Computational Complex Networks

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
Entropy entropy	2.1	4.9	1999	22.3 Days	CHF 2600	Submit
Algorithms algorithms	1.8	4.1	2008	18.9 Days	CHF 1600	Submit
Computation computation	1.9	3.5	2013	18.6 Days	CHF 1800	Submit
Fractal and Fractional fractalfract	3.6	4.6	2017	23.7 Days	CHF 2700	Submit

17 pages, 2734 KiB

Open AccessArticle

An Event–Link Network Model Based on Representation in P-Space

by Wenjun Zhang, Xiangna Chen and Weibing Deng

Entropy 2025, 27(4), 419; https://doi.org/10.3390/e27040419 - 12 Apr 2025

Viewed by 130

The L-space and P-space are two essential representations for studying complex networks that contain different clusters. Existing network models can successfully generate networks in L-space, but generating networks in P-space poses significant challenges. In this study, we present an empirical analysis of the [...] Read more.

The L-space and P-space are two essential representations for studying complex networks that contain different clusters. Existing network models can successfully generate networks in L-space, but generating networks in P-space poses significant challenges. In this study, we present an empirical analysis of the distribution of the number of a line’s nodes and the properties of the networks generated by these data in P-space. To gain insights into the operational mechanisms of the network of these data, we propose an event–link model that incorporates new nodes and links in P-space based on actual data characteristics using real data from marine and public transportation networks. The entire network consists of a series of events that consist of many nodes, and all nodes in an event are connected in the P-space. We conduct simulation experiments to explore the model’s topological features under different parameter conditions, demonstrating that the simulation outcomes are consistent with the theoretical analysis of the model. This model exhibits small-world characteristics, scale-free behavior, and a high clustering coefficient. The event–link model, with its adjustable parameters, effectively generates networks with stable structures that closely resemble the statistical characteristics of real-world networks that share similar growth mechanisms. Moreover, the network’s growth and evolution can be flexibly adjusted by modifying the model parameters. Full article

(This article belongs to the Topic Computational Complex Networks)

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18 pages, 8833 KiB

Open AccessArticle

Finding Key Nodes in Complex Networks Through Quantum Deep Reinforcement Learning

by Juechan Xiong, Xiao-Long Ren and Linyuan Lü

Entropy 2025, 27(4), 382; https://doi.org/10.3390/e27040382 - 3 Apr 2025

Viewed by 252

Abstract

Identifying key nodes in networks is a fundamental problem in network science. This study proposes a quantum deep reinforcement learning (QDRL) framework that integrates reinforcement learning with a variational quantum graph neural network, effectively identifying distributed influential nodes while preserving the network’s fundamental [...] Read more.

Identifying key nodes in networks is a fundamental problem in network science. This study proposes a quantum deep reinforcement learning (QDRL) framework that integrates reinforcement learning with a variational quantum graph neural network, effectively identifying distributed influential nodes while preserving the network’s fundamental topological properties. By leveraging principles of quantum computing, our method is designed to reduce model parameters and computational complexity compared to traditional neural networks. Trained on small networks, it demonstrated strong generalization across diverse scenarios. We compared the proposed algorithm with some classical node ranking and network dismantling algorithms on various synthetical and empirical networks. The results suggest that the proposed algorithm outperforms existing baseline methods. Moreover, in synthetic networks based on Erdős–Rényi and Watts–Strogatz models, QDRL demonstrated its capability to alleviate the issue of localization in network information propagation and node influence ranking. Our research provides insights into addressing fundamental problems in complex networks using quantum machine learning, demonstrating the potential of quantum approaches for network analysis tasks. Full article

(This article belongs to the Topic Computational Complex Networks)

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22 pages, 2595 KiB

Open AccessArticle

Securing Bipartite Nonlinear Fractional-Order Multi-Agent Systems against False Data Injection Attacks (FDIAs) Considering Hostile Environment

by Hanen Louati, Saadia Rehman, Farhat Imtiaz, Nafisa A. AlBasheir, Afrah Y. Al-Rezami, Mohammed M. A. Almazah and Azmat Ullah Khan Niazi

Fractal Fract. 2024, 8(7), 430; https://doi.org/10.3390/fractalfract8070430 - 22 Jul 2024

Cited by 2 | Viewed by 1282

Abstract

This study investigated the stability of bipartite nonlinear fractional-order multi-agent systems (FOMASs) in the presence of false data injection attacks (FDIAs) in a hostile environment. To tackle this problem we used signed graph theory, the Razumikhin methodology, and the Lyapunov function method. The [...] Read more.

