Compared with the optimization-based evaluation, the trained surrogate model significantly reduces the computational time while maintaining accuracy within the tested system sizes and data distributions. These results show that the proposed framework provides an efficient numerical surrogate for rapid approximation of witness-based k-entanglement measures, while extensions to larger systems and experimental data require further validation. Full article
(This article belongs to the Special Issue New Advances in Quantum Communication and Networks, 2nd Edition)
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24 pages, 11347 KB  
Article
Image Encryption via DDPM Skipping-Step Diffusion Based on a Six-Dimensional Hyperchaotic System with Feedback Control
by Songran Wang, Hanqing Zhao, Wei He, Tao Wang, Zhiben Zhuang, Tianfu Zhang and Jiacheng Xu
Entropy 2026, 28(7), 831; https://doi.org/10.3390/e28070831 - 22 Jul 2026
Viewed by 318
Abstract
In this paper, we propose a novel image encryption algorithm that integrates a six-dimensional hyperchaotic system with the forward diffusion process of denoising diffusion probabilistic models (DDPMs). The proposed framework synergistically combines the hyperchaotic system’s high sensitivity to initial conditions with the DDPM’s [...] Read more.
In this paper, we propose a novel image encryption algorithm that integrates a six-dimensional hyperchaotic system with the forward diffusion process of denoising diffusion probabilistic models (DDPMs). The proposed framework synergistically combines the hyperchaotic system’s high sensitivity to initial conditions with the DDPM’s Markov chain-based skipping-step diffusion mechanism, thereby enabling dual-level confusion-diffusion operations at both the pixel and bit levels. This dual-strategy approach significantly enhances plaintext sensitivity and ciphertext randomness. Comprehensive simulations demonstrate that the algorithm achieves superior performance in key space expansion, histogram uniformity, adjacent pixel correlation reduction, information entropy optimization, and resistance to differential attacks. The algorithm exhibits strong resilience against various attack vectors, including brute-force attacks, statistical analysis, chosen-plaintext attacks and common image degradation factors (e.g., noise contamination and cropping). These characteristics establish the proposed method as a highly secure and practical solution for image data protection in cloud-IoT environments. Full article
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19 pages, 941 KB  
Article
Cross-Code Verification for Improved Thermophysical Properties of Argon, Krypton and Xenon Plasmas
by Alberto Vagnoni, Anthony B. Murphy and Emanuele Ghedini
Entropy 2026, 28(7), 830; https://doi.org/10.3390/e28070830 - 22 Jul 2026
Viewed by 506
Abstract
Thermophysical properties of thermal plasmas are essential input data for computational models. The required data are usually taken from the literature without examination of their reliability. Cross-code verifications of properties are rare in the thermal plasma literature, partly due to the complexity of [...] Read more.
Thermophysical properties of thermal plasmas are essential input data for computational models. The required data are usually taken from the literature without examination of their reliability. Cross-code verifications of properties are rare in the thermal plasma literature, partly due to the complexity of the calculation methods, which require a systematic treatment of large datasets, multiple computations and the adoption of different models. Here, a detailed comparison of two computational codes that use different workflows but very similar underlying methods is presented, using the example of thermophysical properties of argon, krypton, and xenon plasmas in local thermodynamic equilibrium at pressures from 1 to 100 atm. The comparison considers plasma composition, collision integrals, thermodynamic properties and, in particular, transport coefficients. The comparison allowed inconsistencies and errors to be identified and corrected, resulting in improved thermophysical properties of argon, krypton, and xenon. Furthermore, transport coefficients obtained from state-of-the-art intermolecular potentials were compared with those obtained from the simpler phenomenological potential, demonstrating good agreement, including at high pressures. Full article
(This article belongs to the Special Issue Thermodynamic and Transport Properties of Plasmas)
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22 pages, 1959 KB  
Article
SPA-QNAS: Improving Search Efficiency and Stability in Evolutionary Quantum Neural Architecture Search
by Linwei Shang, Hao Cao, Yang Wu, Xufeng Niu and Junjie Chen
Entropy 2026, 28(7), 829; https://doi.org/10.3390/e28070829 - 22 Jul 2026
Viewed by 451
Abstract
Quantum neural architecture search (QNAS) has emerged as a promising approach for automatically designing parameterized quantum circuits (PQCs) for near-term quantum machine learning tasks. However, quantum evolutionary algorithm (QEA)-based QNAS methods often suffer from slow distribution concentration and insufficient update adaptivity in high-dimensional [...] Read more.
