Entropy

21 pages, 4176 KB

Open AccessArticle

Anomalous Behavior Induced by a Single Impurity in Non-Hermitian Topological Systems with Nonreciprocal Coupling

by Junjie Wang, Zhenyan Wang, Xie Ma and Xuexi Yi

Entropy 2026, 28(5), 572; https://doi.org/10.3390/e28050572 - 19 May 2026

A remarkable feature of non-Hermitian topological systems with skin effects is that their spectra and eigenstates are strongly dependent on the choice of boundary conditions. Here, we investigate a system where the impurity couples to a nonreciprocal Su–Schrieffer–Heeger (SSH) chain at two points [...] Read more.

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

(This article belongs to the Special Issue Non-Hermitian Quantum Systems: Emergent Phenomena and New Paradigms)

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29 pages, 1883 KB

Open AccessArticle

Does More Flexible Pricing Always Pay? Profit-Driven Pricing and Market Stability Under Platform Regulation

by Le-Bin Wang, Jian Chai and Ying Yang

Entropy 2026, 28(5), 571; https://doi.org/10.3390/e28050571 - 19 May 2026

Abstract

This paper studies a dynamic price adjustment system in platform markets, where sellers continuously revise prices, and examines its implications for market stability. We develop a platform-led discrete-time Stackelberg game model to describe the evolution of sellers’ prices and price adjustment speeds under [...] Read more.

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

(This article belongs to the Section Multidisciplinary Applications)

26 pages, 2397 KB

Open AccessArticle

Delay-Induced Complexity and Chaotic Dynamics in a Network Model of Information Spreading

by Vasyl Martsenyuk and Tomasz Gancarczyk

Entropy 2026, 28(5), 570; https://doi.org/10.3390/e28050570 - 19 May 2026

Abstract

Understanding how information spreads in complex networks is essential for analyzing social influence, opinion formation, and the emergence of collective behavior. In many real-world systems, interactions are not instantaneous but involve delays due to communication, cognition, and response times. Motivated by this observation, [...] Read more.

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

(This article belongs to the Section Complexity)

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27 pages, 904 KB

Open AccessArticle

Fisher–Rao Distance for Finite-Energy Signal Manifolds: Geometric Foundations and Numerical Analysis

by Franck Florin

Entropy 2026, 28(5), 569; https://doi.org/10.3390/e28050569 - 19 May 2026

Abstract

This paper introduces a geometric framework for analyzing finite-energy signals observed with additive noise by representing them as points on statistical manifolds equipped with the Fisher–Rao metric. Each signal is associated with a parameter vector

θ

, which defines a unique probability distribution [...] Read more.

This paper introduces a geometric framework for analyzing finite-energy signals observed with additive noise by representing them as points on statistical manifolds equipped with the Fisher–Rao metric. Each signal is associated with a parameter vector

θ

, which defines a unique probability distribution

p (x | θ)

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

(This article belongs to the Special Issue Methods from Differential Topology and Differential Geometry in Information Geometry)

44 pages, 3489 KB

Open AccessArticle

Research on Error Compensation Methods of Dynamic Gravity Measurement Based on Swarm Cooperation Evolution Strategy and Optimized LSTM

by Xinyu Li, Zhaofa Zhou, Zhili Zhang, Zhe Liang, Zhenjun Chang and Yiyi Li

Entropy 2026, 28(5), 568; https://doi.org/10.3390/e28050568 - 19 May 2026

Abstract

Dynamic gravity measurement, crucial for engineering application, suffers accuracy degradation due to errors induced by carrier maneuvers. To address the limitations of conventional swarm optimization algorithms in precision, stability, and generalizability, this study proposes a Swarm Cooperation Evolution Strategy (SCES) that efficiently integrates [...] Read more.

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

(This article belongs to the Topic AI and Computational Methods for Modelling, Simulations and Optimizing of Advanced Systems: Innovations in Complexity, 2nd Edition)

24 pages, 2468 KB

Open AccessArticle

Exploring Neurofunctional Phase Transition Patterns in Autism Spectrum Disorder via Thermodynamics Parameters

by Dayu Qin, Yuzhe Chen and Ercan E. Kuruoglu

Entropy 2026, 28(5), 567; https://doi.org/10.3390/e28050567 - 19 May 2026

Abstract

Designing informative descriptors for time-varying complex networks is important for characterizing structural reconfiguration in evolving graph data. This paper introduces a thermodynamics-inspired framework for dynamic graph analysis, centered on Spectral Core Entropy (SCE), node energy, internal energy, and a temperature-like reconfiguration index. These [...] Read more.

