Entropy

26 pages, 381 KiB

Open AccessFeature PaperArticle

Bild Conception of Scientific Theory Structuring in Classical and Quantum Physics: From Hertz and Boltzmann to Schrödinger and De Broglie

by Andrei Khrennikov

Entropy 2023, 25(11), 1565; https://doi.org/10.3390/e25111565 - 20 Nov 2023

Cited by 1 | Viewed by 842

Abstract

We start with a methodological analysis of the notion of scientific theory and its interrelation with reality. This analysis is based on the works of Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviews of D’Agostino). Following Helmholtz, Hertz established the “Bild conception” for scientific [...] Read more.

We start with a methodological analysis of the notion of scientific theory and its interrelation with reality. This analysis is based on the works of Helmholtz, Hertz, Boltzmann, and Schrödinger (and reviews of D’Agostino). Following Helmholtz, Hertz established the “Bild conception” for scientific theories. Here, “Bild” (“picture”) carries the meaning “model” (mathematical). The main aim of natural sciences is construction of the causal theoretical models (CTMs) of natural phenomena. Hertz claimed that a CTM cannot be designed solely on the basis of observational data; it typically contains hidden quantities. Experimental data can be described by an observational model (OM), often based on the price of acausality. CTM-OM interrelation can be tricky. Schrödinger used the Bild concept to create a CTM for quantum mechanics (QM), and QM was treated as OM. We follow him and suggest a special CTM for QM, so-called prequantum classical statistical field theory (PCSFT). QM can be considered as a PCSFT image, but not as straightforward as in Bell’s model with hidden variables. The common interpretation of the violation of the Bell inequality is criticized from the perspective of the two-level structuring of scientific theories. Such critical analysis of von Neumann and Bell no-go theorems for hidden variables was performed already by De Broglie (and Lochak) in the 1970s. The Bild approach is applied to the two-level CTM-OM modeling of Brownian motion: the overdamped regime corresponds to OM. In classical mechanics, CTM=OM; on the one hand, this is very convenient; on the other hand, this exceptional coincidence blurred the general CTM-OM structuring of scientific theories. We briefly discuss ontic–epistemic structuring of scientific theories (Primas–Atmanspacher) and its relation to the Bild concept. Interestingly, Atmanspacher as well as Hertz claim that even classical physical theories should be presented on the basic of two-level structuring. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering II)

37 pages, 1823 KiB

Open AccessFeature PaperReview

Percolation Theories for Quantum Networks

by Xiangyi Meng, Xinqi Hu, Yu Tian, Gaogao Dong, Renaud Lambiotte, Jianxi Gao and Shlomo Havlin

Entropy 2023, 25(11), 1564; https://doi.org/10.3390/e25111564 - 20 Nov 2023

Cited by 2 | Viewed by 1620

Abstract

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively [...] Read more.

Quantum networks have experienced rapid advancements in both theoretical and experimental domains over the last decade, making it increasingly important to understand their large-scale features from the viewpoint of statistical physics. This review paper discusses a fundamental question: how can entanglement be effectively and indirectly (e.g., through intermediate nodes) distributed between distant nodes in an imperfect quantum network, where the connections are only partially entangled and subject to quantum noise? We survey recent studies addressing this issue by drawing exact or approximate mappings to percolation theory, a branch of statistical physics centered on network connectivity. Notably, we show that the classical percolation frameworks do not uniquely define the network’s indirect connectivity. This realization leads to the emergence of an alternative theory called “concurrence percolation”, which uncovers a previously unrecognized quantum advantage that emerges at large scales, suggesting that quantum networks are more resilient than initially assumed within classical percolation contexts, offering refreshing insights into future quantum network design. Full article

(This article belongs to the Special Issue Classical and Quantum Networks: Theory, Modeling and Optimization)

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12 pages, 2228 KiB

Open AccessArticle

Machine Learning Approach to Analyze the Heavy Quark Diffusion Coefficient in Relativistic Heavy Ion Collisions

by Rui Guo, Yonghui Li and Baoyi Chen

Entropy 2023, 25(11), 1563; https://doi.org/10.3390/e25111563 - 20 Nov 2023

Cited by 1 | Viewed by 804

Abstract

The diffusion coefficient of heavy quarks in a deconfined medium is examined in this research using a deep convolutional neural network (CNN) that is trained with data from relativistic heavy ion collisions involving heavy flavor hadrons. The CNN is trained using observables such [...] Read more.

The diffusion coefficient of heavy quarks in a deconfined medium is examined in this research using a deep convolutional neural network (CNN) that is trained with data from relativistic heavy ion collisions involving heavy flavor hadrons. The CNN is trained using observables such as the nuclear modification factor

R_{A A}

and the elliptic flow

v_{2}

of non-prompt

J / ψ

from the B-hadron decay in different centralities, where B meson evolutions are calculated using the Langevin equation and the instantaneous coalescence model. The CNN outputs the parameters, thereby characterizing the temperature and momentum dependence of the heavy quark diffusion coefficient. By inputting the experimental data of the non-prompt

J / ψ

(R_{A A}, v_{2})

from various collision centralities into multiple channels of a well-trained network, we derive the values of the diffusion coefficient parameters. Additionally, we evaluate the uncertainty in determining the diffusion coefficient by taking into account the uncertainties present in the experimental data

(R_{A A}, v_{2})

, which serve as inputs to the deep neural network. Full article

(This article belongs to the Section Statistical Physics)

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20 pages, 3572 KiB

Open AccessArticle

Multiscale Cumulative Residual Dispersion Entropy with Applications to Cardiovascular Signals

by Youngjun Kim and Young-Seok Choi

Entropy 2023, 25(11), 1562; https://doi.org/10.3390/e25111562 - 20 Nov 2023

Viewed by 953

Abstract

Heart rate variability (HRV) is used as an index reflecting the adaptability of the autonomic nervous system to external stimuli and can be used to detect various heart diseases. Since HRVs are the time series signal with nonlinear property, entropy has been an [...] Read more.