Quantum neural architecture search (QNAS) has emerged as a promising approach for automatically designing parameterized quantum circuits (PQCs) for near-term quantum machine learning tasks. However, quantum evolutionary algorithm (QEA)-based QNAS methods often suffer from slow distribution concentration and insufficient update adaptivity in high-dimensional discrete search spaces, which limits both search efficiency and final model performance. To address these issues, this paper proposes Structural Probability Adaptive Quantum Neural Architecture Search (SPA-QNAS), a search-dynamics-enhanced QNAS method built upon the EQNAS benchmark framework. SPA-QNAS introduces two complementary mechanisms into the QPV-driven evolutionary search loop: Structural Probability Enhancement (SPE) and Adaptive Evolutionary Control (AEC). SPE reinforces elite structural decisions to accelerate the concentration of the structural sampling distribution toward high-fitness regions, while AEC adaptively regulates the rotation updates of non-elite individuals according to fitness feedback, thereby improving update stability and suppressing ineffective disturbances. Under the same search space, circuit template, and quantum resource budget as EQNAS, SPA-QNAS is evaluated on the MNIST and Warship benchmark datasets. Experimental results across multiple independent runs demonstrate that SPA-QNAS achieves higher classification accuracy and more stable performance compared with EQNAS. In representative experiments, SPA-QNAS achieves classification accuracies of 99.42% on MNIST and 85.33% on Warship under the same search space and quantum resource budget as EQNAS. These results indicate that improving QPV-based evolutionary update dynamics is an effective way to enhance the stability and robustness of QNAS under fixed quantum resource constraints. Full article
(This article belongs to the Section Quantum Information)
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29 pages, 419 KB  
Article
A Class of Robust Estimators for Moment Condition Models
by Amor Keziou and Aida Toma
Entropy 2026, 28(7), 828; https://doi.org/10.3390/e28070828 - 21 Jul 2026
Viewed by 284
Abstract
Moment condition models are popular in statistics and econometrics, as they provide a powerful and flexible framework for estimation. However, estimation procedures based on these models can be sensitive to misspecification or the presence of outliers in the data. In the present paper, [...] Read more.
Moment condition models are popular in statistics and econometrics, as they provide a powerful and flexible framework for estimation. However, estimation procedures based on these models can be sensitive to misspecification or the presence of outliers in the data. In the present paper, we introduce a class of robust estimators for moment condition models, representing robust alternatives to minimum empirical divergence estimators. The estimators are constructed by using truncated orthogonality functions and minimizing divergences in dual form, allowing to limit the impact of outliers or model deviations. We give the expressions of the influence functions of the estimators and prove their robustness. We also prove that the estimators are consistent. These theoretical results together with numerical examples, based on Monte Carlo simulations, show that extreme observations do not disproportionately affect the final estimates. Full article
(This article belongs to the Special Issue Statistical Inference: Theory and Methods)
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16 pages, 584 KB  
Article
Strategic Superposition and Replicator Dynamics: Quantum Collapses in Decision Processes
by Aras Yolusever
Entropy 2026, 28(7), 827; https://doi.org/10.3390/e28070827 - 20 Jul 2026
Viewed by 369
Abstract
Classical evolutionary game theory rests on a hidden assumption: that an actor holds a definite strategy, pure or randomized, before it ever interacts. Yet human and organizational choices routinely violate this premise, displaying interference, order, and framing effects that classical probability cannot accommodate. [...] Read more.
