Entropy

19 pages, 485 KB

Open AccessArticle

Trust-Aware Causal Consistency Routing for Quantum Key Distribution Networks Against Malicious Nodes

by Yi Luo and Qiong Li

Entropy 2025, 27(11), 1100; https://doi.org/10.3390/e27111100 (registering DOI) - 24 Oct 2025

Quantum key distribution (QKD) networks promise information-theoretic security for multiple nodes by leveraging the fundamental laws of quantum mechanics. In practice, QKD networks require dedicated routing protocols to coordinate secure key distribution among distributed nodes. However, most existing routing protocols operate under the [...] Read more.

Quantum key distribution (QKD) networks promise information-theoretic security for multiple nodes by leveraging the fundamental laws of quantum mechanics. In practice, QKD networks require dedicated routing protocols to coordinate secure key distribution among distributed nodes. However, most existing routing protocols operate under the assumption that all relay nodes are honest and fully trustworthy, an assumption that may not hold in realistic scenarios. Malicious nodes may tamper with routing updates, causing inconsistent key-state views or divergent routing plans across the network. Such inconsistencies increase routing failure rates and lead to severe wastage of valuable secret keys. To address these challenges, we propose a distributed routing framework that combines two key components: (i) Causal Consistency Key-State Update, which prevents malicious nodes from propagating inconsistent key states and routing plans; and (ii) Trust-Aware Multi-path Flow Optimization, which incorporates trust metrics derived from discrepancies in reported states into the path-selection objective, penalizing suspicious links and filtering fabricated demands. Across 50-node topologies with up to 30% malicious relays and under all three attack modes, our protocol sustains a high demand completion ratio (DCR) (mean

0.90

, range

0.81

–

0.98

) while keeping key utilization low (

16.6

keys per demand), decisively outperforming the baselines—Multi-Path Planned (DCR

0.48

,

30.8

keys per demand) and OSPF (DCR

\leq 0.12

, 296 keys per demand; max 1601). These results highlight that our framework balances reliability and efficiency, providing a practical and resilient foundation for secure QKD networking in adversarial environments. Full article

(This article belongs to the Section Quantum Information)

23 pages, 2406 KB

Open AccessArticle

Dynamic Hyperbolic Tangent PSO-Optimized VMD for Pressure Signal Denoising and Prediction in Water Supply Networks

by Yujie Shang and Zheng Zhang

Entropy 2025, 27(11), 1099; https://doi.org/10.3390/e27111099 (registering DOI) - 24 Oct 2025

Abstract

Urban water supply networks are prone to complex noise interference, which significantly degrades the performance of data-driven forecasting models. Conventional denoising techniques, such as standard Variational Mode Decomposition (VMD), often rely on empirical parameter selection or optimize only a subset of parameters, lacking [...] Read more.

Urban water supply networks are prone to complex noise interference, which significantly degrades the performance of data-driven forecasting models. Conventional denoising techniques, such as standard Variational Mode Decomposition (VMD), often rely on empirical parameter selection or optimize only a subset of parameters, lacking a robust mechanism for identifying noise-dominant components post-decomposition. To address these issues, this paper proposed a novel denoising framework termed Dynamic Hyperbolic Tangent PSO-optimized VMD (DHTPSO-VMD). The DHTPSO algorithm adaptively adjusts inertia weights and cognitive/social learning factors during iteration, mitigating the local optima convergence typical of traditional PSO and enabling automated VMD parameter selection. Furthermore, a dual-criteria screening strategy based on Variance Contribution Rate (VCR) and Correlation Coefficient Metric (CCM) is employed to accurately identify and eliminate noise-related Intrinsic Mode Functions (IMFs). Validation using pressure data from District A in Zhejiang Province, China, demonstrated that the proposed DHTPSO-VMD method significantly outperforms benchmark approaches (PSO-VMD, EMD, SABO-VMD, GWO-VMD) in terms of Signal-to-Noise Ratio (SNR), Mean Absolute Error (MAE), and Mean Square Error (MSE). Subsequent forecasting experiments using an Informer model showed that signals preprocessed with DHTPSO-VMD achieved superior prediction accuracy (R² = 0.948924), underscoring its practical utility for smart water supply management. Full article