Heart rate variability (HRV) is used as an index reflecting the adaptability of the autonomic nervous system to external stimuli and can be used to detect various heart diseases. Since HRVs are the time series signal with nonlinear property, entropy has been an attractive analysis method. Among the various entropy methods, dispersion entropy (DE) has been preferred due to its ability to quantify the time series’ underlying complexity with low computational cost. However, the order between patterns is not considered in the probability distribution of dispersion patterns for computing the DE value. Here, a multiscale cumulative residual dispersion entropy (MCRDE), which employs a cumulative residual entropy and DE estimation in multiple temporal scales, is presented. Thus, a generalized and fast estimation of complexity in temporal structures is inherited in the proposed MCRDE. To verify the performance of the proposed MCRDE, the complexity of inter-beat interval obtained from ECG signals of congestive heart failure (CHF), atrial fibrillation (AF), and the healthy group was compared. The experimental results show that MCRDE is more capable of quantifying physiological conditions than preceding multiscale entropy methods in that MCRDE achieves more statistically significant cases in terms of p-value from the Mann–Whitney test. Full article

(This article belongs to the Special Issue Entropy in Biomedical Engineering II)

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9 pages, 1127 KiB

Open AccessArticle

Non-Linear Measures of Postural Control in Response to Painful and Non-Painful Visual Stimuli

by Alexandre Vonesch, Cassandre Duhot, Thierry Lelard, Guillaume Léonard, Michalina Błażkiewicz and Harold Mouras

Entropy 2023, 25(11), 1561; https://doi.org/10.3390/e25111561 - 19 Nov 2023

Viewed by 1007

Abstract

Over the past decade, researchers have focused on studying the functional context of perceiving painful stimuli, particularly concerning the posturographic correlates of emotional processing. The aim of this study was to investigate the differential modulation of non-linear measures characterizing postural control in the [...] Read more.

Over the past decade, researchers have focused on studying the functional context of perceiving painful stimuli, particularly concerning the posturographic correlates of emotional processing. The aim of this study was to investigate the differential modulation of non-linear measures characterizing postural control in the context of perceiving painful stimuli. The study involved 36 healthy young participants who, while standing, viewed images depicting feet and hands in painful or non-painful situations, both actively (by imagining themselves affected by the situation) and passively. For Center of Pressure (COP) displacement, three non-linear measures (Sample Entropy, Fractal Dimension, and Lyapunov exponent) were calculated. The results suggest lower values of FD and LyE in response to active stimulation compared to those recorded for passive stimulation. Above all, our results pledge for the usefulness of the Lyapunov exponent for assessing postural modulation dynamics in response to painful stimuli perception. The feasibility of this calculation could provide an interesting insight in the collection of biomarkers related to postural correlates of emotional processes and their modulation in neurological disease where socio-affective functions can be often impaired before cognitive ones. Full article

(This article belongs to the Section Multidisciplinary Applications)

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18 pages, 2447 KiB

Open AccessArticle

How Do Heterogeneous Networks Affect a Firm’s Innovation Performance? A Research Analysis Based on Clustering and Classification

by Liping Zhang, Hanhui Qiu, Jinyi Chen, Wenhao Zhou and Hailin Li

Entropy 2023, 25(11), 1560; https://doi.org/10.3390/e25111560 - 19 Nov 2023

Viewed by 1172

Abstract

Based on authorized patents of China’s artificial intelligence industry from 2013 to 2022, this paper constructs an Industry–University–Research institution (IUR) collaboration network and an Inter-Firm (IF) collaboration network and used the entropy weight method to take both the quantity and quality of patents [...] Read more.

Based on authorized patents of China’s artificial intelligence industry from 2013 to 2022, this paper constructs an Industry–University–Research institution (IUR) collaboration network and an Inter-Firm (IF) collaboration network and used the entropy weight method to take both the quantity and quality of patents into account to calculate the innovation performance of firms. Through the hierarchical clustering algorithm and classification and regression trees (CART) algorithm, in-depth analysis has been conducted on the intricate non-linear influence mechanisms between multiple variables and a firm’s innovation performance. The findings indicate the following: (1) Based on the network centrality (NC), structural hole (SH), collaboration breadth (CB), and collaboration depth (CD) of both IUR and IF collaboration networks, two types of focal firms are identified. (2) For different types of focal firms, the combinations of network characteristics affecting their innovation performance are various. (3) In the IUR collaboration network, focal firms with a wide range of heterogeneous collaborative partners can obtain high innovation performance. However, focal firms in the IF collaboration network can achieve the same aim by maintaining deep collaboration with other focal firms. This paper not only helps firms make scientific decisions for development but also provides valuable suggestions for government policymakers. Full article

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11 pages, 9899 KiB

Open AccessArticle

Evaluating the Adiabatic Invariants in Magnetized Plasmas Using a Classical Ehrenfest Theorem

by Abiam Tamburrini, Sergio Davis and Pablo S. Moya

Entropy 2023, 25(11), 1559; https://doi.org/10.3390/e25111559 - 18 Nov 2023

Viewed by 835

Abstract

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov’s. We introduce the Ehrenfest procedure as an [...] Read more.

In this article, we address the reliance on probability density functions to obtain macroscopic properties in systems with multiple degrees of freedom as plasmas, and the limitations of expensive techniques for solving Equations such as Vlasov’s. We introduce the Ehrenfest procedure as an alternative tool that promises to address these challenges more efficiently. Based on the conjugate variable theorem and the well-known fluctuation-dissipation theorem, this procedure offers a less expensive way of deriving time evolution Equations for macroscopic properties in systems far from equilibrium. We investigate the application of the Ehrenfest procedure for the study of adiabatic invariants in magnetized plasmas. We consider charged particles trapped in a dipole magnetic field and apply the procedure to the study of adiabatic invariants in magnetized plasmas and derive Equations for the magnetic moment, longitudinal invariant, and magnetic flux. We validate our theoretical predictions using a test particle simulation, showing good agreement between theory and numerical results for these observables. Although we observed small differences due to time scales and simulation limitations, our research supports the utility of the Ehrenfest procedure for understanding and modeling the behavior of particles in magnetized plasmas. We conclude that this procedure provides a powerful tool for the study of dynamical systems and statistical mechanics out of equilibrium, and opens perspectives for applications in other systems with probabilistic continuity. Full article

(This article belongs to the Special Issue 180th Anniversary of Ludwig Boltzmann)

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21 pages, 3707 KiB

Open AccessArticle

Modelling and Research on Intuitionistic Fuzzy Goal-Based Attack and Defence Game for Infrastructure Networks

by Zhe Li, Jin Liu, Yibo Dong, Jiaqi Ren and Weili Li

Entropy 2023, 25(11), 1558; https://doi.org/10.3390/e25111558 - 18 Nov 2023

Viewed by 931

Abstract

Network attack and defence games are gradually becoming a new approach through which to study the protection of infrastructure networks such as power grids and transportation networks. Uncertainty factors, such as the subjective decision preferences of attackers and defenders, are not considered in [...] Read more.

