{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T14:24:32Z","timestamp":1781533472771,"version":"3.54.5"},"reference-count":31,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,17]],"date-time":"2023-10-17T00:00:00Z","timestamp":1697500800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Laboratory of Intelligent Manufacturing Technology (Shan-tou University), the Ministry of Education of China","award":["202109242"],"award-info":[{"award-number":["202109242"]}]},{"name":"Key Laboratory of Intelligent Manufacturing Technology (Shan-tou University), the Ministry of Education of China","award":["61673120"],"award-info":[{"award-number":["61673120"]}]},{"name":"National Natural Science Foundation of China","award":["202109242"],"award-info":[{"award-number":["202109242"]}]},{"name":"National Natural Science Foundation of China","award":["61673120"],"award-info":[{"award-number":["61673120"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The mean square synchronization problem of the complex dynamical network (CDN) with the stochastic link dynamics is investigated. In contrast to previous literature, the CDN considered in this paper can be viewed as consisting of two subsystems coupled to each other. One subsystem consists of all nodes, referred to as the nodes subsystem, and the other consists of all links, referred to as the network topology subsystem, where the weighted values can quantitatively reflect changes in the network\u2019s topology. Based on the above understanding of CDN, two vector stochastic differential equations with Brownian motion are used to model the dynamic behaviors of nodes and links, respectively. The control strategy incorporates not only the controller in the nodes but also the coupling term in the links, through which the CDN is synchronized in the mean-square sense. Meanwhile, the dynamic stochastic signal is proposed in this paper, which is regarded as the auxiliary reference tracking target of links, such that the links can track the reference target asymptotically when synchronization occurs in nodes. This implies that the eventual topological structure of CDN is stochastic. Finally, a comparison simulation example confirms the superiority of the control strategy in this paper.<\/jats:p>","DOI":"10.3390\/e25101457","type":"journal-article","created":{"date-parts":[[2023,10,17]],"date-time":"2023-10-17T08:08:33Z","timestamp":1697530113000},"page":"1457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Synchronization of Complex Dynamical Networks with Stochastic Links Dynamics"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0110-8959","authenticated-orcid":false,"given":"Juanxia","family":"Zhao","sequence":"first","affiliation":[{"name":"School of Automation, Guangdong University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yinhe","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Automation, Guangdong University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peitao","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Electronics and Information, Guangdong Polytechnic Normal University, Guangzhou 510665, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shengping","family":"Li","sequence":"additional","affiliation":[{"name":"MOE Key Laboratory of Intelligent Manufacturing, Shantou University, Shantou 515063, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yi","family":"Peng","sequence":"additional","affiliation":[{"name":"School of Automation, Guangdong University of Technology, Guangzhou 510006, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"885","DOI":"10.1142\/S0218127402004802","article-title":"Complex networks: Topology, dynamics and synchronization","volume":"12","author":"Wang","year":"2002","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1109\/TAC.2006.872760","article-title":"Adaptive synchronization of an uncertain complex dynamical network","volume":"51","author":"Zhou","year":"2006","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_3","first-page":"5269","article-title":"Synchronization analysis for complex dynamical networks with coupling delay via event-triggered delayed impulsive control","volume":"51","author":"Cao","year":"2020","journal-title":"IEEE Trans. Cybern."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1334","DOI":"10.1109\/TCYB.2019.2902810","article-title":"PID control for synchronization of complex dynamical networks with directed topologies","volume":"51","author":"Gu","year":"2019","journal-title":"IEEE Trans. Cybern."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1109\/TNNLS.2013.2294727","article-title":"Synchronization of stochastic dynamical networks under impulsive control with time delays","volume":"25","author":"Zhang","year":"2013","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1016\/j.physa.2006.04.105","article-title":"Synchronization control of stochastic delayed neural networks","volume":"373","author":"Yu","year":"2007","journal-title":"Physica A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"113243","DOI":"10.1016\/j.chaos.2023.113243","article-title":"Explosive synchronization dependence on initial conditions: The minimal Kuramoto model","volume":"169","author":"Bayani","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"054304","DOI":"10.1103\/PhysRevE.105.054304","article-title":"Blinking coupling enhances network synchronization","volume":"105","author":"Parastesh","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"60","DOI":"10.1016\/j.neucom.2021.03.095","article-title":"Asymptotical state synchronization for the controlled directed complex dynamic network via links dynamics","volume":"448","author":"Gao","year":"2021","journal-title":"Neurocomputing"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"350","DOI":"10.1016\/j.ins.2023.01.124","article-title":"Tracking control of the nodes for the complex dynamical network with the auxiliary links dynamics","volume":"628","author":"Gao","year":"2023","journal-title":"Inf. Sci."