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

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9 pages, 816 KB

Open AccessTechnical Note

Euclidean-Lorentzian Dichotomy and Algebraic Causality in Finite Ring Continuum

by Yosef Akhtman

Entropy 2025, 27(11), 1098; https://doi.org/10.3390/e27111098 - 24 Oct 2025

Abstract

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells

F_{p}

with

p = 4 t + 1

exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realized within a [...] Read more.

We present a concise and self-contained extension of the Finite Ring Continuum (FRC) program, showing that symmetry-complete prime shells

F_{p}

with

p = 4 t + 1

exhibit a fundamental Euclidean-Lorentzian dichotomy. A genuine Lorentzian quadratic form cannot be realized within a single space-like prime shell

F_{p}

, since to split time from space one requires a time coefficient

c^{2}

in the nonsquare class of

F_{p}^{\times}

, but then

c \notin F_{p}

. An explicit finite-field Lorentz transformation is subsequently derived that preserves the Minkowski form and generates a finite orthogonal group

O (Q_{ν}, F_{p^{2}})

of split type (Witt index 1). These results demonstrate that the essential algebraic features of special relativity—the invariant interval and Lorentz symmetry—emerge naturally within finite-field arithmetic, thereby establishing an intrinsic relativistic algebra within FRC. Finally, this dichotomy implies the algebraic origin of causality: Euclidean invariants reside within a space-like shell

F_{p}

, while Lorentzian structure and causal separation arise in its quadratic spacetime extension

F_{p^{2}}

. Full article

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

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1 pages, 126 KB

Open AccessCorrection

Correction: Zou et al. Geometrical Bounds on Irreversibility in Squeezed Thermal Bath. Entropy 2023, 25, 128

by Chen-Juan Zou, Yue Li, Jia-Kun Xu, Jia-Bin You, Ching Eng Png and Wan-Li Yang

Entropy 2025, 27(11), 1097; https://doi.org/10.3390/e27111097 - 24 Oct 2025

Abstract

In the published article [...] Full article

(This article belongs to the Special Issue Quantum Thermodynamics: Fundamentals and Applications)

16 pages, 280 KB

Open AccessArticle

On Quasi-Cyclic Codes of Index 3

by Kanat Abdukhalikov and Rasha M. Shat

Entropy 2025, 27(11), 1096; https://doi.org/10.3390/e27111096 - 23 Oct 2025

Abstract

Quasi-cyclic codes of index 3 over finite fields are studied. We give a classification of such codes. Their duals with respect to the Euclidean and Hermitian inner products are investigated. We give a characterization of self-orthogonal and dual-containing codes. A quasi-cyclic code of [...] Read more.

Quasi-cyclic codes of index 3 over finite fields are studied. We give a classification of such codes. Their duals with respect to the Euclidean and Hermitian inner products are investigated. We give a characterization of self-orthogonal and dual-containing codes. A quasi-cyclic code of index 3 is generated by at most three elements. We describe conditions when such a code (or its dual) is generated by one element. Full article

(This article belongs to the Special Issue Discrete Math in Coding Theory, 2nd Edition)

18 pages, 908 KB

Open AccessArticle

Bayesian Estimation of Multicomponent Stress–Strength Model Using Progressively Censored Data from the Inverse Rayleigh Distribution

by Asuman Yılmaz

Entropy 2025, 27(11), 1095; https://doi.org/10.3390/e27111095 - 23 Oct 2025

Abstract

This paper presents a comprehensive study on the estimation of multicomponent stress–strength reliability under progressively censored data, assuming the inverse Rayleigh distribution. Both maximum likelihood estimation and Bayesian estimation methods are considered. The loss function and prior distribution play crucial roles in Bayesian [...] Read more.

