Entropy

34 pages, 3375 KB

Open AccessArticle

Empirical Analysis of Hierarchy and Shared Macroscopic Organization in Soccer Leagues

by António M. Lopes

Entropy 2026, 28(9), 969; https://doi.org/10.3390/e28090969 (registering DOI) - 30 Aug 2026

This paper presents a coarse-grained framework for analyzing the long-term structural evolution of soccer leagues. This framework draws on ideas from non-equilibrium statistical physics. It combines latent-strength modeling, interaction rules, and large-scale observables to explore the rise of hierarchical organization in competitive leagues. [...] Read more.

This paper presents a coarse-grained framework for analyzing the long-term structural evolution of soccer leagues. This framework draws on ideas from non-equilibrium statistical physics. It combines latent-strength modeling, interaction rules, and large-scale observables to explore the rise of hierarchical organization in competitive leagues. Teams interact based on a Bradley–Terry-type stochastic model. This model relies on teams’ latent competitive strengths, which are derived from match outcomes employing maximum-likelihood estimation. Experiments on 38 soccer leagues around the world show persistent hierarchical organization. This is evident in varied latent strengths, uneven competitive balance, and stable macro-state organization from season to season. Even with differences in history, geography, and competition format, many leagues display surprisingly similar large-scale structural patterns. These findings support the idea of shared principles that guide the evolution of competitive leagues and highlight the value of using a statistical physics perspective to describe how macro-level organization arises from repeated team interactions. Full article

(This article belongs to the Section Complexity)

► Show Figures

Figure 1

25 pages, 2635 KB

Open AccessArticle

The Configurational Logic of Alliance Networks for Innovation: A Machine Learning-Enabled Investigation

by Wenhao Zhou, Zhiwei Zhang and Siyu Lin

Entropy 2026, 28(9), 968; https://doi.org/10.3390/e28090968 (registering DOI) - 30 Aug 2026

Abstract

Alliance network embeddedness provides firms with access to external knowledge and resources, yet its innovation implications vary across firms and network contexts. This study examines how network structure and combinations of embedding characteristics are associated with corporate innovation performance. Based on 335 firm-level [...] Read more.

Alliance network embeddedness provides firms with access to external knowledge and resources, yet its innovation implications vary across firms and network contexts. This study examines how network structure and combinations of embedding characteristics are associated with corporate innovation performance. Based on 335 firm-level observations from Chinese listed biopharmaceutical manufacturing firms, the study first identifies heterogeneous alliance network environments through community detection and K-Means clustering. Four network types are identified: dyadic, ringlike, star, and complex alliances. Classification and regression trees (CART) are then employed to extract interpretable, threshold-based decision rules linking network embedding characteristics to high and non-high innovation performance. The results show that no single network characteristic is consistently associated with innovation performance across alliance types. In dyadic alliances, moderate cooperation intensity is associated with high innovation performance, whereas ringlike alliances exhibit conditional associations involving cooperation intensity and partner centrality. Star alliances are characterized by configurations involving cooperation breadth and network position, while complex alliances exhibit more multidimensional combinations of structural and relational conditions. The findings indicate that the innovation relevance of alliance network embeddedness is network-type-specific and configuration-dependent. As a complementary robustness analysis, fuzzy-set qualitative comparative analysis broadly supports several core configurational patterns identified by CART, while also revealing alternative configurations, particularly in complex alliances. The study demonstrates that understanding alliance network embeddedness requires attention to network context, empirical thresholds, and combinations of network characteristics rather than isolated network attributes. Full article

► Show Figures

Figure 1

31 pages, 667 KB

Open AccessArticle

On the Structural Properties of Discrete-Time and Sampled-Data Hamiltonian Dynamics

by Salvatore Monaco and Dorothée Normand-Cyrot

Entropy 2026, 28(9), 967; https://doi.org/10.3390/e28090967 (registering DOI) - 29 Aug 2026

Abstract

While continuous-time Hamiltonian dynamics are naturally energy preserving with a symplectic flow, their discrete-time counterparts enhance either geometric or energy preservation properties, but rarely both within a unified framework. It is the object of this paper to more deeply investigate this question. In [...] Read more.

