Entropy

Research

30 pages, 4465 KB

Open AccessArticle

Mapping Vulnerability: Structure, Cascades, and Resilience in the Global Railway Vans Trade Network

by Lingyun Zhou, Langya Zhou, Weiwei Gong, Cheng Chen and Baojing Huang

Entropy 2026, 28(4), 421; https://doi.org/10.3390/e28040421 - 9 Apr 2026

Viewed by 257

Global supply chains face increasing vulnerability to disruptions from geopolitical tensions, natural disasters, and demand shocks. The global trade network for railway vans, critical for transcontinental freight transport, remains understudied despite its foundational role in global logistics. This study addresses the gap in [...] Read more.

Global supply chains face increasing vulnerability to disruptions from geopolitical tensions, natural disasters, and demand shocks. The global trade network for railway vans, critical for transcontinental freight transport, remains understudied despite its foundational role in global logistics. This study addresses the gap in understanding how the railway vans trade network structure evolves and responds to different types of shocks, moving beyond static analyses to capture dynamic vulnerabilities. Using UN Comtrade data (2013–2024), multi-level network analysis examined structural evolution at macroscopic, mesoscopic, and microscopic scales. Three risk propagation models simulated supply disruption, demand shock, and cooperation disruption scenarios to assess systemic vulnerabilities. The network transformed from a polycentric to core-periphery structure, with China dominating exports (67 partners in 2024) and Germany leading European integration. Supply disruptions from Romania and Czechia affected up to 114 countries under low risk absorption capacity (α = 0.1), while demand shocks from the USA impacted 53 countries. The disruption of strategic trade links, such as China–Australia, triggered severe systemic risks. The systemic criticality of risk sources varies by shock type, requiring context-specific resilience strategies. The findings guide policymakers in identifying critical vulnerabilities and designing targeted interventions for enhancing supply chain resilience in infrastructure sectors. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

16 pages, 55907 KB

Open AccessArticle

Exploring Cultural Evolution Through Modular Dynamics in Temporal Hashtag Networks

by Yasuhiro Hashimoto, Hiroki Sato and Takashi Ikegami

Entropy 2026, 28(4), 398; https://doi.org/10.3390/e28040398 - 1 Apr 2026

Viewed by 355

Abstract

Social media platforms offer unprecedented opportunities to study cultural evolution by analyzing digital traces. This study presents a methodological framework for analyzing the temporal dynamics of cultural modules in hashtag co-occurrence networks. We address the inherent challenges of analyzing dense, skewed, and highly [...] Read more.

Social media platforms offer unprecedented opportunities to study cultural evolution by analyzing digital traces. This study presents a methodological framework for analyzing the temporal dynamics of cultural modules in hashtag co-occurrence networks. We address the inherent challenges of analyzing dense, skewed, and highly variable cultural networks by introducing a perturbation ensemble clustering approach that distinguishes stable from unstable structural elements. By applying the Leiden algorithm to a perturbed ensemble of hashtag networks, we identify robust core modules and their stable periphery, and distinguish them from floating elements with unstable associations. Analysis of four years of data from a major photo-sharing platform reveals complex patterns in the evolution of cultural modules, including both stable associations and dynamic reorganizations. Our findings demonstrate how ensemble clustering techniques can effectively capture the interplay between stability and change in evolving cultural systems. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

21 pages, 3665 KB

Open AccessArticle

Coupled Dynamics of Vaccination Behavior and Epidemic Spreading on Multilayer Higher-Order Networks

by Zhishuang Wang, Guoqiang Zeng, Qian Yin, Linyuan Guo and Zhiyong Hong

Entropy 2026, 28(2), 243; https://doi.org/10.3390/e28020243 - 20 Feb 2026

Viewed by 377

Abstract

Vaccination behavior and epidemic spreading are strongly intertwined processes, and their coevolution is often shaped by both individual decision-making and social interactions. However, most existing studies model such interactions at the pairwise level, overlooking the potential impact of higher-order social influence arising from [...] Read more.