This paper presents a comprehensive study on the estimation of multicomponent stress–strength reliability under progressively censored data, assuming the inverse Rayleigh distribution. Both maximum likelihood estimation and Bayesian estimation methods are considered. The loss function and prior distribution play crucial roles in Bayesian inference. Therefore, Bayes estimators of the unknown model parameters are obtained under symmetric (squared error loss function) and asymmetric (linear exponential and general entropy) loss functions using gamma priors. Lindley and MCMC approximation methods are used for Bayesian calculations. Additionally, asymptotic confidence intervals based on maximum likelihood estimators and Bayesian credible intervals constructed via Markov Chain Monte Carlo methods are presented. An extensive Monte Carlo simulation study compares the efficiencies of classical and Bayesian estimators, revealing that Bayesian estimators outperform classical ones. Finally, a real-life data example is provided to illustrate the practical applicability of the proposed methods. Full article

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

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24 pages, 4420 KB

Open AccessArticle

AttSCNs: A Bayesian-Optimized Hybrid Model with Attention-Guided Stochastic Configuration Networks for Robust GPS Trajectory Prediction

by Xue-Bo Jin, Ye-Qing Wang, Jian-Lei Kong, Yu-Ting Bai and Ting-Li Su

Entropy 2025, 27(11), 1094; https://doi.org/10.3390/e27111094 - 23 Oct 2025

Abstract

Trajectory prediction in the Internet of Vehicles (IoV) is crucial for enhancing road safety and traffic efficiency; however, existing methods often fail to address the challenges of colored noise in GPS data and long-term dependency modeling. To overcome these limitations, this paper proposes [...] Read more.

Trajectory prediction in the Internet of Vehicles (IoV) is crucial for enhancing road safety and traffic efficiency; however, existing methods often fail to address the challenges of colored noise in GPS data and long-term dependency modeling. To overcome these limitations, this paper proposes AttSCNs, a probabilistic hybrid framework integrating stochastic configuration networks (SCNs) with an attention-based encoder to model trajectories while quantifying prediction uncertainty. The model leverages SCNs’ stochastic neurons for adaptive noise filtering, attention mechanisms for dependency learning, and Bayesian hyperparameter optimization to infer robust configurations as a posterior distribution. Experimental results on real-world GPS datasets (10,000+ urban/highway trajectories) demonstrate that AttSCNs significantly outperform conventional approaches, reducing RMSE by 36.51% compared to traditional SCNs and lowering MAE by 97.8% compared to Kalman filter baselines. Moreover, compared to the LSTM model, AttSCNs achieve a 52.5% reduction in RMSE and a 68.5% reduction in MAE, with real-time inference speed. These advancements position AttSCNs as a robust, noise-resistant solution for IoV applications, offering superior performance in autonomous driving and smart city systems. Full article

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

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10 pages, 1153 KB

Open AccessArticle

Entanglement Islands in 1D and 2D Lattices with Defects

by Ivan P. Christov

Entropy 2025, 27(11), 1093; https://doi.org/10.3390/e27111093 - 23 Oct 2025

Abstract

We investigate the spatial structure of quantum entanglement in one- and two-dimensional lattice systems containing structural defects, using the Time-Dependent Quantum Monte Carlo (TDQMC) method. By constructing reduced density matrices from ensembles of guide waves, we resolve spatial variations in both Coulomb-mediated entanglement [...] Read more.

We investigate the spatial structure of quantum entanglement in one- and two-dimensional lattice systems containing structural defects, using the Time-Dependent Quantum Monte Carlo (TDQMC) method. By constructing reduced density matrices from ensembles of guide waves, we resolve spatial variations in both Coulomb-mediated entanglement and coherence without requiring full many-body wavefunctions. This approach reveals localized regions, entanglement islands, where quantum correlations are enhanced or suppressed due to the presence of vacancies or interaction inhomogeneities. In 1D systems, entanglement tends to concentrate near defects, while in 2D systems, we observe bridge-like and radially symmetric domains. Our results demonstrate that TDQMC offers a scalable and physically transparent framework for real-space quantum information analysis, with implications for information transfer in atomic-size structures, quantum materials, entanglement-based sensing, and coherent state engineering. Full article