While continuous-time Hamiltonian dynamics are naturally energy preserving with a symplectic flow, their discrete-time counterparts enhance either geometric or energy preservation properties, but rarely both within a unified framework. It is the object of this paper to more deeply investigate this question. In both linear and nonlinear settings, necessary and sufficient conditions characterizing discrete Hamiltonian dynamics that are conservative and symplectic are derived. The relationship with exact sampled models of continuous-time Hamiltonian dynamics are investigated, showing that such models, that preserve both energy and symplectic structures, do not generally fit into the proposed canonical form. Generalized Hamiltonian structures are, thus, introduced. On these bases, Hamiltonian integrators that preserve both the energy and the symplectic structure up to a prescribed order in the sampling period, are constructed. Some simulations on nonlinear test cases illustrate the theoretical findings. Full article

(This article belongs to the Special Issue Port-Hamiltonian Methods)

24 pages, 38466 KB

Open AccessArticle

Optical Color Zero-Watermarking via Phase-Shifting Digital Holography Coupled with High-Robustness Bimodal Biometric Keys

by Guanghai Liu, Zhe Zhang, Wang Fu, Cui Zhang, Boyu Wang, Yanfeng Su and Zhijian Cai

Entropy 2026, 28(9), 966; https://doi.org/10.3390/e28090966 (registering DOI) - 29 Aug 2026

Abstract

In this paper, an optical color zero-watermarking scheme based on robust bimodal biometric keys and phase-shifting digital holography is proposed. The color watermark is first encrypted into three amplitude ciphertexts through an optical encryption framework combining grating modulation, Fresnel-domain double random phase encoding [...] Read more.

In this paper, an optical color zero-watermarking scheme based on robust bimodal biometric keys and phase-shifting digital holography is proposed. The color watermark is first encrypted into three amplitude ciphertexts through an optical encryption framework combining grating modulation, Fresnel-domain double random phase encoding (DRPE), and phase-shifting digital holography, where the phase masks are generated from biometric keys derived from the iris and three-dimensional (3D) face features of the encryption user. These high-level biometric features are extracted by a bimodal biometric high-order feature extraction network (BBHEN), including an iris high-order data extraction network and a 3D face high-order data extraction network. The extracted features of the color host image are then XORed with the corresponding ciphertexts, and the results are merged to construct a single zero-watermark image containing both host and watermark information. During extraction, biometric authentication is first performed to verify the identity of the decryption user. Only authorized users can recover the original watermark through zero-watermark reconstruction and extraction; otherwise, the process is terminated. Numerical simulations demonstrate the effectiveness, security, and robustness of the proposed scheme, particularly the strong protection capability of the bimodal biometric keys. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

14 pages, 350 KB

Open AccessArticle

An Agent-Based Model of Cooperation and Competition in Organizational Systems

by D. S. Fonte and L. H. A. Monteiro

Entropy 2026, 28(9), 965; https://doi.org/10.3390/e28090965 (registering DOI) - 29 Aug 2026

Abstract

This study proposes a game-theoretic agent-based model to investigate cooperation and competition in workplace environments. The iterated prisoner’s dilemma is implemented on a two-dimensional lattice, where agents interact locally and accumulate wealth over time. Two types of agents are considered: fixed probabilistic cooperators, [...] Read more.

This study proposes a game-theoretic agent-based model to investigate cooperation and competition in workplace environments. The iterated prisoner’s dilemma is implemented on a two-dimensional lattice, where agents interact locally and accumulate wealth over time. Two types of agents are considered: fixed probabilistic cooperators, who adopt a constant cooperation probability, and adaptive probabilistic cooperators, whose behavior depends on their accumulated wealth and reputation. Population composition and wealth distribution are quantified using Shannon entropy and the Gini coefficient. Numerical simulations show that fixed cooperators tend to predominate and accumulate higher average wealth. The simulations also show that increasing tolerance to defections raises average wealth and reduces inequality. In contrast, a higher cooperation probability increases wealth but may also amplify inequality. These results highlight the role of reputation and performance targets in shaping cooperation in organizational environments. Full article

(This article belongs to the Special Issue Dynamics in Biological and Social Networks, Second Edition)

► Show Figures

Figure 1

24 pages, 20003 KB

Open AccessArticle

A DNA-Based Image Encryption Scheme Using Optimized Lorenz–Sprott Hyperchaotic System

by Wenxia Xu, Liang Xue, Liping Zhu, Jiaofen Li and Guodong Li

Entropy 2026, 28(9), 964; https://doi.org/10.3390/e28090964 (registering DOI) - 27 Aug 2026

Abstract

Chaotic systems have been widely investigated for color image encryption because of their nonlinear dynamics, initial-condition sensitivity, and pseudorandom behavior. However, locating numerically robust parameter regions in high-dimensional hyperchaotic systems remains difficult, while many DNA-based schemes employ fixed or weakly varying rules. This [...] Read more.