Network attack and defence games are gradually becoming a new approach through which to study the protection of infrastructure networks such as power grids and transportation networks. Uncertainty factors, such as the subjective decision preferences of attackers and defenders, are not considered in existing attack and defence game studies for infrastructure networks. In this paper, we introduce, respectively, the attacker’s and defender’s expectation value, rejection value, and hesitation degree of the target, as well as construct an intuitionistic fuzzy goal-based attack and defence game model for infrastructure networks that are based on the maximum connectivity slice size, which is a network performance index. The intuitionistic fuzzy two-player, zero-sum game model is converted into a linear programming problem for solving, and the results are analysed to verify the applicability and feasibility of the model proposed in this paper. Furthermore, different situations, such as single-round games and multi-round repeated games, are also considered. The experimental results show that, when attacking the network, the attacker rarely attacks the nodes with higher importance in the network, but instead pays more attention to the nodes that are not prominent in the network neutrality and median; meanwhile, the defender is more inclined to protect the more important nodes in the network to ensure the normal performance of the network. Full article

(This article belongs to the Special Issue Uncertainty Management in Intelligent Information Processing)

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9 pages, 597 KiB

Open AccessFeature PaperArticle

Topological Dimensions from Disorder and Quantum Mechanics?

by Ivan Horváth and Peter Markoš

Entropy 2023, 25(11), 1557; https://doi.org/10.3390/e25111557 - 17 Nov 2023

Cited by 4 | Viewed by 790

Abstract

We have recently shown that the critical Anderson electron in

D = 3

dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension

d_{IR} \approx 8 / 3

. Here, we inquire about the dimensional substructure involved. We partition space [...] Read more.

We have recently shown that the critical Anderson electron in

D = 3

dimensions effectively occupies a spatial region of the infrared (IR) scaling dimension

d_{IR} \approx 8 / 3

. Here, we inquire about the dimensional substructure involved. We partition space into regions of equal quantum occurrence probabilities, such that the points comprising a region are of similar relevance, and calculate the IR scaling dimension d of each. This allows us to infer the probability density

p (d)

for dimension d to be accessed by the electron. We find that

p (d)

has a strong peak at d very close to two. In fact, our data suggest that

p (d)

is non-zero on the interval

[d_{\min}, d_{\max}] \approx [4 / 3, 8 / 3]

and may develop a discrete part (

δ

-function) at

d = 2

in the infinite-volume limit. The latter invokes the possibility that a combination of quantum mechanics and pure disorder can lead to the emergence of integer (topological) dimensions. Although

d_{IR}

is based on effective counting, of which

p (d)

has no a priori knowledge,

d_{IR} \geq d_{\max}

is an exact feature of the ensuing formalism. A possible connection of our results to the recent findings of

d_{IR} \approx 2

in Dirac near-zero modes of thermal quantum chromodynamics is emphasized. Full article

(This article belongs to the Special Issue Recent Advances in the Theory of Disordered Systems)

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18 pages, 1211 KiB

Open AccessArticle

Security of the Decoy-State BB84 Protocol with Imperfect State Preparation

by Aleksei Reutov, Andrey Tayduganov, Vladimir Mayboroda and Oleg Fat’yanov

Entropy 2023, 25(11), 1556; https://doi.org/10.3390/e25111556 - 17 Nov 2023

Viewed by 963

Abstract

The quantum key distribution (QKD) allows two remote users to share a common information-theoretic secure secret key. In order to guarantee the security of a practical QKD implementation, the physical system has to be fully characterized and all deviations from the ideal protocol [...] Read more.

The quantum key distribution (QKD) allows two remote users to share a common information-theoretic secure secret key. In order to guarantee the security of a practical QKD implementation, the physical system has to be fully characterized and all deviations from the ideal protocol due to various imperfections of realistic devices have to be taken into account in the security proof. In this work, we study the security of the efficient decoy-state BB84 QKD protocol in the presence of the source flaws, caused by imperfect intensity and polarization modulation. We investigate the non-Poissonian photon-number statistics due to coherent-state intensity fluctuations and the basis-dependence of the source due to non-ideal polarization state preparation. The analysis is supported by the experimental characterization of intensity and phase distributions. Full article

(This article belongs to the Special Issue Quantum Information: From Fundamental Aspects to Practical Applications)

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11 pages, 992 KiB

Open AccessArticle

Connectivity of Random Geometric Hypergraphs

by Henry-Louis de Kergorlay and Desmond J. Higham

Entropy 2023, 25(11), 1555; https://doi.org/10.3390/e25111555 - 17 Nov 2023

Viewed by 818

Abstract

We consider a random geometric hypergraph model based on an underlying bipartite graph. Nodes and hyperedges are sampled uniformly in a domain, and a node is assigned to those hyperedges that lie within a certain radius. From a modelling perspective, we explain how [...] Read more.

We consider a random geometric hypergraph model based on an underlying bipartite graph. Nodes and hyperedges are sampled uniformly in a domain, and a node is assigned to those hyperedges that lie within a certain radius. From a modelling perspective, we explain how the model captures higher-order connections that arise in real data sets. Our main contribution is to study the connectivity properties of the model. In an asymptotic limit where the number of nodes and hyperedges grow in tandem, we give a condition on the radius that guarantees connectivity. Full article

(This article belongs to the Special Issue Models, Topology and Inference of Multilayer and Higher-Order Networks)

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18 pages, 952 KiB

Open AccessArticle

Causal Factor Disentanglement for Few-Shot Domain Adaptation in Video Prediction

by Nathan Cornille, Katrien Laenen, Jingyuan Sun and Marie-Francine Moens

Entropy 2023, 25(11), 1554; https://doi.org/10.3390/e25111554 - 17 Nov 2023

Viewed by 966

Abstract

An important challenge in machine learning is performing with accuracy when few training samples are available from the target distribution. If a large number of training samples from a related distribution are available, transfer learning can be used to improve the performance. This [...] Read more.