Classical evolutionary game theory rests on a hidden assumption: that an actor holds a definite strategy, pure or randomized, before it ever interacts. Yet human and organizational choices routinely violate this premise, displaying interference, order, and framing effects that classical probability cannot accommodate. We propose a framework in which an economic actor is genuinely undecided before market entry, modeled as a strategic superposition of pure strategies in a Hilbert space, and in which interaction acts as a measurement that collapses this state onto a realized strategy with Born-rule probabilities. Populations are described by a density operator whose diagonal carries strategy frequencies and whose off-diagonal coherences encode maintained superposition, evolving under a strategic master equation that couples coherent deliberation, decoherence in the strategy basis, and a replicator selection superoperator. Three results follow. A square-root representation places quantum normalization and evolutionary selection on a common geometric footing; the classical replicator equation emerges exactly as the strong-decoherence limit, with an explicit error bound; and strategy realization becomes basis-dependent through interference that no classical mixture reproduces. In a two-strategy market game, coherent coupling displaces the evolutionarily stable strategy by order Δ2/γ, recovering the classical value as decoherence dominates. The construction formalizes constitutive self-opacity and links bounded rationality to quantum interference, positioning classical evolutionary dynamics as one limiting regime of a broader strategic dynamics. Full article
(This article belongs to the Section Multidisciplinary Applications)
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23 pages, 3479 KB  
Article
From Centralized to Distributed Entropy: Long-Term Resilience and Structural Evolution of Regional Innovation Networks in the Yangtze River Delta
by Ju Yang, Fenglei Zhou, Zhongying Hu, Lintao Zha and Hui Yang
Entropy 2026, 28(7), 826; https://doi.org/10.3390/e28070826 - 20 Jul 2026
Viewed by 264
Abstract
Understanding how regional innovation networks maintain functionality under disruption is critical for regional economic sustainability. This study investigates the structural evolution and topological resilience of inter-city patent cooperation networks in China‘s Yangtze River Delta (YRD) from 2005 to 2024. We construct weighted networks [...] Read more.
Understanding how regional innovation networks maintain functionality under disruption is critical for regional economic sustainability. This study investigates the structural evolution and topological resilience of inter-city patent cooperation networks in China‘s Yangtze River Delta (YRD) from 2005 to 2024. We construct weighted networks of 41 cities and simulate targeted attack scenarios to quantify network resilience as the area under the robustness curve. Using degree distribution entropy to quantify the spatial distribution of innovation activity, we demonstrate that the system’s innovation activity redistributes from a Shanghai-dominated, concentrated, single-core configuration to a more distributed, polycentric architecture. The resilience index increases from 0.0836 in 2005 to 0.5000 in 2015, and full connectivity is achieved by 2024. Notably, removing the top-ranked node in 2010 reduces the largest connected component by 5.3%, whereas removing four core nodes simultaneously in 2015 produces a 9.8% reduction, indicating that the redistribution of innovation activity is associated with systemic robustness. Correlation analysis further reveals strong associations between resilience and network density, weighted-degree entropy, and short path lengths, reflecting a hub-dominated topology that evolved from a single hub (Shanghai) to multiple co-existing hubs (Shanghai, Nanjing, Hangzhou, Hefei). These findings provide empirical evidence consistent with the view that polycentric configurations are associated with innovation ecosystem resilience and offer actionable insights for regional sustainability policies. Full article
(This article belongs to the Special Issue Advances in Complex Networks and Their Applications, from COMPLEX NETWORKS 2025)
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22 pages, 1246 KB  
Article
Asymptotic Thermodynamics for Chemical Reaction Networks with Fast-Slow Kinetics
by Liangrong Peng and Liu Hong
Entropy 2026, 28(7), 825; https://doi.org/10.3390/e28070825 - 20 Jul 2026
Viewed by 283
Abstract
We present a systematic derivation of asymptotic expansions of nonequilibrium thermodynamics for chemical reaction networks (CRNs) based on singular perturbation theory. For a general reversible CRN with fast–slow kinetics, we obtain the first and second laws of thermodynamics for the asymptotic expansion model. [...] Read more.