This study investigated the stability of bipartite nonlinear fractional-order multi-agent systems (FOMASs) in the presence of false data injection attacks (FDIAs) in a hostile environment. To tackle this problem we used signed graph theory, the Razumikhin methodology, and the Lyapunov function method. The main focus of our proposed work is to provide a method of stability for FOMASs against FDIAs. The technique of Razumikhin improves the Lyapunov-based stability analysis by supporting the handling of the intricacies of fractional-order dynamics. Moreover, utilizing signed graph theory, we analyzed both hostile and cooperative interactions between agents within the MASs. We determined the system stability requirements to ensure robustness against erroneous data injections through comprehensive theoretical investigation. We present numerical examples to illustrate the robustness and efficiency of our proposed technique. Full article

(This article belongs to the Topic Computational Complex Networks)

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16 pages, 15509 KiB

Open AccessArticle

HAG-NET: Hiding Data and Adversarial Attacking with Generative Adversarial Network

by Haiju Fan and Jinsong Wang

Entropy 2024, 26(3), 269; https://doi.org/10.3390/e26030269 - 19 Mar 2024

Viewed by 1639

Abstract

Recent studies on watermarking techniques based on image carriers have demonstrated new approaches that combine adversarial perturbations against steganalysis with embedding distortions. However, while these methods successfully counter convolutional neural network-based steganalysis, they do not adequately protect the data of the carrier itself. [...] Read more.

Recent studies on watermarking techniques based on image carriers have demonstrated new approaches that combine adversarial perturbations against steganalysis with embedding distortions. However, while these methods successfully counter convolutional neural network-based steganalysis, they do not adequately protect the data of the carrier itself. Recognizing the high sensitivity of Deep Neural Networks (DNNs) to small perturbations, we propose HAG-NET, a method based on image carriers, which is jointly trained by the encoder, decoder, and attacker. In this paper, the encoder generates Adversarial Steganographic Examples (ASEs) that are adversarial to the target classification network, thereby providing protection for the carrier data. Additionally, the decoder can recover secret data from ASEs. The experimental results demonstrate that ASEs produced by HAG-NET achieve an average success rate of over 99% on both the MNIST and CIFAR-10 datasets. ASEs generated with the attacker exhibit greater robustness in terms of attack ability, with an average increase of about 3.32%. Furthermore, our method, when compared with other generative stego examples under similar perturbation strength, contains significantly more information according to image information entropy measurements. Full article

(This article belongs to the Topic Computational Complex Networks)

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24 pages, 8072 KiB

Open AccessArticle

Research on Risk Contagion in ESG Industries: An Information Entropy-Based Network Approach

by Chenglong Hu and Ranran Guo

Entropy 2024, 26(3), 206; https://doi.org/10.3390/e26030206 - 27 Feb 2024

Cited by 4 | Viewed by 2061

Abstract

Sustainable development is a practical path to optimize industrial structures and enhance investment efficiency. Investigating risk contagion within ESG industries is a crucial step towards reducing systemic risks and fostering the green evolution of the economy. This research constructs ESG industry indices, taking [...] Read more.