An important challenge in machine learning is performing with accuracy when few training samples are available from the target distribution. If a large number of training samples from a related distribution are available, transfer learning can be used to improve the performance. This paper investigates how to do transfer learning more effectively if the source and target distributions are related through a Sparse Mechanism Shift for the application of next-frame prediction. We create Sparse Mechanism Shift-TempoRal Intervened Sequences (SMS-TRIS), a benchmark to evaluate transfer learning for next-frame prediction derived from the TRIS datasets. We then propose to exploit the Sparse Mechanism Shift property of the distribution shift by disentangling the model parameters with regard to the true causal mechanisms underlying the data. We use the Causal Identifiability from TempoRal Intervened Sequences (CITRIS) model to achieve this disentanglement via causal representation learning. We show that encouraging disentanglement with the CITRIS extensions can improve performance, but their effectiveness varies depending on the dataset and backbone used. We find that it is effective only when encouraging disentanglement actually succeeds in increasing disentanglement. We also show that an alternative method designed for domain adaptation does not help, indicating the challenging nature of the SMS-TRIS benchmark. Full article

(This article belongs to the Special Issue Causality and Complex Systems)

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11 pages, 1626 KiB

Open AccessArticle

Evolution Dynamics Model of Private Enterprises under Simultaneous and Sequential Innovation Decisions

by Chi Zhang, Yutong Wang and Tingqiang Chen

Entropy 2023, 25(11), 1553; https://doi.org/10.3390/e25111553 - 17 Nov 2023

Viewed by 617

Abstract

The innovation of private enterprises plays a crucial role. This study focuses on the impacts of market information asymmetry, the technology spillover effect, and the order of innovation research and development (R&D) decisions on the evolution of private enterprises’ innovation. This study constructs [...] Read more.

The innovation of private enterprises plays a crucial role. This study focuses on the impacts of market information asymmetry, the technology spillover effect, and the order of innovation research and development (R&D) decisions on the evolution of private enterprises’ innovation. This study constructs a dynamic model to analyze how the innovation decision-making order of private enterprises influences their profits and intertemporal innovation decision making. First, we derive the equilibrium point under sequential decisions and the stability of the system at the equilibrium point. Second, we investigate the impact of sequential and simultaneous innovation decisions on the evolution of the dynamic system and its economic implications. Finally, we study the evolutionary dynamics of the attractor with the rate of innovation adjustment and point to the existence of multiple equilibria. The results suggest that the speed of the innovation R&D cost change should be moderate, and the asynchronous updating of the innovation R&D strategy can prevent the system evolution from turning into chaos. These conclusions guide innovation policies. Full article

(This article belongs to the Special Issue Complexity, Entropy and the Physics of Information)

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12 pages, 478 KiB

Open AccessArticle

Efficient Quantum Private Comparison without Sharing a Key

by Jian Li, Fanting Che, Zhuo Wang and Anqi Fu

Entropy 2023, 25(11), 1552; https://doi.org/10.3390/e25111552 - 17 Nov 2023

Cited by 1 | Viewed by 778

Abstract

Quantum private comparison (QPC) allows at least two users to compare the equality of their secret information, for which the security is based on the properties of quantum mechanics. To improve the use of quantum resources and the efficiency of private comparison, a [...] Read more.

Quantum private comparison (QPC) allows at least two users to compare the equality of their secret information, for which the security is based on the properties of quantum mechanics. To improve the use of quantum resources and the efficiency of private comparison, a new QPC protocol based on GHZ-like states is proposed. The protocol adopts unitary operations to encode the secret information instead of performing quantum key distribution (QKD), which can reduce the amount of computation required to perform QKD and improve the utilization of quantum resources. The decoy photon technique used to detect channel eavesdropping ensures that the protocol is resistant to external attacks. The quantum efficiency of the protocol reaches 66%. Compared with many previous QPC schemes, the proposed protocol does not need to share a key and has advantages in quantum efficiency and quantum resources. Full article

(This article belongs to the Special Issue Quantum and Classical Physical Cryptography)

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14 pages, 551 KiB

Open AccessArticle

FLPP: A Federated-Learning-Based Scheme for Privacy Protection in Mobile Edge Computing

by Zhimo Cheng, Xinsheng Ji, Wei You, Yi Bai, Yunjie Chen and Xiaogang Qin

Entropy 2023, 25(11), 1551; https://doi.org/10.3390/e25111551 - 16 Nov 2023

Viewed by 920

Abstract

Data sharing and analyzing among different devices in mobile edge computing is valuable for social innovation and development. The limitation to the achievement of this goal is the data privacy risk. Therefore, existing studies mainly focus on enhancing the data privacy-protection capability. On [...] Read more.

Data sharing and analyzing among different devices in mobile edge computing is valuable for social innovation and development. The limitation to the achievement of this goal is the data privacy risk. Therefore, existing studies mainly focus on enhancing the data privacy-protection capability. On the one hand, direct data leakage is avoided through federated learning by converting raw data into model parameters for transmission. On the other hand, the security of federated learning is further strengthened by privacy-protection techniques to defend against inference attack. However, privacy-protection techniques may reduce the training accuracy of the data while improving the security. Particularly, trading off data security and accuracy is a major challenge in dynamic mobile edge computing scenarios. To address this issue, we propose a federated-learning-based privacy-protection scheme, FLPP. Then, we build a layered adaptive differential privacy model to dynamically adjust the privacy-protection level in different situations. Finally, we design a differential evolutionary algorithm to derive the most suitable privacy-protection policy for achieving the optimal overall performance. The simulation results show that FLPP has an advantage of 8∼34% in overall performance. This demonstrates that our scheme can enable data to be shared securely and accurately. Full article

(This article belongs to the Special Issue Methods in Artificial Intelligence and Information Processing II)

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15 pages, 2745 KiB

Open AccessArticle

Advancing Federated Learning through Verifiable Computations and Homomorphic Encryption

by Bingxue Zhang, Guangguang Lu, Pengpeng Qiu, Xumin Gui and Yang Shi

Entropy 2023, 25(11), 1550; https://doi.org/10.3390/e25111550 - 16 Nov 2023

Viewed by 1385

Abstract

Federated learning, as one of the three main technical routes for privacy computing, has been widely studied and applied in both academia and industry. However, malicious nodes may tamper with the algorithm execution process or submit false learning results, which directly affects the [...] Read more.

