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Entropy
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Emergent Dynamics of Complex Systems: From Synchronization to Clustering
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Emergent Dynamics of Complex Systems: From Synchronization to Clustering

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 15 June 2026 | Viewed by 4281

Special Issue Editor

Prof. Dr. Zhigang Zheng

E-Mail Website
Guest Editor
Institute of Systems Science, Huaqiao University, Xiamen 361021, China
Interests: non-equilibrium statistical physics; complex systems; nonlinear dynamics; synchronization; swarming dynamics; neuron dynamics
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Complex systems, which are prevalent in nature and human society, exhibit fascinating emergent behaviors, e.g., phase transition, synchronization, clustering, chimeras, nonlinear waves, and swarming dynamics. Understanding the mechanism embedded in these behaviors is crucial for various fields, including physics, biology, and engineering.

This Special Issue aims to bring together researchers to explore the emergent dynamics of complex systems, focusing on the dynamics and statistical physics of synchronization, clustering, and swarming phenomena.

We invite submissions that investigate the underlying mechanisms, mathematical models, and real-world applications related to these topics. We are interested in a wide range of research areas, such as the synchronization of coupled oscillators, the formation of clusters in social and biological networks, and the swarming dynamics of active particles. Both theoretical and empirical studies are welcome. We also encourage interdisciplinary approaches that combine methods from different disciplines to provide new insights into the emergent dynamics of complex systems.

Manuscripts should be original and not under consideration for publication elsewhere. We invite authors to submit full-length research articles, review papers, and short communications. For more information, please visit the journal’s website or contact the Guest Editors.

We look forward to receiving your contributions and, together, advancing our understanding of emergent dynamics in complex systems.

Prof. Dr. Zhigang Zheng
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • complex systems
  • emergence
  • self-organization
  • synchronization
  • swarming
  • chaos
  • nonlinear dynamics
  • statistical physics
  • phase transition

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Published Papers (5 papers)