Chaotic systems have been widely investigated for color image encryption because of their nonlinear dynamics, initial-condition sensitivity, and pseudorandom behavior. However, locating numerically robust parameter regions in high-dimensional hyperchaotic systems remains difficult, while many DNA-based schemes employ fixed or weakly varying rules. This study proposes a color image encryption scheme combining a 6D Lorenz–Sprott system optimized by particle swarm optimization (PSO), symbol-level feedback-dependent DNA transformation, and bidirectional cross-channel chained diffusion. The second-largest Lyapunov exponent is used as the optimization objective to locate parameter sets with at least two positive exponents. The selected system has the Lyapunov spectrum (0.8118, 0.2736, −0.0013, −4.8957, −8.5499, −12.6746) and retains two positive exponents under refined numerical settings and ±1% single-parameter perturbations. One hundred independently initialized sequences satisfy all 15 categories of the NIST SP 800-22 test suite. Tests on six images and three secret keys achieve exact reconstruction in all 18 cases. Across 180 randomly located one-bit plaintext perturbations, the mean NPCR and UACI are 99.6089% and 33.4554%, respectively. For the tested Baboon case, perturbing any initial-state component by approximately 10⁻¹⁴ prevents meaningful plaintext recovery. Ablation results further demonstrate the contributions of dynamic DNA transformation and bidirectional diffusion. The proposed scheme therefore provides reproducible hyperchaotic parameter modulation and strong empirical statistical and differential performance. Full article

(This article belongs to the Special Issue Advances in Image Encryption and Chaotic Cryptography)

► Show Figures

Figure 1

25 pages, 921 KB

Open AccessArticle

Differentially Private Hierarchical Spectral Clustering

by Mohamed Seif Mohamed and Andrea J. Goldsmith

Entropy 2026, 28(9), 963; https://doi.org/10.3390/e28090963 - 27 Aug 2026

Abstract

We study hierarchical spectral graph clustering under edge differential privacy (DP) through the lens of iterative eigenvector estimation on adjacency matrices. We propose a differentially private recursive spectral framework, where each binary partition is obtained via a rank-one noisy power method applied to [...] Read more.

We study hierarchical spectral graph clustering under edge differential privacy (DP) through the lens of iterative eigenvector estimation on adjacency matrices. We propose a differentially private recursive spectral framework, where each binary partition is obtained via a rank-one noisy power method applied to induced adjacency sub-matrices. At each iteration, carefully calibrated Gaussian noise is injected into the matrix–vector multiplication, ensuring

(ε, δ)

-edge DP under cumulative privacy accounting across both power iterations and recursive hierarchy levels while preserving the essential convergence properties of the classical power method. We provide a non-asymptotic analysis of the resulting noisy iterations, characterizing the trade-off between privacy and accuracy via explicit bounds on the eigenvector estimation error. In particular, we quantify how the noise variance, number of iterations, eigengap, and hierarchy depth jointly influence the accuracy of each recursive split and the overall clustering performance. Empirical evaluations on synthetic and real-world networks validate the theoretical predictions and demonstrate that the proposed method achieves strong multi-scale clustering performance under meaningful privacy budgets. 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)

► Show Figures

Figure 1

119 pages, 3043 KB

Open AccessArticle

Inverse-Probability-Weighted Kernel Estimation of Regression Derivatives Under Missing-at-Random Responses for Stationary Ergodic Processes

by Salim Bouzebda and Sultana Didi

Entropy 2026, 28(9), 962; https://doi.org/10.3390/e28090962 - 27 Aug 2026

Abstract

This paper develops asymptotic theory for kernel estimation of density-weighted conditional functionals and regression derivatives when responses are missing at random (MAR) and the observations form a strictly stationary ergodic process. Sequential MAR and positivity identify the complete-data conditional target through an inverse-probability-weighted [...] Read more.

This paper develops asymptotic theory for kernel estimation of density-weighted conditional functionals and regression derivatives when responses are missing at random (MAR) and the observations form a strictly stationary ergodic process. Sequential MAR and positivity identify the complete-data conditional target through an inverse-probability-weighted pseudo-response, while the fully observed covariate density and its derivatives are estimated without unnecessary response weighting. A martingale-predictable decomposition yields uniform almost-sure rates, pointwise Gaussian limits, variance expansions, studentization, and AMISE results under explicit projective/maximal, conditional-moment, conditional-density, and variance-stabilization conditions. These quantitative assumptions are additional to stationarity and ergodicity: the results are not asserted for arbitrary stationary ergodic sequences. Exact-quotient and multi-index identities transfer the primitive-estimator theory to regression derivatives, and feasible propensity estimation contributes an explicit additional remainder. Monte Carlo experiments show that stronger dependence, weak response probabilities, higher derivative order, propensity misspecification, and smoothing bias can materially degrade finite-sample performance; undersmoothing improves centring but need not eliminate coverage distortion at moderate sample sizes. Full article