Federated learning, as one of the three main technical routes for privacy computing, has been widely studied and applied in both academia and industry. However, malicious nodes may tamper with the algorithm execution process or submit false learning results, which directly affects the performance of federated learning. In addition, learning nodes can easily obtain the global model. In practical applications, we would like to obtain the federated learning results only by the demand side. Unfortunately, no discussion on protecting the privacy of the global model is found in the existing research. As emerging cryptographic tools, the zero-knowledge virtual machine (ZKVM) and homomorphic encryption provide new ideas for the design of federated learning frameworks. We have introduced ZKVM for the first time, creating learning nodes as local computing provers. This provides execution integrity proofs for multi-class machine learning algorithms. Meanwhile, we discuss how to generate verifiable proofs for large-scale machine learning tasks under resource constraints. In addition, we implement the fully homomorphic encryption (FHE) scheme in ZKVM. We encrypt the model weights so that the federated learning nodes always collaborate in the ciphertext space. The real results can be obtained only after the demand side decrypts them using the private key. The innovativeness of this paper is demonstrated in the following aspects: 1. We introduce the ZKVM for the first time, which achieves zero-knowledge proofs (ZKP) for machine learning tasks with multiple classes and arbitrary scales. 2. We encrypt the global model, which protects the model privacy during local computation and transmission. 3. We propose and implement a new federated learning framework. We measure the verification costs under different federated learning rounds on the IRIS dataset. Despite the impact of homomorphic encryption on computational accuracy, the framework proposed in this paper achieves a satisfactory 90% model accuracy. Our framework is highly secure and is expected to further improve the overall efficiency as cryptographic tools continue to evolve. Full article

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

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31 pages, 9698 KiB

Open AccessArticle

Fault Diagnosis of Rotating Machinery Using Kernel Neighborhood Preserving Embedding and a Modified Sparse Bayesian Classification Model

by Lixin Lu, Weihao Wang, Dongdong Kong, Junjiang Zhu and Dongxing Chen

Entropy 2023, 25(11), 1549; https://doi.org/10.3390/e25111549 - 16 Nov 2023

Cited by 2 | Viewed by 808

Abstract

Fault diagnosis of rotating machinery plays an important role in modern industrial machines. In this paper, a modified sparse Bayesian classification model (i.e., Standard_SBC) is utilized to construct the fault diagnosis system of rotating machinery. The features are extracted and adopted as the [...] Read more.

Fault diagnosis of rotating machinery plays an important role in modern industrial machines. In this paper, a modified sparse Bayesian classification model (i.e., Standard_SBC) is utilized to construct the fault diagnosis system of rotating machinery. The features are extracted and adopted as the input of the SBC-based fault diagnosis system, and the kernel neighborhood preserving embedding (KNPE) is proposed to fuse the features. The effectiveness of the fault diagnosis system of rotating machinery based on KNPE and Standard_SBC is validated by utilizing two case studies: rolling bearing fault diagnosis and rotating shaft fault diagnosis. Experimental results show that base on the proposed KNPE, the feature fusion method shows superior performance. The accuracy of case1 and case2 is improved from 93.96% to 99.92% and 98.67% to 99.64%, respectively. To further prove the superiority of the KNPE feature fusion method, the kernel principal component analysis (KPCA) and relevance vector machine (RVM) are utilized, respectively. This study lays the foundation for the feature fusion and fault diagnosis of rotating machinery. Full article

(This article belongs to the Special Issue The Application of Information Theory in Fault Detection and Diagnosis)

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7 pages, 1225 KiB

Open AccessArticle

Experimental Demonstration of Secure Relay in Quantum Secure Direct Communication Network

by Min Wang, Wei Zhang, Jianxing Guo, Xiaotian Song and Guilu Long

Entropy 2023, 25(11), 1548; https://doi.org/10.3390/e25111548 - 16 Nov 2023

Viewed by 921

Abstract

Quantum secure direct communication (QSDC) offers a practical way to realize a quantum network which can transmit information securely and reliably. Practical quantum networks are hindered by the unavailability of quantum relays. To overcome this limitation, a proposal has been made to transmit [...] Read more.

Quantum secure direct communication (QSDC) offers a practical way to realize a quantum network which can transmit information securely and reliably. Practical quantum networks are hindered by the unavailability of quantum relays. To overcome this limitation, a proposal has been made to transmit the messages encrypted with classical cryptography, such as post-quantum algorithms, between intermediate nodes of the network, where encrypted messages in quantum states are read out in classical bits, and sent to the next node using QSDC. In this paper, we report a real-time demonstration of a computationally secure relay for a quantum secure direct communication network. We have chosen CRYSTALS-KYBER which has been standardized by the National Institute of Standards and Technology to encrypt the messages for transmission of the QSDC system. The quantum bit error rate of the relay system is typically below the security threshold. Our relay can support a QSDC communication rate of 2.5 kb/s within a 4 ms time delay. The experimental demonstration shows the feasibility of constructing a large-scale quantum network in the near future. Full article

(This article belongs to the Special Issue New Advances in Quantum Communication and Networks)

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18 pages, 932 KiB

Open AccessArticle

Anti-Jamming Communication Using Imitation Learning

by Zhanyang Zhou, Yingtao Niu, Boyu Wan and Wenhao Zhou

Entropy 2023, 25(11), 1547; https://doi.org/10.3390/e25111547 - 16 Nov 2023

Viewed by 834

Abstract

The communication reliability of wireless communication systems is threatened by malicious jammers. Aiming at the problem of reliable communication under malicious jamming, a large number of schemes have been proposed to mitigate the effects of malicious jamming by avoiding the blocking interference of [...] Read more.