Vaccination behavior and epidemic spreading are strongly intertwined processes, and their coevolution is often shaped by both individual decision-making and social interactions. However, most existing studies model such interactions at the pairwise level, overlooking the potential impact of higher-order social influence arising from group interactions. In this work, we develop a coupled vaccination–epidemic spreading model on multilayer higher-order networks, where vaccination behavior evolves on a simplicial complex and epidemic propagation occurs on a physical contact network. The model incorporates imperfect vaccine efficacy, allowing vaccinated individuals to become infected, and introduces a hybrid vaccination strategy that combines rational cost–benefit evaluation with social influence from both pairwise and higher-order interactions, as well as negative effects induced by vaccine failure. By constructing the coupled dynamical equations, we analytically derive the epidemic outbreak threshold and elucidate how higher-order interactions, behavioral responses, and vaccine-related parameters jointly affect epidemic dynamics. Numerical simulations on networks with different structural properties validate the theoretical results and reveal pronounced structure-dependent effects. The results show that higher-order social interactions can significantly reshape vaccination behavior and epidemic prevalence, while network heterogeneity and vaccine imperfection play crucial roles in determining the outbreak threshold and steady-state infection level. These results emphasize the necessity of incorporating higher-order interactions together with realistic vaccination behavior into epidemic modeling and offer new insights for the design of effective vaccination strategies. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

18 pages, 1035 KB

Open AccessArticle

Narrative Divergence and Disinformation: An Entropic Model for Assessing the Informative Utility of Public Information Sources

by José Ignacio Peláez, Gustavo Fabian Vaccaro and Felix Infante León

Entropy 2026, 28(2), 183; https://doi.org/10.3390/e28020183 - 6 Feb 2026

Viewed by 725

Abstract

In today’s information ecosystem, disinformation threatens civic autonomy and the stability of public discourse. Beyond the intentional spread of false information, it often appears as narrative divergence among sources interpreting shared events, generating fragmentation and measurable losses in structural coherence. This study examines [...] Read more.

In today’s information ecosystem, disinformation threatens civic autonomy and the stability of public discourse. Beyond the intentional spread of false information, it often appears as narrative divergence among sources interpreting shared events, generating fragmentation and measurable losses in structural coherence. This study examines disinformation within an entropic structural framework, defining it as narrative disorder and epistemic incoherence in information systems. The approach moves beyond fact-checking by treating narrative structure and informational order as quantifiable attributes of public communication. We present the QVP-RI (Relational Information Valuation) operator, a computational model that quantifies narrative divergence through informational entropy and normalized structural divergence, without issuing truth assessments. Implemented through state-of-the-art NLP pipelines and entropic analysis, the operator maps narrative structure and epistemic order across plural media environments. Unlike accuracy-driven approaches, it evaluates narrative coherence and informational utility (IU) as complementary indicators of epistemic value. Experimental validation with 500 participants confirms the robustness of the structural–entropic model and identifies high divergence regions, revealing communication vulnerabilities and showing how narrative disorder enables disinformation dynamics. The QVP-RI operator thus offers a computationally grounded tool for analyzing disinformation as narrative divergence and for strengthening epistemic order in open information systems. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

24 pages, 24392 KB

Open AccessArticle

Peer Reporting: Sampling Design and Unbiased Estimates

by Kang Wen, Jianhong Mou and Xin Lu

Entropy 2026, 28(1), 116; https://doi.org/10.3390/e28010116 - 18 Jan 2026

Viewed by 381

Abstract

The Ego-Centric Sampling Method (ECM) leverages individual-level reports about peers to estimate population proportions within social networks, offering strong privacy protection without requiring full network data. However, the conventional ECM estimator is unbiased only under the restrictive assumption of a homogeneous network, where [...] Read more.