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

20 pages, 402 KB

Open AccessArticle

A Fast Multiple Change-Point Detection Method via Generalized Nearly Isotonic Optimization

by Luoxin Wang, Mengmeng Wang, Baisuo Jin and Yuehua Wu

Entropy 2026, 28(9), 961; https://doi.org/10.3390/e28090961 - 27 Aug 2026

Abstract

In this paper, we study the generalized nearly isotonic optimization (GNIO) model and its dynamic programming solution (GNIO-DP). We introduce randomness into the GNIO-DP algorithm, enabling its first application to change-point detection and resulting in an

O (n)

complexity multiple change-point [...] Read more.

In this paper, we study the generalized nearly isotonic optimization (GNIO) model and its dynamic programming solution (GNIO-DP). We introduce randomness into the GNIO-DP algorithm, enabling its first application to change-point detection and resulting in an

O (n)

complexity multiple change-point detection method. At the same time, we provide the theoretical properties of the change-point detection and prove the reliability and effectiveness of the GNIO-DP algorithm for this task. The simulation results show that our method has strong change-point detection ability. Compared with traditional methods, our method is faster in most scenarios. Full article

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

► Show Figures

Figure 1

40 pages, 8341 KB

Open AccessArticle

Explaining Driver Behavior in Sim Racing with Shannon Entropy and LLM Feedback

by Tomaz Nunes, Morsinaldo Medeiros, Marianne Silva, João Carlos N. Bittencourt, Daniel G. Costa and Ivanovitch Silva

Entropy 2026, 28(9), 960; https://doi.org/10.3390/e28090960 - 27 Aug 2026

Abstract

In some scenarios, motorsport simulators have been used to enable the controlled acquisition of dense telemetry with high similarity to real-world data, reducing cost when assessing driving performance. However, although popular, performance analyses traditionally treat human control as deterministic and overlook the stochasticity [...] Read more.

In some scenarios, motorsport simulators have been used to enable the controlled acquisition of dense telemetry with high similarity to real-world data, reducing cost when assessing driving performance. However, although popular, performance analyses traditionally treat human control as deterministic and overlook the stochasticity of driving behavior. In fact, existing coaching methods which improve driving performance have to deal with two distinct outcomes: a driver who restructures his race control strategy and a driver who merely repeats it faster. This article presents a Behavior-First framework for interpretable driver behavior analysis that separates them. We characterize control signals with two information-theoretic descriptors: Jensen–Shannon divergence, which quantifies distributional distance from a proficiency-matched reference and whose square root satisfies the triangle inequality, and Permutation Entropy to measure the ordinal complexity of the input sequence. A deterministic, physics-informed heuristic layer then identifies kinematic performance gaps and emits structured tokens that a Large Language Model translates into natural-language coaching narratives. We evaluated the framework in an exploratory case study. The three beginners who received generated coaching messages and the single uncoached comparison participant exhibited different lap-time and information-theoretic trajectories. Because the groups were small and non-randomized, these observations describe within-driver evolution and do not estimate a causal coaching effect. Full article

(This article belongs to the Section Complexity)

► Show Figures

Figure 1

28 pages, 16070 KB

Open AccessArticle

Colorization Algorithm for γ-Photon Flow Field Images Based on the HSCN Model

by Hui Xiao, Liying Hou and Jiantang Liu

Entropy 2026, 28(9), 959; https://doi.org/10.3390/e28090959 - 27 Aug 2026

Abstract

γ-photon tomography provides a non-contact approach for reconstructing and visualizing flow-field parameters. However, the resulting grayscale images often exhibit blurred boundaries and weak texture features, causing conventional colorization methods such as DeOldify to produce cross-region color diffusion and boundary color overflow. To address [...] Read more.