The communication reliability of wireless communication systems is threatened by malicious jammers. Aiming at the problem of reliable communication under malicious jamming, a large number of schemes have been proposed to mitigate the effects of malicious jamming by avoiding the blocking interference of jammers. However, the existing anti-jamming schemes, such as fixed strategy, Reinforcement learning (RL), and deep Q network (DQN) have limited use of historical data, and most of them only pay attention to the current state changes and cannot gain experience from historical samples. In view of this, this manuscript proposes anti-jamming communication using imitation learning. Specifically, this manuscript addresses the problem of anti-jamming decisions for wireless communication in scenarios with malicious jamming and proposes an algorithm that consists of three steps: First, the heuristic-based Expert Trajectory Generation Algorithm is proposed as the expert strategy, which enables us to obtain the expert trajectory from historical samples. The trajectory mentioned in this algorithm represents the sequence of actions undertaken by the expert in various situations. Then obtaining a user strategy by imitating the expert strategy using an imitation learning neural network. Finally, adopting a functional user strategy for efficient and sequential anti-jamming decisions. Simulation results indicate that the proposed method outperforms the RL-based anti-jamming method and DQN-based anti-jamming method regarding solving continuous-state spectrum anti-jamming problems without causing “curse of dimensionality” and providing greater robustness against channel fading and noise as well as when the jamming pattern changes. Full article

(This article belongs to the Special Issue Information Theory in Emerging Wireless Communication Systems and Networks)

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17 pages, 14131 KiB

Open AccessArticle

Denoising Non-Stationary Signals via Dynamic Multivariate Complex Wavelet Thresholding

by Kim C. Raath, Katherine B. Ensor, Alena Crivello and David W. Scott

Entropy 2023, 25(11), 1546; https://doi.org/10.3390/e25111546 - 16 Nov 2023

Viewed by 815

Abstract

Over the past few years, we have seen an increased need to analyze the dynamically changing behaviors of economic and financial time series. These needs have led to significant demand for methods that denoise non-stationary time series across time and for specific investment [...] Read more.

Over the past few years, we have seen an increased need to analyze the dynamically changing behaviors of economic and financial time series. These needs have led to significant demand for methods that denoise non-stationary time series across time and for specific investment horizons (scales) and localized windows (blocks) of time. Wavelets have long been known to decompose non-stationary time series into their different components or scale pieces. Recent methods satisfying this demand first decompose the non-stationary time series using wavelet techniques and then apply a thresholding method to separate and capture the signal and noise components of the series. Traditionally, wavelet thresholding methods rely on the discrete wavelet transform (DWT), which is a static thresholding technique that may not capture the time series of the estimated variance in the additive noise process. We introduce a novel continuous wavelet transform (CWT) dynamically optimized multivariate thresholding method (

W a v e L_{2} E

). Applying this method, we are simultaneously able to separate and capture the signal and noise components while estimating the dynamic noise variance. Our method shows improved results when compared to well-known methods, especially for high-frequency signal-rich time series, typically observed in finance. Full article

(This article belongs to the Special Issue Robust Methods in Complex Scenarios and Data Visualization)

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18 pages, 733 KiB

Open AccessArticle

Complexity Reduction in Analyzing Independence between Statistical Randomness Tests Using Mutual Information

by Jorge Augusto Karell-Albo, Carlos Miguel Legón-Pérez, Raisa Socorro-Llanes, Omar Rojas and Guillermo Sosa-Gómez

Entropy 2023, 25(11), 1545; https://doi.org/10.3390/e25111545 - 15 Nov 2023

Viewed by 1356

Abstract

The advantages of using mutual information to evaluate the correlation between randomness tests have recently been demonstrated. However, it has been pointed out that the high complexity of this method limits its application in batteries with a greater number of tests. The main [...] Read more.

The advantages of using mutual information to evaluate the correlation between randomness tests have recently been demonstrated. However, it has been pointed out that the high complexity of this method limits its application in batteries with a greater number of tests. The main objective of this work is to reduce the complexity of the method based on mutual information for analyzing the independence between the statistical tests of randomness. The achieved complexity reduction is estimated theoretically and verified experimentally. A variant of the original method is proposed by modifying the step in which the significant values of the mutual information are determined. The correlation between the NIST battery tests was studied, and it was concluded that the modifications to the method do not significantly affect the ability to detect correlations. Due to the efficiency of the newly proposed method, its use is recommended to analyze other batteries of tests. Full article

(This article belongs to the Special Issue Information Theory in Multi-Agent Systems: Methods and Applications)

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21 pages, 1116 KiB

Open AccessArticle

Design of Closed-Loop Control Schemes Based on the GA-PID and GA-RBF-PID Algorithms for Brain Dynamic Modulation

by Chengxia Sun, Lijun Geng, Xian Liu and Qing Gao

Entropy 2023, 25(11), 1544; https://doi.org/10.3390/e25111544 - 15 Nov 2023

Viewed by 809

Abstract

Neurostimulation can be used to modulate brain dynamics of patients with neuropsychiatric disorders to make abnormal neural oscillations restore to normal. The control schemes proposed on the bases of neural computational models can predict the mechanism of neural oscillations induced by neurostimulation, and [...] Read more.

Neurostimulation can be used to modulate brain dynamics of patients with neuropsychiatric disorders to make abnormal neural oscillations restore to normal. The control schemes proposed on the bases of neural computational models can predict the mechanism of neural oscillations induced by neurostimulation, and then make clinical decisions that are suitable for the patient’s condition to ensure better treatment outcomes. The present work proposes two closed-loop control schemes based on the improved incremental proportional integral derivative (PID) algorithms to modulate brain dynamics simulated by Wendling-type coupled neural mass models. The introduction of the genetic algorithm (GA) in traditional incremental PID algorithm aims to overcome the disadvantage that the selection of control parameters depends on the designer’s experience, so as to ensure control accuracy. The introduction of the radial basis function (RBF) neural network aims to improve the dynamic performance and stability of the control scheme by adaptively adjusting control parameters. The simulation results show the high accuracy of the closed-loop control schemes based on GA-PID and GA-RBF-PID algorithms for modulation of brain dynamics, and also confirm the superiority of the scheme based on the GA-RBF-PID algorithm in terms of the dynamic performance and stability. This research of making hypotheses and predictions according to model data is expected to improve and perfect the equipment of early intervention and rehabilitation treatment for neuropsychiatric disorders in the biomedical engineering field. Full article

(This article belongs to the Section Entropy and Biology)

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14 pages, 298 KiB

Open AccessArticle

Bulk Operator Reconstruction in Topological Tensor Network and Generalized Free Fields

by Xiangdong Zeng and Ling-Yan Hung

Entropy 2023, 25(11), 1543; https://doi.org/10.3390/e25111543 - 15 Nov 2023

Cited by 2 | Viewed by 726

Abstract

In this paper, we study operator reconstruction in a class of holographic tensor networks describing renormalization group flows studied in arXiv:2210.12127. We study examples of 2D bulk holographic tensor networks constructed from Dijkgraaf–Witten theories and find that for both the

Z_{n}

group [...] Read more.