We present a systematic derivation of asymptotic expansions of nonequilibrium thermodynamics for chemical reaction networks (CRNs) based on singular perturbation theory. For a general reversible CRN with fast–slow kinetics, we obtain the first and second laws of thermodynamics for the asymptotic expansion model. We derive composite expansions of the enthalpy, entropy, entropy production rate, and relative entropy. The slow-varying outer parts of these thermodynamic quantities capture the long-time trend, while the fast-varying corrected inner parts decay to zero as the fast time variable tends to infinity. The convergence order of these quantities is determined by the local Lipschitz properties of the respective functions. The enthalpy retains the same convergence order as the kinetic variables, whereas the entropy, relative entropy, and entropy production rate involve logarithmic terms that cause their gradients to diverge when some concentrations approach zero, reducing their theoretical convergence order by one. The general theory is validated on the reversible Michaelis–Menten reaction, for which both leading-order and first-order matched asymptotic expansions are obtained analytically. Numerical simulations confirm the uniform accuracy of the composite thermodynamic approximations and further reveal that the entropy production rate converges with a higher order than theoretically predicted. The results demonstrate that the composite expansion provides a rigorous and physically consistent tool for analyzing energy and entropy balances in multiscale CRNs. Full article
(This article belongs to the Section Thermodynamics)
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16 pages, 736 KB  
Article
A Reduced-Complexity Iterative Bounded Distance Decoder with Random Flipping for Product Codes
by Guoming Song, Dongming Pi and Shancheng Zhao
Entropy 2026, 28(7), 824; https://doi.org/10.3390/e28070824 - 20 Jul 2026
Viewed by 346
Abstract
Product codes (PCs) are widely used in high-speed communication systems due to their attractive trade-off between error-correction performance and complexity. To further meet the rapidly growing demand for higher data rates, soft-aided hard-decision decoders (SA-HDDs) have been developed. In this paper, we present [...] Read more.
Product codes (PCs) are widely used in high-speed communication systems due to their attractive trade-off between error-correction performance and complexity. To further meet the rapidly growing demand for higher data rates, soft-aided hard-decision decoders (SA-HDDs) have been developed. In this paper, we present a reduced-complexity iterative bounded distance decoder with random flipping (RC-iBDD-RF) for PCs, an SA-HDD that improves the decoding performance while preserving low decoding complexity. RC-iBDD-RF introduces an enhanced reliability metric that incorporates channel log-likelihood ratios (LLRs) and memory from previous iterations to guide random flipping. In addition, an error-and-erasure decoding (EaED) module employing fixed filling patterns, rather than randomly generated ones, is used for post-processing to further reduce the residual error rate. The comparisons for BCH-based PCs show that RC-iBDD-RF achieves both performance gain and complexity reduction with only a slight increase in memory requirement. Moreover, the proposed complexity-storage weighted SNR metric confirms its superior complexity-performance trade-off over iBDD-RF. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
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31 pages, 21575 KB  
Article
Structural Entropy, Modal Diversity Entropy, and Accessibility Differentiation: A Study of the Western China–Central Asia Cross-Border Multimodal Transportation Network
by Ruifen Sun, Ying Xin, Yang Shao and Peilun Ju
Entropy 2026, 28(7), 823; https://doi.org/10.3390/e28070823 - 20 Jul 2026
Viewed by 352
Abstract
Cross-border multimodal transportation systems are essential for regional connectivity, yet their structural concentration, modal imbalance, community organization, and accessibility differentiation remain insufficiently understood from an entropy perspective. Based on 2024 data, this study constructs railway, highway, aviation, and integrated transportation networks between Western [...] Read more.
Cross-border multimodal transportation systems are essential for regional connectivity, yet their structural concentration, modal imbalance, community organization, and accessibility differentiation remain insufficiently understood from an entropy perspective. Based on 2024 data, this study constructs railway, highway, aviation, and integrated transportation networks between Western China and the five Central Asian countries. It integrates complex network analysis, structural entropy, modal diversity entropy, Louvain community detection, and accessibility assessment within a structure–organization–function framework. The results reveal a core–periphery pattern, with Xi’an, Urumqi, Almaty, and Tashkent serving as hubs. Structural entropy shows that highway connections are balanced, aviation links are concentrated around core hubs, and the integrated network reflects the coexistence of core-hub agglomeration and multimodal coverage expansion. Modal diversity entropy indicates that high connectivity does not necessarily imply balanced modal configuration, and 41.8% of nodes remain dependent on a single mode. Community detection reveals local cohesion and global segmentation, while accessibility analysis identifies a Western China core and Central Asian periphery. Sensitivity analyses based on common-node normalization, travel-time weighting, and alternative modal weights confirm the robustness of the findings. These results provide an integrated diagnostic framework for identifying structural concentration, modal imbalance, and accessibility inequality in cross-border multimodal transportation networks. Full article
(This article belongs to the Special Issue Insight into Entropy)
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22 pages, 1945 KB  
Article
Carbon Market Price Forecasting Using a Bidirectional Temporal Convolution Exogenous-Enhanced Time-Series Model
by Xinyu Tang, Mingzhu Tang, Na Li and Shumei Zhang
Entropy 2026, 28(7), 822; https://doi.org/10.3390/e28070822 - 19 Jul 2026
Viewed by 448
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Complexity Features and Blockchain Foundations of the Digital Instruments Market)
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20 pages, 1020 KB  
Article
Exact Combinatorial Density of States for the Critical 1D Ising Model
by Bastian Castorene, Francisco J. Peña, Martin HvE Groves and Patricio Vargas
Entropy 2026, 28(7), 821; https://doi.org/10.3390/e28070821 - 19 Jul 2026
Viewed by 317
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Ising Model—100 Years Old and Still Attractive)
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19 pages, 4987 KB  
Article
Fastformer: An Efficient Attention-Based Framework for Rapid Multi-Class Fault Diagnosis in High-End Equipment Vibration Signals
by Xiaohan Zhang, Hailun Dai, Chong Zhou and Qi Shen
Entropy 2026, 28(7), 820; https://doi.org/10.3390/e28070820 - 19 Jul 2026
Viewed by 306
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Failure Diagnosis of Complex Systems)
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50 pages, 1785 KB  
Article
Statistic Maximal Leakage
by Shuaiqi Wang, Zinan Lin and Giulia Fanti
Entropy 2026, 28(7), 819; https://doi.org/10.3390/e28070819 - 18 Jul 2026
Viewed by 265
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Information Theory and Differential Privacy)
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31 pages, 3789 KB  
Article
A Dynamic Optimization Algorithm Based on Energy Level Collaboration Mechanism
by Quan Tang, Yazhi Yang and Jing Liu
Entropy 2026, 28(7), 818; https://doi.org/10.3390/e28070818 - 17 Jul 2026
Viewed by 363
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Entropy-Guided Evolutionary Intelligence: Information Dynamics for Adaptive and Explainable Machine Learning)
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15 pages, 272 KB  
Article
On the Algorithm Complexity of Generating Discrete Uniform Distribution from a Biased Coin
by Mengqi Zhang, Guangqiang Teng and Xiaoyu Lei
Entropy 2026, 28(7), 817; https://doi.org/10.3390/e28070817 - 17 Jul 2026
Viewed by 266
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Information Theory, Probability and Statistics)
23 pages, 790 KB  
Article
Structural Phylogenetic Signal Fails at Deep Time: A Bayesian Treebank Analysis of the Transeurasian Languages
by Wenchao Li and Haitao Liu
Entropy 2026, 28(7), 816; https://doi.org/10.3390/e28070816 - 17 Jul 2026
Viewed by 405
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Multidisciplinary Applications)
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23 pages, 602 KB  
Article
Prior-Assisted Hierarchical ADMM Decoding for Punctured Globally Coupled LDPC Codes
by Wenbo Shi, Wenlong Xie, Jiashen Hu and Lishan Liu
Entropy 2026, 28(7), 815; https://doi.org/10.3390/e28070815 - 17 Jul 2026
Viewed by 306
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Key Technologies Towards Future Wireless Networks)
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19 pages, 9459 KB  
Article
Transfer Entropy Causal Networks for Interconnectedness Analysis of Global Banking and Green Markets: A CEEMDAN-SE-KM Approach