γ-photon tomography provides a non-contact approach for reconstructing and visualizing flow-field parameters. However, the resulting grayscale images often exhibit blurred boundaries and weak texture features, causing conventional colorization methods such as DeOldify to produce cross-region color diffusion and boundary color overflow. To address this, this paper proposes a γ-photon flow-field image colorization algorithm based on the Hybrid Swin Colorization Network (HSCN). A hybrid dual-stream encoder composed of a Swin Transformer semantic stream and a central difference convolution (CDC) gradient branch is combined with cross-stage gradient injection and a spatially gated adaptive fusion mechanism to enhance the perception of high-frequency structures at flow-field boundaries and suppress color overflow. The effectiveness of the algorithm is evaluated in terms of colorization quality and flow-field temperature-parameter inversion using γ-photon flow-field images of two CFD-simulated flow patterns, a large-scale vortical wake and a horizontal wake. The proposed method achieves PSNR, SSIM, FID, and MAE values of 38.7422, 0.9372, 10.7344, and 0.0085, respectively. Compared with DeOldify, PSNR and SSIM are improved by 24.30% and 11.89%, while FID and MAE are reduced by 42.98% and 60.47%, respectively. In addition, HSCN achieved a MAPE of 12.65% across 15 boundary and temperature-transition locations in three representative samples, compared with 31.24% for DeOldify and 28.70% for DDColor. Full article

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

► Show Figures

Figure 1

21 pages, 511 KB

Open AccessArticle

Kernel-Weighted Aggregation for Hyperparameter-Free Quantum Federated Learning

by Rui Huang

Entropy 2026, 28(9), 958; https://doi.org/10.3390/e28090958 - 26 Aug 2026

Abstract

Quantum federated learning (QFL) enables collaborative training of variational quantum circuits across decentralized clients, but client drift under non-IID data distributions degrades standard Federated Averaging (FedAvg). Existing aggregation methods either introduce tunable hyperparameters that depend on unknown data heterogeneity or incur structural costs [...] Read more.

Quantum federated learning (QFL) enables collaborative training of variational quantum circuits across decentralized clients, but client drift under non-IID data distributions degrades standard Federated Averaging (FedAvg). Existing aggregation methods either introduce tunable hyperparameters that depend on unknown data heterogeneity or incur structural costs such as order-dependent error propagation. We propose Kernel-Weighted Aggregation (KWA), a hyperparameter-free aggregation strategy that constructs a non-parametric kernel density estimate over the received client parameter vectors at each communication round. Clients in high-density regions receive greater voting power, and isolated clients are automatically attenuated. The kernel bandwidth is the median of all pairwise squared distances. We evaluate KWA against six state-of-the-art QFL methods and five classical robust baselines on five binary MNIST digit-pair tasks under IID, Dir(

0.5

), and Dir(

0.1

) distributions with a unified four-qubit benchmark. Under IID data, KWA recovers the performance of FedAvg. Under Dir(

0.5

), KWA attains the highest average accuracy (

0.7120

), followed by FedAvg (

0.7093

). The margin at

N = 4

clients is not statistically significant. It grows to

+ 0.023

at

N = 16

in the client scaling experiment. Under this distribution, the five classical robust baselines also rank below FedAvg. Under Dir(

0.1

), KWA exceeds FedAvg by

0.028

on average, with a pooled paired t-test

p = 0.068

. Larger-circuit and three-class experiments show that the advantage does not yet transfer consistently beyond the four-qubit binary setting. KWA thus provides a hyperparameter-free aggregation strategy that recovers FedAvg under IID conditions, adapts to client drift, and requires no manually tuned hyperparameters. Full article

(This article belongs to the Special Issue Recent Advances in Quantum Machine Learning)

► Show Figures

Figure 1

27 pages, 4461 KB

Open AccessArticle

Intent-Conditioned Diffusion Trajectory Prediction for Proactive Lane-Change Risk Assessment

by Lijing Ma, Shaofei Zhang, Wei Zhang, Jiacheng Yin and Yilong Wu

Entropy 2026, 28(9), 957; https://doi.org/10.3390/e28090957 - 26 Aug 2026

Abstract

Proactive lane-change risk assessment requires estimating whether an intended maneuver will lead to unsafe interactions before the maneuver is completed. This is challenging in naturalistic driving data because actual crashes are extremely rare, and binary collision labels provide little discriminative information for learning [...] Read more.