(This article belongs to the Special Issue Editorial Board Members' Collection Series on Quantum Entanglement)

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4 pages, 146 KB

Open AccessEditorial

Semantic Information Theory and Applications

by Meixia Tao, Kai Niu and Youlong Wu

Entropy 2025, 27(11), 1092; https://doi.org/10.3390/e27111092 - 23 Oct 2025

Abstract

Traditional information theory provides a rigorous foundation for information compression and reliable symbol transmission [...] Full article

(This article belongs to the Special Issue Semantic Information Theory)

17 pages, 1775 KB

Open AccessArticle

Self-Diffusion in Two-Dimensional Colloidal Systems: A Computer Simulation Study

by Piotr Polanowski and Andrzej Sikorski

Entropy 2025, 27(11), 1091; https://doi.org/10.3390/e27111091 - 22 Oct 2025

Abstract

The dynamics of dense colloidal systems are not fully understood. In the study of these types of systems, computer simulations based on the so-called hard sphere model play a significant role. In the presented work, we consider a system of hard spheres of [...] Read more.

The dynamics of dense colloidal systems are not fully understood. In the study of these types of systems, computer simulations based on the so-called hard sphere model play a significant role. In the presented work, we consider a system of hard spheres of the same size but different mobilities (molecules with high mobility correspond to solvent molecules, while molecules with reduced mobility are colloid particles) at varying concentrations. For this purpose, a two-dimensional lattice and an thermal model of such systems was designed. In order to determine the properties of such systems, a Monte Carlo computer simulation was used, employing the Dynamic Lattice Liquid (DLL) algorithm. Our main aim was to determine how the dynamic behavior of the system in the short time affects the long-time behavior. For this purpose, we investigated the cross-ratios of the diffusion coefficients in the short and long time of the considered system elements. It was found that the reduction in the solvent mobility with increasing concentration of colloidal particles in a short time leads to a very similar reduction in the mobility of the colloid particles in a long time, but we do not observe such behavior in the case of the solvent, i.e., there is a decrease in the value of the solvent diffusion coefficient in the long time with the change in the concentration of colloid particles, but it is difficult to connect it in a simple way with the decrease in the diffusion coefficient in the short time. Full article

(This article belongs to the Special Issue Statistical Mechanics of Lattice Gases)

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27 pages, 1592 KB

Open AccessArticle

Information-Theoretic Reliability Analysis of Consecutive r-out-of-n:G Systems via Residual Extropy

by Anfal A. Alqefari, Ghadah Alomani, Faten Alrewely and Mohamed Kayid

Entropy 2025, 27(11), 1090; https://doi.org/10.3390/e27111090 - 22 Oct 2025

Abstract

This paper develops an information-theoretic reliability inference framework for consecutive r-out-of-n:G systems by employing the concept of residual extropy, a dual measure to entropy. Explicit analytical representations are established in tractable cases, while novel bounds are derived for more complex [...] Read more.

This paper develops an information-theoretic reliability inference framework for consecutive r-out-of-n:G systems by employing the concept of residual extropy, a dual measure to entropy. Explicit analytical representations are established in tractable cases, while novel bounds are derived for more complex lifetime models, providing effective tools when closed-form expressions are unavailable. Preservation properties under classical stochastic orders and aging notions are examined, together with monotonicity and characterization results that offer deeper insights into system uncertainty. A conditional formulation, in which all components are assumed operational at a given time, is also investigated, yielding new theoretical findings. From an inferential perspective, we propose a maximum likelihood estimator of residual extropy under exponential lifetimes, supported by simulation studies and real-world reliability data. These contributions highlight residual extropy as a powerful information-theoretic tool for modeling, estimation, and decision-making in multicomponent reliability systems, thereby aligning with the objectives of statistical inference through entropy-like measures. Full article

(This article belongs to the Special Issue Recent Progress in Uncertainty Measures)

26 pages, 2220 KB

Open AccessArticle

Lindbladian Decoherence in Quantum Universal Gates: An Insight Analysis for Digital Noise and Thermalisation

by José Carlos Rebón and Francisco Delgado

Entropy 2025, 27(11), 1089; https://doi.org/10.3390/e27111089 - 22 Oct 2025

Abstract

Quantum computing is an emergent field promising the improvement of processing speed in key algorithms by reducing their exponential scaling to polynomial, thus enabling solutions to problems that exceed classical computational capabilities. Gate-based quantum computing is the most common approach but still faces [...] Read more.

