Entropy

Research

14 pages, 3645 KiB

Open AccessArticle

Learning Pathways and Students Performance: A Dynamic Complex System

by Pilar Ortiz-Vilchis and Aldo Ramirez-Arellano

Entropy 2023, 25(2), 291; https://doi.org/10.3390/e25020291 - 3 Feb 2023

Cited by 5 | Viewed by 1745

In this study, learning pathways are modelled by networks constructed from the log data of student–LMS interactions. These networks capture the sequence of reviewing the learning materials by the students enrolled in a given course. In previous research, the networks of successful students [...] Read more.

In this study, learning pathways are modelled by networks constructed from the log data of student–LMS interactions. These networks capture the sequence of reviewing the learning materials by the students enrolled in a given course. In previous research, the networks of successful students showed a fractal property; meanwhile, the networks of students who failed showed an exponential pattern. This research aims to provide empirical evidence that students’ learning pathways have the properties of emergence and non-additivity from a macro level; meanwhile, equifinality (same end of learning process but different learning pathways) is presented at a micro level. Furthermore, the learning pathways of 422 students enrolled in a blended course are classified according to learning performance. These individual learning pathways are modelled by networks from which the relevant learning activities (nodes) are extracted in a sequence by a fractal-based method. The fractal method reduces the number of nodes to be considered relevant. A deep learning network classifies these sequences of each student into passed or failed. The results show that the accuracy of the prediction of the learning performance was 94%, the area under the receiver operating characteristic curve was 97%, and the Matthews correlation was 88%, showing that deep learning networks can model equifinality in complex systems. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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18 pages, 2322 KiB

Open AccessEditor’s ChoiceArticle

Reconstructing Sparse Multiplex Networks with Application to Covert Networks

by Jin-Zhu Yu, Mincheng Wu, Gisela Bichler, Felipe Aros-Vera and Jianxi Gao

Entropy 2023, 25(1), 142; https://doi.org/10.3390/e25010142 - 10 Jan 2023

Cited by 1 | Viewed by 2243

Abstract

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we [...] Read more.

Network structure provides critical information for understanding the dynamic behavior of complex systems. However, the complete structure of real-world networks is often unavailable, thus it is crucially important to develop approaches to infer a more complete structure of networks. In this paper, we integrate the configuration model for generating random networks into an Expectation–Maximization–Aggregation (EMA) framework to reconstruct the complete structure of multiplex networks. We validate the proposed EMA framework against the Expectation–Maximization (EM) framework and random model on several real-world multiplex networks, including both covert and overt ones. It is found that the EMA framework generally achieves the best predictive accuracy compared to the EM framework and the random model. As the number of layers increases, the performance improvement of EMA over EM decreases. The inferred multiplex networks can be leveraged to inform the decision-making on monitoring covert networks as well as allocating limited resources for collecting additional information to improve reconstruction accuracy. For law enforcement agencies, the inferred complete network structure can be used to develop more effective strategies for covert network interdiction. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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17 pages, 1609 KiB

Open AccessArticle

Modeling the Evolution of Biological Neural Networks Based on Caenorhabditis elegans Connectomes across Development

by Hongfei Zhao, Zhiguo Shi, Zhefeng Gong and Shibo He

Entropy 2023, 25(1), 51; https://doi.org/10.3390/e25010051 - 27 Dec 2022

Cited by 1 | Viewed by 1819

Abstract

Knowledge of the structural properties of biological neural networks can help in understanding how particular responses and actions are generated. Recently, Witvliet et al. published the connectomes of eight isogenic Caenorhabditis elegans hermaphrodites at different postembryonic ages, from birth to adulthood. We analyzed [...] Read more.

Knowledge of the structural properties of biological neural networks can help in understanding how particular responses and actions are generated. Recently, Witvliet et al. published the connectomes of eight isogenic Caenorhabditis elegans hermaphrodites at different postembryonic ages, from birth to adulthood. We analyzed the basic structural properties of these biological neural networks. From birth to adulthood, the asymmetry between in-degrees and out-degrees over the C. elegans neuronal network increased with age, in addition to an increase in the number of nodes and edges. The degree distributions were neither Poisson distributions nor pure power-law distributions. We have proposed a model of network evolution with different initial attractiveness for in-degrees and out-degrees of nodes and preferential attachment, which reproduces the asymmetry between in-degrees and out-degrees and similar degree distributions via the tuning of the initial attractiveness values. In this study, we present the well-preserved structural properties of C. elegans neuronal networks across development, and provide some insight into understanding the evolutionary processes of biological neural networks through a simple network model. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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11 pages, 1145 KiB

Open AccessArticle

Robustness of Interdependent Networks with Weak Dependency Based on Bond Percolation

by Yingjie Qiang, Xueming Liu and Linqiang Pan

Entropy 2022, 24(12), 1801; https://doi.org/10.3390/e24121801 - 9 Dec 2022

Cited by 1 | Viewed by 1384

Abstract

Real-world systems interact with one another via dependency connectivities. Dependency connectivities make systems less robust because failures may spread iteratively among systems via dependency links. Most previous studies have assumed that two nodes connected by a dependency link are strongly dependent on each [...] Read more.