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Li, X.X., Wang, Y.H., and Li, S.P. (2023). Double Model Following Adaptive Control for a Complex Dynamical Network. Entropy, 25.","DOI":"10.3390\/e25010115"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Winkler, L., Ojeda, C., and Opper, M. (2022). Stochastic Control for Bayesian Neural Network Training. Entropy, 24.","DOI":"10.3390\/e24081097"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1016\/j.automatica.2019.01.004","article-title":"Globally adaptive control for stochastic nonlinear time-delay systems with perturbations and its application","volume":"102","author":"Min","year":"2019","journal-title":"Automatica"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"20200236","DOI":"10.1098\/rsta.2020.0236","article-title":"Stochastic and vibrational resonance in complex networks of neurons","volume":"379","author":"Calim","year":"2021","journal-title":"Philos. Trans. R. Soc. A"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2017","DOI":"10.1016\/j.neucom.2008.12.006","article-title":"Mean square exponential stability of stochastic fuzzy hopfield neural networks with discrete and distribute time-varying delay","volume":"72","author":"Li","year":"2009","journal-title":"Neurocomputing"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3631","DOI":"10.1016\/j.apm.2010.03.012","article-title":"Finite-time stochastic synchronization of complex networks","volume":"34","author":"Yang","year":"2010","journal-title":"Appl. Math. Model."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1038\/nnano.2016.70","article-title":"Stochastic phase-change neurons","volume":"11","author":"Tuma","year":"2016","journal-title":"Nat. Nanotechnol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"241","DOI":"10.3389\/fnins.2016.00241","article-title":"Stochastic synapses enable efficient brain-inspired learning machines","volume":"10","author":"Neftci","year":"2016","journal-title":"Front. Neurosci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2571","DOI":"10.1038\/s41467-022-30305-8","article-title":"Neural sampling machine with stochastic synapse allows brain-like learning and inference","volume":"13","author":"Dutta","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1002\/asjc.180","article-title":"Adaptive synchronization of uncertain coupled stochastic complex networks","volume":"13","author":"Yu","year":"2011","journal-title":"Asian J. Control"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"135101","DOI":"10.1088\/1751-8113\/42\/13\/135101","article-title":"Global synchronization for delayed complex networks with randomly occurring nonlinearities and multiple stochastic disturbances","volume":"42","author":"Wang","year":"2009","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1109\/TNN.2009.2033599","article-title":"Global synchronization for discrete-time stochastic complex networks with randomly occurred nonlinearities and mixed time delays","volume":"21","author":"Wang","year":"2009","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"023705","DOI":"10.1063\/1.1938627","article-title":"Complete synchronization of the noise-perturbed Chua\u2019s circuits","volume":"15","author":"Lin","year":"2005","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.physd.2003.12.006","article-title":"Noise induced synchronization in a neuronal oscillator","volume":"92","author":"Pakdamana","year":"2004","journal-title":"Physica D"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1233","DOI":"10.1007\/s11063-020-10301-z","article-title":"Synchronization of stochastic complex dynamical networks with mixed time-varying coupling delays","volume":"52","author":"Ali","year":"2020","journal-title":"Neural Process. Lett."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3779","DOI":"10.1109\/TCYB.2019.2907279","article-title":"Synchronization of stochastic complex dynamical networks subject to consecutive packet dropouts","volume":"51","author":"Hu","year":"2021","journal-title":"IEEE Trans. Cybern."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2861","DOI":"10.1002\/rnc.3716","article-title":"Synchronization of stochastic complex dynamical networks under self-triggered control","volume":"27","author":"Xing","year":"2017","journal-title":"Int. J. Robust Nonlinear Control"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"101200","DOI":"10.1016\/j.nahs.2022.101200","article-title":"Stability for multi-linked stochastic delayed complex networks with stochastic hybrid impulses by Dupire It\u00f4\u2019s formula","volume":"45","author":"Zhang","year":"2022","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_29","unstructured":"Slotine, J.J.E., and Li, W. (1991). Applied Nonlinear Control, Prentice-Hall."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2252","DOI":"10.1016\/j.nonrwa.2011.01.007","article-title":"Synchronization of delayed complex dynamical networks with impulsive and stochastic effects","volume":"12","author":"Yang","year":"2011","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.isatra.2018.09.007","article-title":"Tracking control of an underactuated ship by modified dynamic inversion","volume":"83","author":"Ye","year":"2018","journal-title":"ISA Trans."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1457\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:08:20Z","timestamp":1760130500000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1457"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,17]]},"references-count":31,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["e25101457"],"URL":"https:\/\/doi.org\/10.3390\/e25101457","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,17]]}}}