The Ego-Centric Sampling Method (ECM) leverages individual-level reports about peers to estimate population proportions within social networks, offering strong privacy protection without requiring full network data. However, the conventional ECM estimator is unbiased only under the restrictive assumption of a homogeneous network, where node degrees are uniform and uncorrelated with attributes. To overcome this limitation, we introduce the Activity Ratio Corrected ECM estimator (

{ECM}_{ac}

), which exploits network reciprocity to recast the population–proportion problem into an equivalent formulation in edge space. This reformulation relies solely on ego–peer data and explicitly corrects for degree–attribute dependencies, yielding unbiased and stable estimates even in highly heterogeneous networks. Simulations and analyses on real-world networks show that

{ECM}_{ac}

reduces estimation error by up to 70% compared with the conventional ECM. Our results establish a theoretically grounded and practically scalable framework for unbiased inference in network-based sampling designs. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

18 pages, 3445 KB

Open AccessArticle

Narrative Co-Evolution in Hybrid Social Networks: A Longitudinal Computational Analysis of Confucius Institutes

by Ming Huang, Jun-Ling Wang and Zi-Ke Zhang

Entropy 2025, 27(12), 1240; https://doi.org/10.3390/e27121240 - 8 Dec 2025

Cited by 1 | Viewed by 752

Abstract

This study investigates the complex dynamics of public discourse surrounding Confucius Institutes (CIs) across the hybrid social networks of mainstream news and social platforms from 2010 to 2023. Employing a longitudinal, multi-platform design, we analyzed news articles and tweets using a computational framework [...] Read more.

This study investigates the complex dynamics of public discourse surrounding Confucius Institutes (CIs) across the hybrid social networks of mainstream news and social platforms from 2010 to 2023. Employing a longitudinal, multi-platform design, we analyzed news articles and tweets using a computational framework combining topic modeling and sentiment analysis. Our results reveal a shared cross-platform narrative evolution from a “culture-first” to a “politics-central” orientation. However, the trajectory differed significantly: mainstream media underwent a gradual, policy-oriented shift, while social media exhibited an abrupt, nonlinear transition. Crucially, we identify an asymmetric interdependence: Twitter sentiment reliably Granger-causes mainstream media sentiment, establishing its role as a leading indicator, and systematic asymmetries in thematic framing reflect the divergent logics of each platform. The study demonstrates that public discourse on contested, state-linked institutions operates as a complex adaptive system, where bottom-up affective reactions and top-down editorial processes continuously interact in a dynamic equilibrium, ultimately co-constructing a fragmented public understanding. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

28 pages, 3196 KB

Open AccessArticle

The Impact of Blame Attribution on Moral Contagion in Controversial Events

by Hua Li, Qifang Wang and Renmeng Cao

Entropy 2025, 27(10), 1052; https://doi.org/10.3390/e27101052 - 10 Oct 2025

Viewed by 1528

Abstract

Controversial events are social incidents that trigger wide discussion and strong emotions, often touching on public interests, moral judgment, or social values. Their diffusion typically involves moral evaluations and affect-laden language. Prior work has mostly examined how the quantity of moral and emotional [...] Read more.

Controversial events are social incidents that trigger wide discussion and strong emotions, often touching on public interests, moral judgment, or social values. Their diffusion typically involves moral evaluations and affect-laden language. Prior work has mostly examined how the quantity of moral and emotional words shapes diffusion, while largely overlooking blame attribution—that is, whether audiences locate the cause of a controversial event in individual actions or in social structures, across different contexts. Using 189,872 original Weibo posts covering 105 events in three domains— street-level bureaucracy (SLB; individual attribution), education governance (EG; structural attribution), and gender-based violence (GBV; mixed attribution)—we estimate negative binomial models with an interaction between word type and account verification and report incidence rate ratios (IRR). Moral contagion is strongest for SLB (IRR = 1.337) and attenuated for EG (IRR = 1.037). For GBV, moral-emotional language decreases reposts (IRR = 0.844). Unverified accounts amplify the diffusion advantage of moral-emotional wording for both individually and structurally attributed issues, with the largest gains in SLB. When disaggregating by valence and discrete emotions, fear-type moral-emotional words are positively associated with reposts in GBV (IRR = 1.314). Theoretically, we shift the question from whether moral contagion occurs to when it operates, highlighting attribution tendencies and verification status as key moderators. Empirically, we provide cross-issue evidence from large-scale Chinese social media. Methodologically, we offer a replicable workflow that combines length-normalized lexical measures with negative binomial models, including interaction terms. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