Real-world systems interact with one another via dependency connectivities. Dependency connectivities make systems less robust because failures may spread iteratively among systems via dependency links. Most previous studies have assumed that two nodes connected by a dependency link are strongly dependent on each other; that is, if one node fails, its dependent partner would also immediately fail. However, in many real scenarios, nodes from different networks may be weakly dependent, and links may fail instead of nodes. How interdependent networks with weak dependency react to link failures remains unknown. In this paper, we build a model of fully interdependent networks with weak dependency and define a parameter

α

in order to describe the node-coupling strength. If a node fails, its dependent partner has a probability of failing of

1 - α

. Then, we develop an analytical tool for analyzing the robustness of interdependent networks with weak dependency under link failures, with which we can accurately predict the system robustness when

1 - p

fractions of links are randomly removed. We find that as the node coupling strength increases, interdependent networks show a discontinuous phase transition when

α < α_{c}

and a continuous phase transition when

α > α_{c}

. Compared to site percolation with nodes being attacked, the crossover points

α_{c}

are larger in the bond percolation with links being attacked. This finding can give us some suggestions for designing and protecting systems in which link failures can happen. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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11 pages, 1832 KiB

Open AccessArticle

The Analysis of the Power Law Feature in Complex Networks

by Xiaojun Zhang, Zheng He, Liwei Zhang, Lez Rayman-Bacchus, Shuhui Shen and Yue Xiao

Entropy 2022, 24(11), 1561; https://doi.org/10.3390/e24111561 - 29 Oct 2022

Cited by 1 | Viewed by 1966

Abstract

Consensus about the universality of the power law feature in complex networks is experiencing widespread challenges. In this paper, we propose a generic theoretical framework in order to examine the power law property. First, we study a class of birth-and-death networks that are [...] Read more.

Consensus about the universality of the power law feature in complex networks is experiencing widespread challenges. In this paper, we propose a generic theoretical framework in order to examine the power law property. First, we study a class of birth-and-death networks that are more common than BA networks in the real world, and then we calculate their degree distributions; the results show that the tails of their degree distributions exhibit a distinct power law feature. Second, we suggest that in the real world two important factors—network size and node disappearance probability—will affect the analysis of power law characteristics in observation networks. Finally, we suggest that an effective way of detecting the power law property is to observe the asymptotic (limiting) behavior of the degree distribution within its effective intervals. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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15 pages, 14251 KiB

Open AccessArticle

An Internet-Oriented Multilayer Network Model Characterization and Robustness Analysis Method

by Yongheng Zhang, Yuliang Lu, Guozheng Yang, Dongdong Hou and Zhihao Luo

Entropy 2022, 24(8), 1147; https://doi.org/10.3390/e24081147 - 18 Aug 2022

Cited by 3 | Viewed by 1869

Abstract

The Internet creates multidimensional and complex relationships in terms of the composition, application and mapping of social users. Most of the previous related research has focused on the single-layer topology of physical device networks but ignored the study of service access relationships and [...] Read more.

The Internet creates multidimensional and complex relationships in terms of the composition, application and mapping of social users. Most of the previous related research has focused on the single-layer topology of physical device networks but ignored the study of service access relationships and the social structure of users on the Internet. Here, we propose a composite framework to understand how the interaction between the physical devices network, business application network, and user role network affects the robustness of the entire Internet. In this paper, a multilayer network consisting of a physical device layer, business application layer and user role layer is constructed by collecting experimental network data. We characterize the disturbance process of the entire multilayer network when a physical entity device fails by designing nodal disturbance to investigate the interactions that exist between the different network layers. Meanwhile, we analyze the characteristics of the Internet-oriented multilayer network structure and propose a heuristic multilayer network topology generation algorithm based on the initial routing topology and networking pattern, which simulates the evolution process of multilayer network topology. To further analyze the robustness of this multilayer network model, we combined a total of six target node ranking indicators including random strategy, degree centrality, betweenness centrality, closeness centrality, clustering coefficient and network constraint coefficient, performed node deletion simulations in the experimental network, and analyzed the impact of component types and interactions on the robustness of the overall multilayer network based on the maximum component change in the network. These results provide new insights into the operational processes of the Internet from a multi-domain data fusion perspective, reflecting that the coupling relationships that exist between the different interaction layers are closely linked to the robustness of multilayer networks. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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17 pages, 2217 KiB