In this paper, we study operator reconstruction in a class of holographic tensor networks describing renormalization group flows studied in arXiv:2210.12127. We study examples of 2D bulk holographic tensor networks constructed from Dijkgraaf–Witten theories and find that for both the

Z_{n}

group and the

S_{3}

group, the number of bulk operators behaving like a generalized free field in the bulk scales as the order of the group. We also generalize our study to 3D bulks and find the same scaling for

Z_{n}

theories. However, there is no generalized free field when the bulk comes from more generic fusion categories such as the Fibonacci model. Full article

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25 pages, 3785 KiB

Open AccessReview

Enhancing Predictability Assessment: An Overview and Analysis of Predictability Measures for Time Series and Network Links

by Alexandra Bezbochina, Elizaveta Stavinova, Anton Kovantsev and Petr Chunaev

Entropy 2023, 25(11), 1542; https://doi.org/10.3390/e25111542 - 15 Nov 2023

Viewed by 939

Abstract

Driven by the variety of available measures intended to estimate predictability of diverse objects such as time series and network links, this paper presents a comprehensive overview of the existing literature in this domain. Our overview delves into predictability from two distinct perspectives: [...] Read more.

Driven by the variety of available measures intended to estimate predictability of diverse objects such as time series and network links, this paper presents a comprehensive overview of the existing literature in this domain. Our overview delves into predictability from two distinct perspectives: the intrinsic predictability, which represents a data property independent of the chosen forecasting model and serves as the highest achievable forecasting quality level, and the realized predictability, which represents a chosen quality metric for a specific pair of data and model. The reviewed measures are used to assess predictability across different objects, starting from time series (univariate, multivariate, and categorical) to network links. Through experiments, we establish a noticeable relationship between measures of realized and intrinsic predictability in both generated and real-world time series data (with the correlation coefficient being statistically significant at a 5% significance level). The discovered correlation in this research holds significant value for tasks related to evaluating time series complexity and their potential to be accurately predicted. Full article

(This article belongs to the Section Complexity)

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22 pages, 2701 KiB

Open AccessArticle

Bio-Inspired Intelligent Systems: Negotiations between Minimum Manifest Task Entropy and Maximum Latent System Entropy in Changing Environments

by Stephen Fox, Tapio Heikkilä, Eric Halbach and Samuli Soutukorva

Entropy 2023, 25(11), 1541; https://doi.org/10.3390/e25111541 - 14 Nov 2023

Viewed by 842

Abstract

In theoretical physics and theoretical neuroscience, increased intelligence is associated with increased entropy, which entails potential access to an increased number of states that could facilitate adaptive behavior. Potential to access a larger number of states is a latent entropy as it refers [...] Read more.

In theoretical physics and theoretical neuroscience, increased intelligence is associated with increased entropy, which entails potential access to an increased number of states that could facilitate adaptive behavior. Potential to access a larger number of states is a latent entropy as it refers to the number of states that could possibly be accessed, and it is also recognized that functioning needs to be efficient through minimization of manifest entropy. For example, in theoretical physics, the importance of efficiency is recognized through the observation that nature is thrifty in all its actions and through the principle of least action. In this paper, system intelligence is explained as capability to maintain internal stability while adapting to changing environments by minimizing manifest task entropy while maximizing latent system entropy. In addition, it is explained how automated negotiation relates to balancing adaptability and stability; and a mathematical negotiation model is presented that enables balancing of latent system entropy and manifest task entropy in intelligent systems. Furthermore, this first principles analysis of system intelligence is related to everyday challenges in production systems through multiple simulations of the negotiation model. The results indicate that manifest task entropy is minimized when maximization of latent system entropy is used as the criterion for task allocation in the simulated production scenarios. Full article

(This article belongs to the Collection Maximum Entropy and Its Applications)

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11 pages, 2112 KiB

Open AccessArticle

A Model for Material Metrics in Thermoelectric Thomson Coolers

by Mona Zebarjadi and Omid Akbari

Entropy 2023, 25(11), 1540; https://doi.org/10.3390/e25111540 - 14 Nov 2023

Viewed by 824

Abstract

Thomson heat absorption corresponding to changes in the Seebeck coefficient with respect to temperature enables the design of thermoelectric coolers wherein Thomson cooling is the dominant term, i.e., the Thomson coolers. Thomson coolers extend the working range of Peltier coolers to larger temperature [...] Read more.

Thomson heat absorption corresponding to changes in the Seebeck coefficient with respect to temperature enables the design of thermoelectric coolers wherein Thomson cooling is the dominant term, i.e., the Thomson coolers. Thomson coolers extend the working range of Peltier coolers to larger temperature differences and higher electrical currents. The Thomson coefficient is small in most materials. Recently, large Thomson coefficient values have been measured attributed to thermally induced phase change during magnetic and structural phase transitions. The large Thomson coefficient observed can result in the design of highly efficient Thomson coolers. This work analyzes the performance of Thomson coolers analytically and sets the metrics for evaluating the performance of materials as their constituent components. The maximum heat flux when the Thomson coefficient is constant is obtained and the performance is compared to Peltier coolers. Three dimensionless parameters are introduced which determine the performance of the Thomson coolers and can be used to analyze the coefficient of performance, the maximum heat flux, and the maximum temperature difference of a Thomson cooler. Full article

(This article belongs to the Special Issue Heat Transfer in Thermoelectric Modules)

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19 pages, 2575 KiB

Open AccessArticle

A Multi-Featured Factor Analysis and Dynamic Window Rectification Method for Remaining Useful Life Prognosis of Rolling Bearings

by Cheng Peng, Yuanyuan Zhao, Changyun Li, Zhaohui Tang and Weihua Gui

Entropy 2023, 25(11), 1539; https://doi.org/10.3390/e25111539 - 13 Nov 2023

Cited by 1 | Viewed by 830

Abstract

Currently, the research on the predictions of remaining useful life (RUL) of rotating machinery mainly focuses on the process of health indicator (HI) construction and the determination of the first prediction time (FPT). In complex industrial environments, the influence of environmental factors such [...] Read more.