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

(This article belongs to the Special Issue Artificial Intelligence and Machine Learning for Biomedical Applications: Entropy and Information-Theoretic Perspectives)

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21 pages, 421 KB

Open AccessReview

Multi-Stream Quickest Change Detection: Foundations and Recent Advances

by Topi Halme and Visa Koivunen

Entropy 2026, 28(5), 566; https://doi.org/10.3390/e28050566 - 18 May 2026

Abstract

This paper provides an overview of recent developments in quickest change detection (QCD) for high-dimensional multi-sensor systems, with an emphasis on settings involving structural constraints and limited sensing resources. Classical QCD methodologies, while well understood in low-dimensional and fully observed regimes, face significant [...] Read more.

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

(This article belongs to the Special Issue Foundations and Frontiers of Information Theory—Dedicated to Professor H. Vincent Poor on the Occasion of His 75th Birthday)

17 pages, 2639 KB

Open AccessArticle

Uncertainty-Aware Remaining Useful Life Prediction via Synergizing TCN–Transformer Networks and Fractional Brownian Motion

by Yiming Geng, Tianshuo Yu, Yan Liu and Jiayin Zhao

Entropy 2026, 28(5), 565; https://doi.org/10.3390/e28050565 - 18 May 2026

Abstract

Accurate Remaining Useful Life (RUL) prediction is pivotal for the intelligent operation and maintenance of high-precision equipment. However, existing deep learning-based prognostic methods predominantly focus on point estimations and often overlook the non-Markovian characteristics and stochastic uncertainties inherent in complex mechanical degradation. To [...] Read more.

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

(This article belongs to the Section Signal and Data Analysis)

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9 pages, 622 KB

Open AccessArticle

The Role of Information Entropy in Symmetry of Euclidean Polygons

by Melvin M. Vopson

Entropy 2026, 28(5), 564; https://doi.org/10.3390/e28050564 - 18 May 2026

Abstract

In this paper we investigate the relationship between Shannon information entropy and symmetry in closed Euclidean polygons within the framework of the second law of information dynamics. Using Lagrange multiplier formalism, we derive the condition for minimum entropy in a system of fixed [...] Read more.

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

(This article belongs to the Special Issue Modern Perspectives on the Second Law of Thermodynamics and Infodynamics)

22 pages, 3159 KB

Open AccessArticle

Autoencoding-Assisted Quantum Cloning Machine

by Qian Jun Beh, Moritz Straeter, Zeen Sun, Leong Chuan Kwek and Yuancheng Zhan

Entropy 2026, 28(5), 563; https://doi.org/10.3390/e28050563 - 18 May 2026

Abstract

Quantum cloning machines are essential in quantum information processing, finding applications in areas such as quantum communication and cryptographic protocols. However, the fidelity of universal quantum cloning machines diminishes as the dimension of the Hilbert space increases, resulting in significantly lower efficiency when [...] Read more.

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

(This article belongs to the Section Quantum Information)

19 pages, 4172 KB

Open AccessArticle

A Comparative Analysis of Explainable AI (XAI) Techniques for Transparent and Reliable Image Classification

by Sovon Chakraborty, Shakib Mahmud Dipto, Kevin R. Pilkiewicz, Michael L. Mayo and Pratip Rana

Entropy 2026, 28(5), 562; https://doi.org/10.3390/e28050562 - 18 May 2026

Abstract

Evaluating the trustworthiness of black-box machine learning models remains a significant methodological challenge. Their lack of transparency and interpretability limits applicability, because stakeholders often seek transparency before trusting the results of black-box machine learning models. Explainable AI (XAI) methods provide for human-understandable justifications [...] Read more.