17 pages, 1824 KB

Open AccessArticle

Evolving Public Attitudes Towards the HPV Vaccine in China: A Fine-Grained Emotion Analysis of Sina Weibo (2016 vs. 2024)

by Bowen Shi, Ruibo Chen, Xinyue Yuan and Junran Wu

Entropy 2025, 27(9), 887; https://doi.org/10.3390/e27090887 - 22 Aug 2025

Viewed by 2187

Abstract

In the digital age, social media significantly shapes public attitudes and emotional responses towards health interventions, such as HPV vaccination, which is critical in developing countries. This study employed a deep learning model to identify fine-grained emotions of 38,615 HPV-related tweets from 2016 [...] Read more.

In the digital age, social media significantly shapes public attitudes and emotional responses towards health interventions, such as HPV vaccination, which is critical in developing countries. This study employed a deep learning model to identify fine-grained emotions of 38,615 HPV-related tweets from 2016 to 2024, revealing significant shifts in public emotions. Notably, skepticism about vaccine commercialization motives heightened anger, while university outreach initiatives fostered positive emotions. Structural entropy analysis highlighted polarized emotional communication networks: the network of joy exhibited lower entropy with centralized information flow, whereas other emotions displayed higher entropy, fragmented dissemination, and enhanced cross-community communication efficiency. New communicators, such as campus accounts and music bloggers, played pivotal roles in spreading positive emotions, while individual bloggers in specific fields amplified negative emotions like anger, particularly in closed networks. This research underscores the intricate dynamics of online health communication and the need for targeted interventions to address stigma and enhance public awareness of HPV vaccination, providing valuable insights for future public health policy. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

22 pages, 1305 KB

Open AccessArticle

Influences of Language Functions on Linguistic Features: Multi-Dimensional and Entropy Analyses of Academic and Entertainment Registers

by Changwei Hu, Yu Zhu and Liangjie Yuan

Entropy 2025, 27(8), 783; https://doi.org/10.3390/e27080783 - 24 Jul 2025

Viewed by 3092

Abstract

This study examines how language functions impact linguistic features in academic and entertainment registers. Using multi-dimensional analysis (MDA) and computing entropy values, we analyze a large-scale Chinese corpus consisting of over 19 million tokens from 1000 texts, including academic journals, dissertations, entertainment magazines, [...] Read more.

This study examines how language functions impact linguistic features in academic and entertainment registers. Using multi-dimensional analysis (MDA) and computing entropy values, we analyze a large-scale Chinese corpus consisting of over 19 million tokens from 1000 texts, including academic journals, dissertations, entertainment magazines, and novellas. We identify key language functions that shape linguistic features within these registers. Our results reveal five core dimensions of linguistic functional variation, narrative versus rational discourse, modification, reference, uncertainty, and prudence, which account for over 52% of the variance in language use. Certain linguistic features systematically co-occur in each dimension, forming language functions that underpin broader social networks. Entropy values further confirm the findings of multi-dimensional analysis. This study emphasizes the associations between linguistic features and language functions, offering a theoretical perspective for understanding how language functions impact linguistic features and shape different registers. The findings suggest a language variation perspective on social networks’ communication. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

36 pages, 4216 KB

Open AccessArticle

Research on the Tail Risk Spillover Effect of Cryptocurrencies and Energy Market Based on Complex Network

by Xiao-Li Gong and Xue-Ting Wang

Entropy 2025, 27(7), 704; https://doi.org/10.3390/e27070704 - 30 Jun 2025

Cited by 2 | Viewed by 2608

Abstract

As the relationship between cryptocurrency mining activities and electricity consumption becomes increasingly close, the risk spillover effect is steadily drawing a lot of attention to the energy and cryptocurrency markets. For the purpose of studying the risk contagion between the cryptocurrency and energy [...] Read more.