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Research

21 pages, 3863 KB  
Article
K-Means Community Detection Algorithm Based on Density Peaks
by Hongyan Gao, Jing Han, Yue Liu, Peng Zhang, Bo Yang, Yanqing Zu, Fei Liu and Yu Qian
Entropy 2026, 28(2), 152; https://doi.org/10.3390/e28020152 - 29 Jan 2026
Viewed by 516
Abstract
The identification of community structure is pivotal for understanding the functional characteristics of complex networks. To address the limitations of most existing community detection algorithms, which often require predefining the number of communities and lack robustness, this paper proposes a novel community detection [...] Read more.
The identification of community structure is pivotal for understanding the functional characteristics of complex networks. To address the limitations of most existing community detection algorithms, which often require predefining the number of communities and lack robustness, this paper proposes a novel community detection algorithm named D-means (K-means community detection algorithm based on density peaks). This algorithm integrates the concept of density peak clustering with K-means spectral clustering, employing Chebyshev’s inequality to automatically determine the number of community centers, thereby enabling unsupervised identification of community quantities. By designing a multi-dimensional evaluation framework, the comparative experiments were conducted on LFR benchmark networks (Lancichinetti-Fortunato-Radicchi benchmark networks) and real-world social network datasets. The results demonstrate that the D-means algorithm outperforms traditional algorithms in terms of ACC (accuracy), ARI (adjusted rand index), and NMI (normalized mutual information) metrics, while also achieving improvements in runtime efficiency, showcasing strong robustness. Finally, the D-means algorithm was applied to the public transportation network of Urumqi. Empirical analysis identified 12 functionally significant transportation communities, providing theoretical support for urban rail transit optimization and commercial facility layout planning. Full article
(This article belongs to the Special Issue Emergent Dynamics of Complex Systems: From Synchronization to Clustering)
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12 pages, 27122 KB  
Article
Orientation-Modulated Hyperuniformity in Frustrated Vicsek–Kuramoto Systems
by Yichen Lu, Tong Zhu, Yingshan Guo, Yunyun Li and Zhigang Zheng
Entropy 2026, 28(1), 126; https://doi.org/10.3390/e28010126 - 21 Jan 2026
Cited by 1 | Viewed by 442
Abstract
In the study of disordered hyperuniformity, which bridges ordered and disordered states and has broad implications in physics and biology, active matter systems offer a rich platform for spontaneous pattern formation. This work investigates frustrated Vicsek–Kuramoto systems, where frustration induces complex collective behaviors, [...] Read more.
In the study of disordered hyperuniformity, which bridges ordered and disordered states and has broad implications in physics and biology, active matter systems offer a rich platform for spontaneous pattern formation. This work investigates frustrated Vicsek–Kuramoto systems, where frustration induces complex collective behaviors, to explore how hyperuniform states arise. We numerically analyze the phase diagram via the structure factor S(q) and the density variance ⟨δρ2R⟩. Results show that recessive lattice states exhibit Class I hyperuniformity under high coupling strength and intermediate frustration, emerging from the interplay of frustration-induced periodicity and active motion, characterized by dynamic, drifting rotation centers rather than static order. Notably, global hyperuniformity emerges from the spatial complementarity of orientation subgroups that are individually non-hyperuniform, a phenomenon termed “orientation-modulated hyperuniformity”. This work establishes frustration as a novel mechanism for generating hyperuniform states in active matter, highlighting how anisotropic interactions can yield global order from disordered components, with potential relevance to biological systems and material science. Full article
(This article belongs to the Special Issue Emergent Dynamics of Complex Systems: From Synchronization to Clustering)
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23 pages, 2095 KB  
Article
From Agent-Based Markov Dynamics to Hierarchical Closures on Networks: Emergent Complexity and Epidemic Applications
by A. Y. Klimenko, A. Rozycki and Y. Lu
Entropy 2026, 28(1), 63; https://doi.org/10.3390/e28010063 - 5 Jan 2026
Viewed by 433
Abstract
We explore a rigorous formulation of agent-based SIR epidemic dynamics as a discrete-state Markov process, capturing the stochastic propagation of infection or an invading agent on networks. Using indicator functions and corresponding marginal probabilities, we derive a hierarchy of evolution equations that resembles [...] Read more.
We explore a rigorous formulation of agent-based SIR epidemic dynamics as a discrete-state Markov process, capturing the stochastic propagation of infection or an invading agent on networks. Using indicator functions and corresponding marginal probabilities, we derive a hierarchy of evolution equations that resembles the classical BBGKY hierarchy in statistical mechanics. The structure of these equations clarifies the challenges of closure and highlights the principal problem of systemic complexity arising from stochastic but generally not fully chaotic interactions. Monte Carlo simulations are used to validate simplified closures and approximations, offering a unified perspective on the interplay between network topology, stochasticity, and infection dynamics. We also explore the impact of lockdown measures within a networked agent framework, illustrating how SIR dynamics and structural complexity of the network shape epidemic with propagation of the COVID-19 pandemic in Northern Italy taken as an example. Full article
(This article belongs to the Special Issue Emergent Dynamics of Complex Systems: From Synchronization to Clustering)
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12 pages, 4220 KB  
Article
Experimental and Numerical Study of Coupled Metronomes on a Floating Platform
by Xiaolongzi Wu, Caiyi Zheng, Zhao Lei, Yu Qian, Zengru Di and Xiaohua Cui
Entropy 2025, 27(9), 908; https://doi.org/10.3390/e27090908 - 27 Aug 2025
Viewed by 1345
Abstract
We investigated synchronization behavior using an experimental setup consisting of two metronomes placed on a platform floating over water. By setting the metronomes to oscillate perpendicular to the line between them, we observed three distinct modes of movement: in-phase synchronization, anti-phase synchronization, and [...] Read more.
We investigated synchronization behavior using an experimental setup consisting of two metronomes placed on a platform floating over water. By setting the metronomes to oscillate perpendicular to the line between them, we observed three distinct modes of movement: in-phase synchronization, anti-phase synchronization, and synchronization with a fixed phase difference. While this last mode resembles phase-locking, it is important to distinguish that phase-locking typically refers to an oscillator’s response to external pacing, whereas the fixed phase difference observed in our study emerges from the mutual interaction between two metronomes. The frequencies of oscillations, and the placement of the metronomes are also changed to check the reliability of the new phenomenon. Even if we changed the material of the platform to a heavier one or turned around one of the metronomes, synchronization with a fixed time delay still was still observed. Drawing on previous research, we developed mathematical equations to model the coupled metronomes and performed numerical simulations that successfully reproduced all three observed phenomena. The simulation results showed excellent agreement with our experimental observations. These findings contribute to our understanding of coupled oscillators and may have potential applications in various fields. Full article
(This article belongs to the Special Issue Emergent Dynamics of Complex Systems: From Synchronization to Clustering)
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10 pages, 2269 KB  
Article
Impact of Calcium and Potassium Currents on Spiral Wave Dynamics in the LR1 Model
by Xiaoping Yuan and Qianqian Zheng
Entropy 2025, 27(7), 690; https://doi.org/10.3390/e27070690 - 27 Jun 2025
Viewed by 863
Abstract
Spiral wave dynamics in cardiac tissue are critically implicated in the pathogenesis of arrhythmias. This study investigates the effects of modulating calcium and potassium currents on spiral wave stability in a two-dimensional cardiac model. The gate variable that dynamically regulates the opening probability [...] Read more.
Spiral wave dynamics in cardiac tissue are critically implicated in the pathogenesis of arrhythmias. This study investigates the effects of modulating calcium and potassium currents on spiral wave stability in a two-dimensional cardiac model. The gate variable that dynamically regulates the opening probability of ion channels also plays a significant role in the control of the spiral wave dynamics. We demonstrate that reducing gate variables accelerates wave propagation, thins spiral arms, and shortens action potential duration, ultimately inducing dynamic instability. Irregular electrocardiogram (ECG) patterns and altered action potential morphology further suggest an enhanced arrhythmogenic potential. These findings elucidate the ionic mechanisms underlying spiral wave breakup, providing both theoretical insights and practical implications for the development of targeted arrhythmia treatments. Full article
(This article belongs to the Special Issue Emergent Dynamics of Complex Systems: From Synchronization to Clustering)
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