Currently, the research on the predictions of remaining useful life (RUL) of rotating machinery mainly focuses on the process of health indicator (HI) construction and the determination of the first prediction time (FPT). In complex industrial environments, the influence of environmental factors such as noise may affect the accuracy of RUL predictions. Accurately estimating the remaining useful life of bearings plays a vital role in reducing costly unscheduled maintenance and increasing machine reliability. To overcome these problems, a health indicator construction and prediction method based on multi-featured factor analysis are proposed. Compared with the existing methods, the advantages of this method are the use of factor analysis, to mine hidden common factors from multiple features, and the construction of health indicators based on the maximization of variance contribution after rotation. A dynamic window rectification method is designed to reduce and weaken the stochastic fluctuations in the health indicators. The first prediction time was determined by the cumulative gradient change in the trajectory of the HI. A regression-based adaptive prediction model is used to learn the evolutionary trend of the HI and estimate the RUL of the bearings. The experimental results of two publicly available bearing datasets show the advantages of the method. Full article

(This article belongs to the Special Issue New Trends in Fault Diagnosis and Prognosis for Engineering Applications: From Signal Processing to Machine Learning and Deep Learning)

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36 pages, 2677 KiB

Open AccessArticle

Dimensionless Groups by Entropic Similarity: II—Wave Phenomena and Information-Theoretic Flow Regimes

by Robert K. Niven

Entropy 2023, 25(11), 1538; https://doi.org/10.3390/e25111538 - 11 Nov 2023

Viewed by 763

Abstract

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form

Π_{info} = U / c

, where U is a [...] Read more.

The aim of this study is to explore the insights of the information-theoretic definition of similarity for a multitude of flow systems with wave propagation. This provides dimensionless groups of the form

Π_{info} = U / c

, where U is a characteristic flow velocity and c is a signal velocity or wave celerity, to distinguish different information-theoretic flow regimes. Traditionally, dimensionless groups in science and engineering are defined by geometric similarity, based on ratios of length scales; kinematic similarity, based on ratios of velocities or accelerations; and dynamic similarity, based on ratios of forces. In Part I, an additional category of entropic similarity was proposed based on ratios of (i) entropy production terms; (ii) entropy flow rates or fluxes; or (iii) information flow rates or fluxes. In this Part II, the information-theoretic definition is applied to a number of flow systems with wave phenomena, including acoustic waves, blast waves, pressure waves, surface or internal gravity waves, capillary waves, inertial waves and electromagnetic waves. These are used to define the appropriate Mach, Euler, Froude, Rossby or other dimensionless number(s)—including new groups for internal gravity, inertial and electromagnetic waves—to classify their flow regimes. For flows with wave dispersion, the coexistence of different celerities for individual waves and wave groups—each with a distinct information-theoretic group—is shown to imply the existence of more than two information-theoretic flow regimes, including for some acoustic wave systems (subsonic/mesosonic/supersonic flow) and most systems with gravity, capillary or inertial waves (subcritical/mesocritical/supercritical flow). For electromagnetic wave systems, the additional vacuum celerity implies the existence of four regimes (subluminal/mesoluminal/transluminal/superluminal flow). In addition, entropic analyses are shown to provide a more complete understanding of frictional behavior and sharp transitions in compressible and open channel flows, as well as the transport of entropy by electromagnetic radiation. The analyses significantly extend the applications of entropic similarity for the analysis of flow systems with wave propagation. Full article

(This article belongs to the Section Multidisciplinary Applications)

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16 pages, 1516 KiB

Open AccessArticle

Robustness and Complexity of Directed and Weighted Metabolic Hypergraphs

by Pietro Traversa, Guilherme Ferraz de Arruda, Alexei Vazquez and Yamir Moreno

Entropy 2023, 25(11), 1537; https://doi.org/10.3390/e25111537 - 11 Nov 2023

Viewed by 1333

Abstract

Metabolic networks are probably among the most challenging and important biological networks. Their study provides insight into how biological pathways work and how robust a specific organism is against an environment or therapy. Here, we propose a directed hypergraph with edge-dependent vertex weight [...] Read more.

Metabolic networks are probably among the most challenging and important biological networks. Their study provides insight into how biological pathways work and how robust a specific organism is against an environment or therapy. Here, we propose a directed hypergraph with edge-dependent vertex weight as a novel framework to represent metabolic networks. This hypergraph-based representation captures higher-order interactions among metabolites and reactions, as well as the directionalities of reactions and stoichiometric weights, preserving all essential information. Within this framework, we propose the communicability and the search information as metrics to quantify the robustness and complexity of directed hypergraphs. We explore the implications of network directionality on these measures and illustrate a practical example by applying them to a small-scale E. coli core model. Additionally, we compare the robustness and the complexity of 30 different models of metabolism, connecting structural and biological properties. Our findings show that antibiotic resistance is associated with high structural robustness, while the complexity can distinguish between eukaryotic and prokaryotic organisms. Full article

(This article belongs to the Special Issue Models, Topology and Inference of Multilayer and Higher-Order Networks)

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19 pages, 8212 KiB

Open AccessArticle

Distributed Formation Control of Multi-Robot Systems with Path Navigation via Complex Laplacian

by Xiru Wu, Rili Wu, Yuchong Zhang and Jiansheng Peng

Entropy 2023, 25(11), 1536; https://doi.org/10.3390/e25111536 - 11 Nov 2023

Viewed by 827

Abstract

This paper focuses on the formation control of multi-robot systems with leader–follower network structure in directed topology to guide a system composed of multiple mobile robot agents to achieve global path navigation with a desired formation. A distributed linear formation control strategy based [...] Read more.

This paper focuses on the formation control of multi-robot systems with leader–follower network structure in directed topology to guide a system composed of multiple mobile robot agents to achieve global path navigation with a desired formation. A distributed linear formation control strategy based on the complex Laplacian matrix is employed, which enables the robot agents to converge into a similar formation of the desired formation, and the size and orientation of the formation are determined by the positions of two leaders. Additionally, in order to ensure that all robot agents in the formation move at a common velocity, the distributed control approach also includes a velocity consensus component. Based on the realization of similar formation control of a multi-robot system, the path navigation algorithm is combined with it to realize the global navigation of the system as a whole. Furthermore, a controller enabling the scalability of the formation size is introduced to enhance the overall maneuverability of the system in specific scenarios like narrow corridors. The simulation results demonstrate the feasibility of the proposed approach. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Behaviors in Complex Systems)

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Entropy, Volume 25, Issue 11 (November 2023) – 90 articles

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