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

(This article belongs to the Special Issue Deciphering the Link Between Information and Interpretability in Deep Learning and Artificial Intelligence)

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30 pages, 1591 KB

Open AccessArticle

Joint Optimization of User Association and Dynamic Multi-UAV Deployment for Maritime Emergency Communications

by Xiaonan Ma, Hua Yang, Yanli Xu and Naoki Wakamiya

Entropy 2026, 28(5), 561; https://doi.org/10.3390/e28050561 - 17 May 2026

Abstract

Maritime emergency response requires broadband and reliable communications in sea areas where shore coverage is limited or emergency connectivity is temporarily unavailable, making rapid on-demand aerial networking essential. Unmanned aerial vehicles (UAVs) acting as aerial base stations can be rapidly deployed to provide [...] Read more.

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

(This article belongs to the Special Issue Space-Air-Ground-Sea Integrated Communication Networks, Second Edition)

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36 pages, 933 KB

Open AccessArticle

A Deep Prompt-Based Chain-of-Thought Approach to Harmful Euphemism Detection in Social Networks

by Siyu Xie, Gang Zhou and Haizhou Wang

Entropy 2026, 28(5), 560; https://doi.org/10.3390/e28050560 - 17 May 2026

Abstract

In recent years, cyberspace governance has become a critical component of national security strategies worldwide. Although social network platforms provide users with convenient channels for expression and information acquisition, unregulated, harmful euphemisms have become increasingly prevalent. These euphemisms disrupt the order of the [...] Read more.

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

(This article belongs to the Special Issue Complexity of Social Networks)

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30 pages, 3075 KB

Open AccessArticle

Metabolic Saliency as KL-Divergence Estimator: Information-Geometric Attribution of Systemic Stress in JSE Equity Network

by Ntebogang Dinah Moroke

Entropy 2026, 28(5), 559; https://doi.org/10.3390/e28050559 - 15 May 2026

Abstract

The attribution of systemic financial stress to specific market sectors requires metrics that are faithful to the model’s computations, statistically consistent, and connected to a physically meaningful measure of directed information flow. This paper addresses all three requirements through information geometry, contributing to [...] Read more.

The attribution of systemic financial stress to specific market sectors requires metrics that are faithful to the model’s computations, statistically consistent, and connected to a physically meaningful measure of directed information flow. This paper addresses all three requirements through information geometry, contributing to SDGs 7, 8, 9, and 17 through an entropic causal chain linking energy infrastructure failure to financial market stress. We conjecture and empirically verify the Entropy–Saliency Equivalence: Metabolic Saliency is an asymptotically unbiased estimator of the local Kullback–Leibler divergence between stressed and resting sector return distributions, with bias decaying at a parametric rate under Gaussian regularity conditions. The finite-sample bias–variance decomposition of the Kraskov–Stögbauer–Grassberger transfer entropy estimator is derived, establishing a minimax-optimal convergence rate. A novel metric, the Spatio-Temporal Information Flux (STIF), quantifies directed inter-sector stress transmission in bits per trading day, providing a bootstrap-calibrated audit trail aligned with the South African Financial Sector Regulation Act and MiFID II. Empirical validation on the JSE canonical panel (87 securities, 2857 trading days, 2015–2026) with Eskom load-shedding stages as exogenous stress injectors confirms the equivalence (

R^{2} = 0.810

,

\hat{ρ} = 0.90

), with walk-forward

R^{2} = 0.789

and placebo

R^{2} = 0.081

ruling out estimation artefacts. The energy sector is identified as the primary stress transmitter during Stage 4+ Eskom events (STIF rising from 0.14 to 0.43 bits/day, directional asymmetry ratio 4.7). Robustness checks confirm stability across non-Gaussian securities and rolling transfer entropy windows. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

19 pages, 1387 KB

Open AccessArticle

Uniform in Bandwidth Consistency of the L¹-Modal Regression Estimator for High-Dimensional Data

by Fatimah A. Almulhim, Mohammed B. Alamari and Ali Laksaci

Entropy 2026, 28(5), 558; https://doi.org/10.3390/e28050558 - 15 May 2026

Abstract

We propose a new nonparametric estimator of the conditional mode in a regression framework where the covariates are functional in nature. The estimator is constructed through a quantile regression approach, which provides a robust alternative to classical density-based procedures. It is well documented [...] Read more.