Quantum computing is an emergent field promising the improvement of processing speed in key algorithms by reducing their exponential scaling to polynomial, thus enabling solutions to problems that exceed classical computational capabilities. Gate-based quantum computing is the most common approach but still faces high levels of noise and decoherence. Gates play the role of probability mixers codifying information settled in quantum systems. However, they are deviated from their programmed behaviour due to those decoherent effects as a hidden source modifies the desired probability flux. Their quantification of such unavoidable behaviours becomes crucial for quantum error correction or mitigation. This work presents an approach to decoherence in quantum circuits using the Lindblad master equation to model the impact of noise and thermalisation underlying the ideal programmed behaviour expected for processing gates. The Lindblad approach then provides a comprehensive tool to model both probability fluxes being present in the process, thus regarding the gate and the environment. It analyses the deviation of resulting noisy states from the ideal unitary evolution of some gates considered as universal, setting some operating regimes. Thermalisation considers a radiation bath where gates are immersed as a feasible model of decoherence. Numerical simulations track the information loss as a function of the decay rate magnitude. It also exhibits the minimal impact on decoherence coming from particular quantum states being processed, but a higher impact on the number of qubits being processed by the gate. The methodology provides a unified framework to characterise the processing probability transport in quantum gates, including noise or thermalisation effects. Full article

(This article belongs to the Special Issue Probability Theory and Quantum Information)

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24 pages, 1998 KB

Open AccessArticle

NetTopoBFT: Network Topology-Aware Byzantine Fault Tolerance for High-Coverage Consortium Blockchains

by Runyu Chen, Rangang Zhu and Lunwen Wang

Entropy 2025, 27(11), 1088; https://doi.org/10.3390/e27111088 - 22 Oct 2025

Abstract

The Practical Byzantine Fault Tolerance (PBFT) algorithm, while fundamental to consortium blockchains, suffers from performance degradation and vulnerability of leader nodes in large-scale scenarios. Existing improvements often prioritize performance while lacking systematic consideration of the structural characteristics of the nodes and network coverage. [...] Read more.

The Practical Byzantine Fault Tolerance (PBFT) algorithm, while fundamental to consortium blockchains, suffers from performance degradation and vulnerability of leader nodes in large-scale scenarios. Existing improvements often prioritize performance while lacking systematic consideration of the structural characteristics of the nodes and network coverage. In this paper, a new network topology-aware Byzantine fault-tolerant algorithm NetTopoBFT is proposed for the supply chain and other application scenarios that require strict transaction finality but moderate throughput. Firstly, it innovatively combines the weighted signed network with the consortium chain, constructs a two-layer Bayesian smoothing node evaluation model, and evaluates the nodes through the two-dimensional evaluation of ‘behavioral reputation plus structural importance’. Then, to reduce the risk of being attacked, it uses Verifiable Random Function (VRF) to decide the leader. Furthermore, it uses a duplicate coverage-driven waitlisting mechanism to enhance the robustness and connectivity of the system. Theoretical analysis and experiment results show that NetTopoBFT significantly improves the quality of consensus nodes under the premise of guaranteeing decentralization, realizes the simultaneous optimization of communication overhead, security and network coverage. It provides a new idea for designing consensus mechanism of consortium blockchains. Full article

(This article belongs to the Section Complexity)

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17 pages, 552 KB

Open AccessArticle

Winning Opinion in the Voter Model: Following Your Friends’ Advice or That of Their Friends?

by Francisco J. Muñoz and Juan Carlos Nuño

Entropy 2025, 27(11), 1087; https://doi.org/10.3390/e27111087 - 22 Oct 2025

Viewed by 95

Abstract

We investigate a variation of the classical voter model where the set of influencing agents depends on an individual’s current opinion. The initial population is made up of a random sample of equally sized sub-populations for each state, and two types of interactions [...] Read more.