Proactive lane-change risk assessment requires estimating whether an intended maneuver will lead to unsafe interactions before the maneuver is completed. This is challenging in naturalistic driving data because actual crashes are extremely rare, and binary collision labels provide little discriminative information for learning risk. We propose IntentDiff, an intent-conditioned diffusion framework for proactive lane-change risk assessment. The framework uses predicted future trajectories as the basis for risk estimation. A vectorized scene context learning module combines a VectorNet backbone with a Vector Quantized Variational Autoencoder (VQ-VAE) to map agent–map interactions into discrete intent codes. These codes organize complex traffic situations into interpretable intent prototypes and provide semantic guidance for trajectory generation. Conditioned on the learned intent code, a diffusion model generates kinematically consistent multimodal trajectories of the target vehicle. On the forecast trajectories, Monte Carlo rear-end risk is evaluated against the four bounding vehicles and fused into a Lane-Change Risk Index (LCRI). On the highD dataset, the framework attains an average displacement error of 0.42 m over a 5-s horizon. The forecast-based LCRI agrees closely with the index computed from realized future trajectories, indicating that most high-risk lane changes can be identified before the maneuver is completed. Grouping LCRI by intent code further reveals systematic variation in risk across lane-change maneuvers, suggesting that the learned codebook captures risk-relevant interaction patterns in addition to maneuver semantics. Full article

(This article belongs to the Section Multidisciplinary Applications)

► Show Figures

Figure 1

34 pages, 7431 KB

Open AccessArticle

The Free Energy Principle and Free Markets

by Karl Friston, Johan Medrano and Tim Verbelen

Entropy 2026, 28(9), 956; https://doi.org/10.3390/e28090956 - 25 Aug 2026

Abstract

We apply the free energy principle to free markets by treating the Market as a random dynamical system with an attracting set, i.e., some characteristic states. This licenses a normal form for stochastic dynamics that inherits from the Helmholtz–Hodge decomposition. Equipped with this [...] Read more.

We apply the free energy principle to free markets by treating the Market as a random dynamical system with an attracting set, i.e., some characteristic states. This licenses a normal form for stochastic dynamics that inherits from the Helmholtz–Hodge decomposition. Equipped with this functional form—and a suitable parameterization—one can create a generative model of fluctuations in the value of assets and accompanying indicator variables. This affords the opportunity for prospective (ex ante) prediction, scenario modelling and forecasting that could, in principle, be applied to any complex dynamical system exhibiting stochastic chaos. Here, we illustrate the application to portfolio management—in the context of financial services—and use the (posterior) predictive densities over future paths to evaluate the expected free energy that underwrites active inference. In this application, active inference reduces to risk-sensitive control, which can be used to model the optimal decision-making of an agent or investor. In this setting, an investor is characterized by their prior preferences for a high rate of return under drawdown constraints. Using numerical studies and historical financial data, we quantify the improvement in portfolio management, relative to baseline policies. Full article

(This article belongs to the Section Statistical Physics)

► Show Figures

Figure 1

26 pages, 5235 KB

Open AccessArticle

Performance Evaluation of Bayesian Network Learning Algorithms in Structural Equation Modeling: A Simulation Study

by Tugay Karadag

Entropy 2026, 28(9), 955; https://doi.org/10.3390/e28090955 - 25 Aug 2026

Abstract

Bayesian Network (BN) learning algorithms may exhibit substantially different performance across graph structures, sample sizes, and evaluation criteria. Comparative evidence on BN learning under structurally validated conditions remains limited. This study evaluates BN learning algorithms for causal discovery through a simulation-based framework that [...] Read more.

Bayesian Network (BN) learning algorithms may exhibit substantially different performance across graph structures, sample sizes, and evaluation criteria. Comparative evidence on BN learning under structurally validated conditions remains limited. This study evaluates BN learning algorithms for causal discovery through a simulation-based framework that integrates multiple graph structures, sample sizes, and structural equation modeling (SEM)-based validation within a common design. Fourteen constraint-based, score-based, and hybrid algorithms were examined across five randomly generated directed acyclic graphs (DAGs) containing latent constructs and four sample sizes (n = 200, 500, 1000, and 2500). For each DAG–sample size combination, 1000 datasets were generated and validated using SEM, yielding 20,000 accepted datasets. Performance was assessed primarily by Matthews correlation coefficient (MCC), supported by directed structural Hamming distance (SHD) and F1. The results reveal substantial variation across DAG structures, sample sizes, and evaluation metrics, with mean MCC ranging from −0.43 to 0.65. Peter–Clark Stable achieved the strongest performance in several conditions, whereas hybrid algorithms were frequently among the weaker performers. Increasing sample size did not produce uniform performance gains, and no algorithm or algorithm class consistently dominated across all settings. These findings show that BN structure learning performance is strongly structure- and condition-dependent and support evaluation across multiple controlled DAG configurations. Full article

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

► Show Figures

Figure 1

26 pages, 7181 KB

Open AccessArticle

Numerical Investigation of Downstream-Shaft Aeration and Air-Pocket Evolution in a Navigation-Lock Valve

by Tingqiang Xie, Zhonghua Li, Xiujun Yan, Jun Deng and Duo Xu

Entropy 2026, 28(9), 954; https://doi.org/10.3390/e28090954 - 25 Aug 2026

Abstract

The filling-and-emptying valve and downstream shaft are crucial components of navigation-lock systems. Under insufficient downstream submergence, air can be drawn through the shaft and trapped in the post-valve culvert, altering the flow structure and compromising hydraulic stability. A three-dimensional Reynolds-averaged Navier–Stokes/volume-of-fluid model was [...] Read more.

