{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T14:58:25Z","timestamp":1777733905702,"version":"3.51.4"},"reference-count":39,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04n05","funder":[{"name":"The Major Research Plan of the National Natural Science Foundation of China","award":["91530320"],"award-info":[{"award-number":["91530320"]}]},{"DOI":"10.13039\/501100001809","name":"The Chinese National Natural Science Foundation","doi-asserted-by":"crossref","award":["61672388"],"award-info":[{"award-number":["61672388"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advs. Complex Syst."],"published-print":{"date-parts":[[2017,6]]},"abstract":"<jats:p> The controllability of multilayer networks has become increasingly important in many areas of science and engineering. In this paper, we identify the general rules that determine the controllability and control energy cost of multilayer networks. First, we quantitatively estimate the control energy cost of multilayer networks and investigate the impacts of different coupling strength and coupling patterns on the control energy cost for multilayer networks. The results indicate that the average energy and the coupling strength have an approximately linear relationship in multilayer networks with two layers. Second, we study how the coupling strength and the connection patterns between different layers affect the controllability of multilayer networks from both theoretical and numerical aspects. The obtained piecewise functional relations between the controllability\u2019s measure and coupling strength reveal the existence of an optimal coupling strength for the different interconnection strategies in multilayer networks. In particular, the numerical experiments demonstrate that there exists a tradeoff between the optimal controllability and optimal control energy for selecting interlayer connection patterns in multilayer networks. These results provide a comprehensive understanding of the impact of interlayer couplings on the controllability and control energy cost for multilayer networks and provide a methodology for selecting the control nodes and coupling strength to maximize the controllability and minimize the control energy cost. <\/jats:p>","DOI":"10.1142\/s0219525917500084","type":"journal-article","created":{"date-parts":[[2017,8,29]],"date-time":"2017-08-29T03:25:34Z","timestamp":1503977134000},"page":"1750008","source":"Crossref","is-referenced-by-count":17,"title":["CONTROL ENERGY AND CONTROLLABILITY OF MULTILAYER NETWORKS"],"prefix":"10.1142","volume":"20","author":[{"given":"DINGJIE","family":"WANG","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China"},{"name":"Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, P. R. China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"XIUFEN","family":"ZOU","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China"},{"name":"Computational Science Hubei Key Laboratory, Wuhan University, Wuhan 430072, P. R. 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