We investigate a variation of the classical voter model where the set of influencing agents depends on an individual’s current opinion. The initial population is made up of a random sample of equally sized sub-populations for each state, and two types of interactions are considered: (i) direct neighbors and (ii) second neighbors (friends of direct neighbors, excluding the direct neighbors themselves). The neighborhood size, reflecting regular network connectivity, remains constant across all agents. Our findings show that varying the interaction range introduces asymmetries that affect the probability of consensus and convergence time. At low connectivity, direct neighbor interactions dominate, leading to consensus. As connectivity increases, the probability of either state reaching consensus becomes equal, reflecting symmetric dynamics. This asymmetric effect on the probability of consensus is shown to be independent of network topology in small-world and scale-free networks. Asymmetry also influences convergence time: while symmetric cases display decreasing times with increased connectivity, asymmetric cases show an almost linear increase. Unlike the probability of reaching consensus, the impact of asymmetry on convergence time depends on the network topology. The introduction of stubborn agents further magnifies these effects, especially when they favor the less dominant state, significantly lengthening the time to consensus. We conclude by discussing the implications of these findings for decision-making processes and political campaigns in human populations. Full article

(This article belongs to the Special Issue Entropy-Based Applications in Sociophysics II)

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22 pages, 376 KB

Open AccessArticle

CSCVAE-NID: A Conditionally Symmetric Two-Stage CVAE Framework with Cost-Sensitive Learning for Imbalanced Network Intrusion Detection

by Zhenyu Wang and Xuejun Yu

Entropy 2025, 27(11), 1086; https://doi.org/10.3390/e27111086 - 22 Oct 2025

Viewed by 149

Abstract

With the increasing complexity and diversity of network threats, developing high-performance Network Intrusion Detection Systems (NIDSs) has become a critical challenge. A primary obstacle in this domain is the pervasive issue of class imbalance, where the scarcity of minority attack samples and the [...] Read more.

With the increasing complexity and diversity of network threats, developing high-performance Network Intrusion Detection Systems (NIDSs) has become a critical challenge. A primary obstacle in this domain is the pervasive issue of class imbalance, where the scarcity of minority attack samples and the varying costs of misclassification severely limit the effectiveness of traditional models, often leading to a difficult trade-off between high False Positive Rates (FPRs) and low Recall. To address this challenge, this paper proposes a novel, conditionally symmetric two-stage framework, termed CSCVAE-NID (Conditionally Symmetric Two-Stage CVAE for Network Intrusion Detection). The framework operates in two synergistic stages: Firstly, a Data Augmentation Conditional Variational Autoencoder (DA-CVAE) is introduced to tackle the data imbalance problem at the data level. By conditioning on attack categories, the DA-CVAE generates high-quality and diverse synthetic samples for underrepresented classes, providing a more balanced training dataset. Secondly, the core of our framework, a Cost-Sensitive Multi-Class Classification CVAE (CSMC-CVAE), is proposed. This model innovatively reframes the classification task as a probabilistic distribution matching problem and integrates a cost-sensitive learning strategy at the algorithm level. By incorporating a predefined cost matrix into its loss function, the CSMC-CVAE is compelled to prioritize the correct classification of high-cost, minority attack classes. Comprehensive experiments conducted on the public CICIDS-2017 and UNSW-NB15 datasets demonstrate the superiority of the proposed CSCVAE-NID framework. Compared to several state-of-the-art methods, our approach achieves exceptional performance in both binary and multi-class classification tasks. Notably, the DA-CVAE module is designed to be independent and extensible, allowing the effective data that it generates to support any advanced intrusion detection methodology. Full article

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