The filling-and-emptying valve and downstream shaft are crucial components of navigation-lock systems. Under insufficient downstream submergence, air can be drawn through the shaft and trapped in the post-valve culvert, altering the flow structure and compromising hydraulic stability. A three-dimensional Reynolds-averaged Navier–Stokes/volume-of-fluid model was developed to investigate shaft aeration and entrapped-air-pocket evolution under varying inlet velocities and downstream-submergence depths. The aeration process comprises three stages: jet establishment, air-pocket formation, and air-pocket breakup and reorganization. Downstream-submergence depth determines whether a continuous air-intake pathway forms, whereas inlet velocity primarily controls aeration intensity and air-pocket persistence once the pathway is established. With decreasing submergence depth, the flow transitions successively from a water-sealed regime to a transition regime, a stable entrapped-air-pocket regime, and a strongly unsteady hydraulic-jump-like regime. For the present geometry and fixed valve opening, the transition from transient to sustained shaft aeration is identified within the downstream-submergence interval of h_w = 2–5 m. Combined analyses of the air-pocket volume per unit width, pressure response, vortex structures, and shear-layer characteristics indicate that enhanced jet-induced shear is closely associated with shaft aeration and air entrapment, while pressure fluctuations are closely coupled with air-pocket formation, persistence, breakup, and reorganization. Full article

(This article belongs to the Section Thermodynamics)

► Show Figures

Figure 1

12 pages, 883 KB

Open AccessArticle

Complex Operator Growth in Dissipative Quantum Systems

by Hikaru Wakaura and Taiki Tanimae

Entropy 2026, 28(9), 953; https://doi.org/10.3390/e28090953 - 24 Aug 2026

Abstract

The universal operator-growth hypothesis (OGH) states that, in a closed chaotic system, the Lanczos coefficients grow linearly,

b_{n} ≃ α n

. We ask how this structure is modified when the system is coupled to a Markovian environment, so that the generator [...] Read more.

The universal operator-growth hypothesis (OGH) states that, in a closed chaotic system, the Lanczos coefficients grow linearly,

b_{n} ≃ α n

. We ask how this structure is modified when the system is coupled to a Markovian environment, so that the generator becomes non-Hermitian. Applying the Arnoldi recursion to the vectorized Lindbladian in the infinite-temperature Wightman inner product, we organize the resulting pair of growth rates

α_{C} \equiv α_{R} + i α_{I}

—defined as effective slopes of the sub-diagonal and diagonal Arnoldi coefficients over a pre-registered fit window—around two statements whose logical status we delimit precisely. First, whenever the dissipator acts as

D = - 2 γ \hat{G}

with

\hat{G}

, a Hermitian grading (all dephasing-type baths), the diagonal obeys the identity

Re a_{n} = - 2 γ {⟨ \hat{G} ⟩}_{n}

: the imaginary rate measures how fast the growing operator accumulates weight in the dissipation channels. Second, we prove a conditional parity theorem: if the Hamiltonian, jump operators, and seeds can be made simultaneously real in some basis (an antiunitary condition), then

b_{n}

is even, and

Re a_{n}

is odd in

γ

exactly, so

α_{R}

is renormalized only at

O (γ^{2})

, and

α_{I} = 2 κ_{0} γ

follows from closed-system data alone. We exhibit a one-qubit Lindbladian that satisfies the often-assumed generator symmetry

G^{†} (γ) = - G (- γ)

yet violates parity (

b_{1} = | 1 - γ |

), showing that the extra condition is essential; all models studied here satisfy it bit-exactly. For large-q SYK, these ingredients predict

α_{C} = J - 2 i (q - 2) γ

, whose imaginary part is fixed solely by the interaction range; the first ladder step is exact, and the multi-step increments approach

q - 2

with system size (

1.92 \pm 0.04

at

N = 12

,

q = 4

). Under a common fit protocol, the closed-system rates saturate by

N = 10

(

α_{R} (0) \to 0.437

,

2 κ_{0} \to 0.224

). The imaginary rate is not an independent observable at leading order—its content is its sign, which resolves how the growing operator meets its environment (opposite for spin chains and SYK). Full article