As the relationship between cryptocurrency mining activities and electricity consumption becomes increasingly close, the risk spillover effect is steadily drawing a lot of attention to the energy and cryptocurrency markets. For the purpose of studying the risk contagion between the cryptocurrency and energy market, this paper constructs a risk contagion network between cryptocurrency and China’s energy market using complex network methods. The tail risk spillover effects under various time and frequency domains were captured by the spillover index, which was assessed by the leptokurtic quantile vector autoregression (QVAR) model. Considering the spatial heterogeneity of energy companies, the spatial Durbin model was used to explore the impact mechanism of risk spillovers. The research showed that the framework of this paper more accurately reflects the tail risk spillover effect between China’s energy market and cryptocurrency market under various shock scales, with the extreme state experiencing a much higher spillover effect than the normal state. Furthermore, this study found that the tail risk contagion between cryptocurrency and China’s energy market exhibits notable dynamic variation and cyclical features, and the long-term risk spillover effect is primarily responsible for the total spillover. At the same time, the study found that the company with the most significant spillover effect does not necessarily have the largest company size, and other factors, such as geographical location and business composition, need to be considered. Moreover, there are spatial spillover effects among listed energy companies, and the connectedness between cryptocurrency and the energy market network generates an obvious impact on risk spillover effects. The research conclusions have an important role in preventing cross-contagion of risks between cryptocurrency and the energy market. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

22 pages, 397 KB

Open AccessArticle

Echo Chambers and Homophily in the Diffusion of Risk Information on Social Media: The Case of Genetically Modified Organisms (GMOs)

by Xiaoxiao Cheng and Jianbin Jin

Entropy 2025, 27(7), 699; https://doi.org/10.3390/e27070699 - 29 Jun 2025

Cited by 1 | Viewed by 1756

Abstract

This study investigates the mechanisms underlying the diffusion of risk information about genetically modified organisms (GMOs) on the Chinese social media platform Weibo. Drawing upon social contagion theory, we examine how endogenous and exogenous mechanisms shape users’ information-sharing behaviors. An analysis of 388,722 [...] Read more.

This study investigates the mechanisms underlying the diffusion of risk information about genetically modified organisms (GMOs) on the Chinese social media platform Weibo. Drawing upon social contagion theory, we examine how endogenous and exogenous mechanisms shape users’ information-sharing behaviors. An analysis of 388,722 reposts from 2444 original GMO risk-related texts enabled the construction of a comprehensive sharing network, with computational text-mining techniques employed to detect users’ attitudes toward GMOs. To bridge the gap between descriptive and inferential network analysis, we employ a Shannon entropy-based approach to quantify the uncertainty and concentration of attitudinal differences and similarities among sharing and non-sharing dyads, providing an information-theoretic foundation for understanding positional and differential homophily. The entropy-based analysis reveals that information-sharing ties are characterized by lower entropy in attitude differences, indicating greater attitudinal alignment among sharing users, especially among GMO opponents. Building on these findings, the Exponential Random Graph Model (ERGM) further demonstrates that both endogenous network mechanisms (reciprocity, preferential attachment, and triadic closure) and positional homophily influence GMO risk information sharing and dissemination. A key finding is the presence of a differential homophily effect, where GMO opponents exhibit stronger homophilic tendencies than non-opponents. Despite the prevalence of homophily, this paper uncovers substantial cross-attitude interactions, challenging simplistic notions of echo chambers in GMO risk communication. By integrating entropy and ERGM analyses, this study advances a more nuanced, information-theoretic understanding of how digital platforms mediate public perceptions and debates surrounding controversial socio-scientific issues, offering valuable implications for developing effective risk communication strategies in increasingly polarized online spaces. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