We propose a new nonparametric estimator of the conditional mode in a regression framework where the covariates are functional in nature. The estimator is constructed through a quantile regression approach, which provides a robust alternative to classical density-based procedures. It is well documented that employing the

L^{1}

-structure in quantile regression, the estimation procedure improves robustness properties, particularly resistance to outliers and heavy-tailed error distributions. This feature makes the

L^{1}

estimation of the conditional mode more stable and reliable in complex and high-variability functional data settings. The main objective of this paper is to establish strong consistency, with explicit convergence rates, for the associated kernel estimators, uniformly over a range of bandwidth parameters. The latter is developed under general regularity conditions involving the concentration distribution of the functional regressor, smoothness assumptions on the structural components of the model, and entropy conditions ensuring adequate control of the functional class complexity. Uniformity in bandwidth is essential both from a theoretical and practical issues, as it guarantees stability of the estimator under data-driven smoothing parameter selection. Beyond its theoretical contribution, this paper has direct implications for applied statistics. Specifically, it provides mathematical support for the automatic bandwidth selection procedures in the high-dimensional data context. Furthermore, the main theoretical novelty is highlighted through simulation experiments and applications to real data. Full article

49 pages, 538 KB

Open AccessArticle

Finite-Capacity Thermodynamics of Causal Horizons

by Cristián Alberto Antiba

Entropy 2026, 28(5), 557; https://doi.org/10.3390/e28050557 (registering DOI) - 15 May 2026

Abstract

This work proposes that fundamental physics remains unchanged, but its local classical representability is limited by the finite entropic capacity of causal horizons. Geometric entanglement entropy is treated as a thermodynamic potential, and horizons as finite-capacity information systems. When this capacity is saturated, [...] Read more.

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

(This article belongs to the Special Issue Geometry in Thermodynamics, 4th Edition)

25 pages, 3056 KB

Open AccessArticle

On Intention and Fluctuations in the Coordination Dynamics of Animate Movement

by Amaury Dechaux, Aliza T. Sloan and J. A. Scott Kelso

Entropy 2026, 28(5), 556; https://doi.org/10.3390/e28050556 - 15 May 2026

Abstract

Many of life’s biggest dilemmas can be summed up as a tension between holding on and letting go. The very language evokes a notion of intentionality which, for the most part, has evaded scientific understanding. How might we even get a window into [...] Read more.

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

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12 pages, 2981 KB

Open AccessArticle

Quantum Capacity of Continuously Observed Ion Channels

by Paulina Trybek and Jerzy Dajka

Entropy 2026, 28(5), 555; https://doi.org/10.3390/e28050555 - 15 May 2026

Abstract

A quantum model describing ion channels from an information-theoretic perspective is considered. The information

χ

-capacity of an ion channel, treated as an information channel whose properties are modified by continuous quantum measurements, is investigated. The behavior of the

χ

-capacity is analyzed [...] Read more.

A quantum model describing ion channels from an information-theoretic perspective is considered. The information

χ

-capacity of an ion channel, treated as an information channel whose properties are modified by continuous quantum measurements, is investigated. The behavior of the

χ

-capacity is analyzed as a function of the measurement parameters, in particular the type of measured observable, the measurement duration, and the measurement strength. It is shown that the information

χ

-capacity exhibits qualitatively different behaviors depending on the measurement conditions, including regimes of rapid decay as well as regimes where it remains finite for long observation times. These results indicate that, within the considered model, continuous observation may significantly influence the information-theoretic properties of the effective ion-channel dynamics. Full article

(This article belongs to the Special Issue Mathematical Modeling for Ion Channels)

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16 pages, 2346 KB

Open AccessArticle

Semantic Algorithmic Information Theory: From Kolmogorov Complexity to Semantic Equivalence

by Jiatong Wu, Sen Wang, Kai Niu, Yifei She and Ping Zhang

Entropy 2026, 28(5), 554; https://doi.org/10.3390/e28050554 - 14 May 2026

Abstract

Classical Algorithmic Information Theory (AIT) provides a rigorous foundation for information-based similarity measurement, but classical formulations and their compression-based approximations largely operate at the syntactic level, making them sensitive to surface-level variation and insufficient for semantic equivalence. To address this limitation, this paper [...] Read more.

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