Sustainable development is a practical path to optimize industrial structures and enhance investment efficiency. Investigating risk contagion within ESG industries is a crucial step towards reducing systemic risks and fostering the green evolution of the economy. This research constructs ESG industry indices, taking into account the possibility of extreme tail risks, and employs VaR and CoVaR as measures of tail risk. The TENET network approach is integrated to to capture the structural evolution and direction of information flow among ESG industries, employing information entropy to quantify the topological characteristics of the network model, exploring the risk transmission paths and evolution patterns of ESG industries in an extreme tail risk event. Finally, Mantel tests are conducted to examine the existence of significant risk spillover effects between ESG and traditional industries. The research finds strong correlations among ESG industry indices during stock market crash, Sino–US trade frictions, and the COVID-19 pandemic, with industries such as the COAL, CMP, COM, RT, and RE playing key roles in risk transmission within the network, transmitting risks to other industries. Affected by systemic risk, the information entropy of the TENET network significantly decreases, reducing market information uncertainty and leading market participants to adopt more uniform investment strategies, thus diminishing the diversity of market behaviors. ESG industries show resilience in the face of extreme risks, demonstrating a lack of significant risk contagion with traditional industries. Full article

(This article belongs to the Topic Computational Complex Networks)

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17 pages, 3278 KiB

Open AccessEditor’s ChoiceArticle

Identifying Influential Nodes in Complex Networks Based on Information Entropy and Relationship Strength

by Ying Xi and Xiaohui Cui

Entropy 2023, 25(5), 754; https://doi.org/10.3390/e25050754 - 5 May 2023

Cited by 12 | Viewed by 3448

Abstract

Identifying influential nodes is a key research topic in complex networks, and there have been many studies based on complex networks to explore the influence of nodes. Graph neural networks (GNNs) have emerged as a prominent deep learning architecture, capable of efficiently aggregating [...] Read more.

Identifying influential nodes is a key research topic in complex networks, and there have been many studies based on complex networks to explore the influence of nodes. Graph neural networks (GNNs) have emerged as a prominent deep learning architecture, capable of efficiently aggregating node information and discerning node influence. However, existing graph neural networks often ignore the strength of the relationships between nodes when aggregating information about neighboring nodes. In complex networks, neighboring nodes often do not have the same influence on the target node, so the existing graph neural network methods are not effective. In addition, the diversity of complex networks also makes it difficult to adapt node features with a single attribute to different types of networks. To address the above problems, the paper constructs node input features using information entropy combined with the node degree value and the average degree of the neighbor, and proposes a simple and effective graph neural network model. The model obtains the strength of the relationships between nodes by considering the degree of neighborhood overlap, and uses this as the basis for message passing, thereby effectively aggregating information about nodes and their neighborhoods. Experiments are conducted on 12 real networks, using the SIR model to verify the effectiveness of the model with the benchmark method. The experimental results show that the model can identify the influence of nodes in complex networks more effectively. Full article

(This article belongs to the Topic Computational Complex Networks)

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19 pages, 1822 KiB

Open AccessArticle

Exploiting Dual-Attention Networks for Explainable Recommendation in Heterogeneous Information Networks

by Xianglin Zuo, Tianhao Jia, Xin He, Bo Yang and Ying Wang

Entropy 2022, 24(12), 1718; https://doi.org/10.3390/e24121718 - 24 Nov 2022

Cited by 1 | Viewed by 2169

Abstract

The aim of explainable recommendation is not only to provide recommended items to users, but also to make users aware of why these items are recommended. Traditional recommendation methods infer user preferences for items using user–item rating information. However, the expressive power of [...] Read more.

The aim of explainable recommendation is not only to provide recommended items to users, but also to make users aware of why these items are recommended. Traditional recommendation methods infer user preferences for items using user–item rating information. However, the expressive power of latent representations of users and items is relatively limited due to the sparseness of the user–item rating matrix. Heterogeneous information networks (HIN) provide contextual information for improving recommendation performance and interpreting the interactions between users and items. However, due to the heterogeneity and complexity of context information in HIN, it is still a challenge to integrate this contextual information into explainable recommendation systems effectively. In this paper, we propose a novel framework—the dual-attention networks for explainable recommendation (DANER) in HINs. We first used multiple meta-paths to capture high-order semantic relations between users and items in HIN for generating similarity matrices, and then utilized matrix decomposition on similarity matrices to obtain low-dimensional sparse representations of users and items. Secondly, we introduced two-level attention networks, namely a local attention network and a global attention network, to integrate the representations of users and items from different meta-paths for obtaining high-quality representations. Finally, we use a standard multi-layer perceptron to model the interactions between users and items, which predict users’ ratings of items. Furthermore, the dual-attention mechanism also contributes to identifying critical meta-paths to generate relevant explanations for users. Comprehensive experiments on two real-world datasets demonstrate the effectiveness of DANER on recommendation performance as compared with the state-of-the-art methods. A case study illustrates the interpretability of DANER. Full article