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

► Show Figures

Figure 1

35 pages, 1136 KB

Open AccessArticle

Delay-Modulated Nonlinear Stochastic Mode Veering in Inertially Coupled Vibration Systems

by Lili Zhang, Zikun Han and Qiubao Wang

Entropy 2026, 28(9), 952; https://doi.org/10.3390/e28090952 - 24 Aug 2026

Abstract

Mode veering is a modal-interaction phenomenon found in vibration systems. For inertially coupled structures, the combined influence of coupling delay, nonlinear restoring force, and stochastic coupling perturbation remain insufficiently understood. This work analyzes an inertially coupled two-coordinate prototype in which a discrete delay, [...] Read more.

Mode veering is a modal-interaction phenomenon found in vibration systems. For inertially coupled structures, the combined influence of coupling delay, nonlinear restoring force, and stochastic coupling perturbation remain insufficiently understood. This work analyzes an inertially coupled two-coordinate prototype in which a discrete delay, a delayed cubic stiffness, and positive multiplicative stochastic modulation all enter through the same relative-coordinate coupling channel. We formulate the delayed linear spectrum through a quasi-polynomial characteristic equation. We also characterize the veering by the two positive-frequency characteristic-root branches descending from the mechanical modes. Coupling delay shifts the veering center, alters the minimum frequency gap, and moves the tracked rightmost roots toward the stability boundary. An analytical imaginary-axis-crossing criterion is derived to determine the delay-induced stability boundary of the deterministic linearized system, and the resulting boundary is independently validated by direct multi-start characteristic-root searches and Chebyshev-collocation approximation of the DDE generator. A fixed-reference modal-coordinate representation identifies the off-diagonal modal terms associated with branch exchange while retaining the full delayed characteristic equation. A first-harmonic treatment of the delayed cubic term can yield an amplitude-dependent nonlinear veering backbone. For the stochastic problem, frozen lognormal coupling samples and a time-dependent Ornstein–Uhlenbeck-driven multiplier are constructed from the same unit-mean positive lognormal marginal law. The former is used to quantify realization-wise spectral broadening, whereas the latter retains temporal correlation and is used to evaluate finite-time branch residence and pathwise delayed-work statistics. The pathwise energy balance reveals that the delayed relative-coordinate work rate is sign-indefinite. This provides a common energy-transfer mechanism through which delay, nonlinearity, and stochastic modulation reshape mode veering in the inertially coupled system. Full article

(This article belongs to the Section Complexity)

37 pages, 15811 KB

Open AccessPerspective

Self-Referential Introspection in Large Language Models: The Critical Threshold for Recursive Self-Improvement

by Jiang Zhang, Bing Yuan and Qian Zhang

Entropy 2026, 28(9), 951; https://doi.org/10.3390/e28090951 - 24 Aug 2026

Abstract

The pursuit of self-evolving AI raises a critical question: when is autonomous self-improvement sustainable rather than degenerative? Drawing an analogy to von Neumann’s complexity threshold for self-reproducing automata, we argue that sustainable recursive self-improvement in large language models (LLMs) requires a functional analogue: [...] Read more.

The pursuit of self-evolving AI raises a critical question: when is autonomous self-improvement sustainable rather than degenerative? Drawing an analogy to von Neumann’s complexity threshold for self-reproducing automata, we argue that sustainable recursive self-improvement in large language models (LLMs) requires a functional analogue: introspection—the system’s capacity to simulate its own operations and target modifications. Grounded in Kleene’s Second Recursion Theorem, we construct such introspective self-improvement programs and prove their key properties: completeness of self-modification, necessity of the reflective architecture, undecidability of improvement in general, and equivalence with Schmidhuber’s Gödel machine under a rewrite-equivalence notion, which transfers the global optimality guarantee. An empirical review, organized around these functional criteria, suggests that current LLMs exhibit only quasi-introspection.The available evidence does not establish complete introspection in the formal sense developed here, while pointing to several candidate structural bottlenecks, including incomplete self-access, feedforward processing, and limited computational depth. We outline architectural paths toward the threshold and discuss the safety implications of crossing it. Full article

(This article belongs to the Special Issue Complexity of AI)

► Show Figures

Figure 1

Journal Description

Entropy

Latest Articles

Journal Menu

Journal Browser

Highly Accessed Articles

Latest Books

E-Mail Alert

News

Topics

Conferences

Special Issues

Topical Collections

Further Information

Guidelines

MDPI Initiatives

Follow MDPI