by Qiuyang Xue, Xiu Jin, Jinming Yu and Yueli Liu
Entropy 2026, 28(7), 814; https://doi.org/10.3390/e28070814 - 17 Jul 2026
Viewed by 252
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Section Multidisciplinary Applications)
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38 pages, 6041 KB  
Article
A Wave–Particle Model of Energy Transfer Between Two Atoms in a Transactional Interpretation of Quantum Mechanics
by Lloyd Watts and Carver Mead
Entropy 2026, 28(7), 813; https://doi.org/10.3390/e28070813 - 17 Jul 2026
Viewed by 312
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Time in Quantum Mechanics)
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49 pages, 936 KB  
Article
Time Series Correlations and Kolmogorov Complexity: A Hausdorff Dimension Perspective
by Boumediene Hamzi, Marianne Clausel, Kamal Dingle, Marcus Hutter and Mohammed Terry Jack
Entropy 2026, 28(7), 812; https://doi.org/10.3390/e28070812 - 16 Jul 2026
Viewed by 328
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Kolmogorov Complexity and Applications—Dedicated to Professor Paul Vitanyi on the Occasion of His 80th Birthday)
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48 pages, 574 KB  
Review
Entropy Regularization in Deep Reinforcement Learning: A Structured Review Across Classical Control, Generative Policies, and Reasoning Language Models
by Giorgio Taricco
Entropy 2026, 28(7), 811; https://doi.org/10.3390/e28070811 - 16 Jul 2026
Viewed by 568
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Section Entropy Reviews)
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24 pages, 18515 KB  
Article
Wind Turbine Blade Fault Diagnosis Integrating Multi-Scale Enhanced Hierarchical Fuzzy Entropy, Isolation Forest and GWO-GRU
by Min Wang, Xiao-Fei Zhang, Guo-Jun Qin and Ming Liu
Entropy 2026, 28(7), 810; https://doi.org/10.3390/e28070810 - 16 Jul 2026
Cited by 1 | Viewed by 329
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Multivariate Entropy-Informed Fault Diagnosis and Structural Health Monitoring)
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32 pages, 4094 KB  
Article
From Global to Local: Semantic-Aware Instance-Wise Feature Selection
by Zihan Wang, Yue Zhang, Hengpeng Xu, Zhenglu Yang and Jun Wang
Entropy 2026, 28(7), 809; https://doi.org/10.3390/e28070809 - 16 Jul 2026
Viewed by 291
Abstract
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 [...] Read more.
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. Full article
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26 pages, 1157 KB  
Article
The Measurement Problem in the Thermodynamics of Black Holes
by Jeroen Schoenmaker
Entropy 2026, 28(7), 808; https://doi.org/10.3390/e28070808 - 15 Jul 2026
Viewed by 680
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue The Landauer Principle in Physics, Biophysics, Engineering and Computer Science: From Foundations of Thermodynamics to Computer Engineering)
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24 pages, 3811 KB  
Article
Modelling Cumulative Seismic Damage at the Urban Scale
by Rosa Maria Sava, Annalisa Greco, Alessandro Pluchino and Andrea Rapisarda
Entropy 2026, 28(7), 807; https://doi.org/10.3390/e28070807 - 15 Jul 2026
Viewed by 704
Abstract
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 [...] Read more.
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. Full article
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16 pages, 3016 KB  
Article
A Deng Entropy-Based Heuristic Method to Determine Discounting Coefficient in Dempster-Shafer Evidence Fusion
by Siyao Huang and Yong Deng
Entropy 2026, 28(7), 806; https://doi.org/10.3390/e28070806 - 15 Jul 2026
Viewed by 327
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Entropy Method for Decision Making with Uncertainty, 2nd Edition)
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25 pages, 2799 KB  
Article
Novel Support Routing Algorithm for Quantum Satellite Networks with Finite Quantum Memory
by András Mihály and László Bacsárdi
Entropy 2026, 28(7), 805; https://doi.org/10.3390/e28070805 - 15 Jul 2026
Viewed by 344
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Space Quantum Communication)
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41 pages, 14965 KB  
Article
Detecting Unusual Trading Patterns on Cryptocurrency Exchanges by Means of Complexity Measures
by Jakub Zwydak, Marcin Wątorek, Jarosław Kwapień and Stanisław Drożdż
Entropy 2026, 28(7), 804; https://doi.org/10.3390/e28070804 - 15 Jul 2026
Viewed by 836
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Complexity Features and Blockchain Foundations of the Digital Instruments Market)
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23 pages, 1305 KB  
Article
Semantic Communication for Intelligent Transmission and Recognition of High-Resolution Satellite Images in Satellite-to-Ground Systems
by Jiaxin Liu, Qiwang Chen and Yijun Chen
Entropy 2026, 28(7), 803; https://doi.org/10.3390/e28070803 - 14 Jul 2026
Viewed by 400
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue AI for Wireless Communication Systems: From Semantic Communications to 6G)
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