(This article belongs to the Topic Computational Complex Networks)

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15 pages, 6744 KiB

Open AccessArticle

Research on Fractal Evolution Characteristics and Safe Mining Technology of Overburden Fissures under Gully Water Body

by Kaijun Miao, Shihao Tu, Hongsheng Tu, Xun Liu, Wenlong Li, Hongbin Zhao, Long Tang, Jieyang Ma and Yan Li

Fractal Fract. 2022, 6(9), 486; https://doi.org/10.3390/fractalfract6090486 - 30 Aug 2022

Cited by 14 | Viewed by 1799

Abstract

A fractal realizes the quantitative characterization of complex and disordered mining fracture networks, and it is of great significance to grasp the fractal characteristics of rock movement law to guide mine production. To prevent the water-conducting fracture (WF) under the gullies from conducting [...] Read more.

A fractal realizes the quantitative characterization of complex and disordered mining fracture networks, and it is of great significance to grasp the fractal characteristics of rock movement law to guide mine production. To prevent the water-conducting fracture (WF) under the gullies from conducting the surface water body, and to realize the purpose of safe production and surface water body protection. The evolution of overburden fissures in the working face with shallow buried gulley landform and thick bedrock conditions is studied. The development height of water-conducting fracture (DHWF) is theoretically analyzed. The evolution characteristics of overlying fissures with different mining heights were observed by similarity simulation, and the observation results were analyzed by fractal theory. The results show that the main factor that determines the height of WF is mining height. The working face is mined at different mining heights, and the corresponding indexes such as the height of the WF, the area of the caving zone and the fractal dimension are related to engineering phenomena. In particular, the appearance and disappearance of the separation space correspond to the fractal dimension fluctuation phase. The safe mining technology under a gully water body, which mainly reduces mining height, is adopted, and the fissures of the working face are not connected to the surface water body after mining. Full article

(This article belongs to the Topic Computational Complex Networks)

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19 pages, 4979 KiB

Open AccessArticle

Successive Trajectory Privacy Protection with Semantics Prediction Differential Privacy

by Jing Zhang, Yanzi Li, Qian Ding, Liwei Lin and Xiucai Ye

Entropy 2022, 24(9), 1172; https://doi.org/10.3390/e24091172 - 23 Aug 2022

Cited by 9 | Viewed by 2705

Abstract

The publication of trajectory data provides critical information for various location-based services, and it is critical to publish trajectory data safely while ensuring its availability. Differential privacy is a promising privacy protection technology for publishing trajectory data securely. Most of the existing trajectory [...] Read more.

The publication of trajectory data provides critical information for various location-based services, and it is critical to publish trajectory data safely while ensuring its availability. Differential privacy is a promising privacy protection technology for publishing trajectory data securely. Most of the existing trajectory privacy protection schemes do not take into account the user’s preference for location and the influence of semantic location. Besides, differential privacy for trajectory protection still has the problem of balance between the privacy budget and service quality. In this paper, a semantics- and prediction-based differential privacy protection scheme for trajectory data is proposed. Firstly, trajectory data are transformed into a prefix tree structure to ensure that they satisfy differential privacy. Secondly, considering the influence of semantic location on trajectory, semantic sensitivity combined with location check-in frequency is used to calculate the sensitivity of each position in the trajectory. The privacy level of the position is classified by setting thresholds. Moreover, the corresponding privacy budget is allocated according to the location privacy level. Finally, a Markov chain is used to predict the attack probability of each position in the trajectory. On this basis, the allocation of the privacy budget is further adjusted and its utilization rate is improved. Thus, the problem of the balance between the privacy budget and service quality is solved. Experimental results show that the proposed scheme is able to ensure data availability while protecting data privacy. Full article

(This article belongs to the Topic Computational Complex Networks)

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