Open AccessArticle

Conserved Control Path in Multilayer Networks

by Bingbo Wang, Xiujuan Ma, Cunchi Wang, Mingjie Zhang, Qianhua Gong and Lin Gao

Entropy 2022, 24(7), 979; https://doi.org/10.3390/e24070979 - 15 Jul 2022

Viewed by 1666

Abstract

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that [...] Read more.

The determination of directed control paths in complex networks is important because control paths indicate the structure of the propagation of control signals through edges. A challenging problem is to identify them in complex networked systems characterized by different types of interactions that form multilayer networks. In this study, we describe a graph pattern called the conserved control path, which allows us to model a common control structure among different types of relations. We present a practical conserved control path detection method (CoPath), which is based on a maximum-weighted matching, to determine the paths that play the most consistent roles in controlling signal transmission in multilayer networks. As a pragmatic application, we demonstrate that the control paths detected in a multilayered pan-cancer network are statistically more consistent. Additionally, they lead to the effective identification of drug targets, thereby demonstrating their power in predicting key pathways that influence multiple cancers. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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20 pages, 651 KiB

Open AccessArticle

A Network Structure Entropy Considering Series-Parallel Structures

by Meng Cai, Jiaqi Liu and Ying Cui

Entropy 2022, 24(7), 852; https://doi.org/10.3390/e24070852 - 21 Jun 2022

Cited by 6 | Viewed by 2011

Abstract

Entropy is an important indicator to measure network heterogeneity. We propose a new network structure entropy, SP (series-parallel) structure entropy, based on the global network topology while adding a medial measure that considers the series-parallel structure. First, the results of special networks show [...] Read more.

Entropy is an important indicator to measure network heterogeneity. We propose a new network structure entropy, SP (series-parallel) structure entropy, based on the global network topology while adding a medial measure that considers the series-parallel structure. First, the results of special networks show that SP structure entropy can overcome other structure’s entropy deficiencies to some extent. Then, through simulation analysis of typical networks, the validity and applicability of SP structure entropy in describing general networks are verified. Finally, we analyze an enterprise consulting network to demonstrate the superiority of the SP structure entropy for real network analysis. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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34 pages, 23413 KiB

Open AccessArticle

Finite-Time Pinning Synchronization Control for T-S Fuzzy Discrete Complex Networks with Time-Varying Delays via Adaptive Event-Triggered Approach

by Xiru Wu, Yuchong Zhang, Qingming Ai and Yaonan Wang

Entropy 2022, 24(5), 733; https://doi.org/10.3390/e24050733 - 21 May 2022

Cited by 1 | Viewed by 2328

Abstract

This paper is concerned with the adaptive event-triggered finite-time pinning synchronization control problem for T-S fuzzy discrete complex networks (TSFDCNs) with time-varying delays. In order to accurately describe discrete dynamical behaviors, we build a general model of discrete complex networks via T-S fuzzy [...] Read more.

This paper is concerned with the adaptive event-triggered finite-time pinning synchronization control problem for T-S fuzzy discrete complex networks (TSFDCNs) with time-varying delays. In order to accurately describe discrete dynamical behaviors, we build a general model of discrete complex networks via T-S fuzzy rules, which extends a continuous-time model in existing results. Based on an adaptive threshold and measurement errors, a discrete adaptive event-triggered approach (AETA) is introduced to govern signal transmission. With the hope of improving the resource utilization and reducing the update frequency, an event-based fuzzy pinning feedback control strategy is designed to control a small fraction of network nodes. Furthermore, by new Lyapunov–Krasovskii functionals and the finite-time analysis method, sufficient criteria are provided to guarantee the finite-time bounded stability of the closed-loop error system. Under an optimization condition and linear matrix inequality (LMI) constraints, the desired controller parameters with respect to minimum finite time are derived. Finally, several numerical examples are conducted to show the effectiveness of obtained theoretical results. For the same system, the average triggering rate of AETA is significantly lower than existing event-triggered mechanisms and the convergence rate of synchronization errors is also superior to other control strategies. Full article

(This article belongs to the Special Issue Dynamics of Complex Networks)

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