22 pages, 1467 KB

Open AccessArticle

DDML: Multi-Student Knowledge Distillation for Hate Speech

by Ze Liu, Zerui Shao, Haizhou Wang and Beibei Li

Entropy 2025, 27(4), 417; https://doi.org/10.3390/e27040417 - 11 Apr 2025

Viewed by 1600

Abstract

Recent studies have shown that hate speech on social media negatively impacts users’ mental health and is a contributing factor to suicide attempts. On a broader scale, online hate speech can undermine social stability. With the continuous growth of the internet, the prevalence [...] Read more.

Recent studies have shown that hate speech on social media negatively impacts users’ mental health and is a contributing factor to suicide attempts. On a broader scale, online hate speech can undermine social stability. With the continuous growth of the internet, the prevalence of online hate speech is rising, making its detection an urgent issue. Recent advances in natural language processing, particularly with transformer-based models, have shown significant promise in hate speech detection. However, these models come with a large number of parameters, leading to high computational requirements and making them difficult to deploy on personal computers. To address these challenges, knowledge distillation offers a solution by training smaller student networks using larger teacher networks. Recognizing that learning also occurs through peer interactions, we propose a knowledge distillation method called Deep Distill–Mutual Learning (DDML). DDML employs one teacher network and two or more student networks. While the student networks benefit from the teacher’s knowledge, they also engage in mutual learning with each other. We trained numerous deep neural networks for hate speech detection based on DDML and demonstrated that these networks perform well across various datasets. We tested our method across ten languages and nine datasets. The results demonstrate that DDML enhances the performance of deep neural networks, achieving an average F1 score increase of 4.87% over the baseline. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

20 pages, 2113 KB

Open AccessArticle

Identifying Influential Nodes Based on Evidence Theory in Complex Network

by Fu Tan, Xiaolong Chen, Rui Chen, Ruijie Wang, Chi Huang and Shimin Cai

Entropy 2025, 27(4), 406; https://doi.org/10.3390/e27040406 - 10 Apr 2025

Cited by 6 | Viewed by 1914

Abstract

Influential node identification is an important and hot topic in the field of complex network science. Classical algorithms for identifying influential nodes are typically based on a single attribute of nodes or the simple fusion of a few attributes. However, these methods perform [...] Read more.

Influential node identification is an important and hot topic in the field of complex network science. Classical algorithms for identifying influential nodes are typically based on a single attribute of nodes or the simple fusion of a few attributes. However, these methods perform poorly in real networks with high complexity and diversity. To address this issue, a new method based on the Dempster–Shafer (DS) evidence theory is proposed in this paper, which improves the efficiency of identifying influential nodes through the following three aspects. Firstly, Dempster–Shafer evidence theory quantifies uncertainty through its basic belief assignment function and combines evidence from different information sources, enabling it to effectively handle uncertainty. Secondly, Dempster–Shafer evidence theory processes conflicting evidence using Dempster’s rule of combination, enhancing the reliability of decision-making. Lastly, in complex networks, information may come from multiple dimensions, and the Dempster–Shafer theory can effectively integrate this multidimensional information. To verify the effectiveness of the proposed method, extensive experiments are conducted on real-world complex networks. The results show that, compared to the other algorithms, attacking the influential nodes identified by the DS method is more likely to lead to the disintegration of the network, which indicates that the DS method is more effective for identifying the key nodes in the network. To further validate the reliability of the proposed algorithm, we use the visibility graph algorithm to convert the GBP futures time series into a complex network and then rank the nodes in the network using the DS method. The results show that the top-ranked nodes correspond to the peaks and troughs of the time series, which represents the key turning points in price changes. By conducting an in-depth analysis, investors can uncover major events that influence price trends, once again confirming the effectiveness of the algorithm. Full article

(This article belongs to the Special Issue Complexity of Social Networks)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Complexity of Social Networks

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (13 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI