{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:32:23Z","timestamp":1760236343790,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2021,11,21]],"date-time":"2021-11-21T00:00:00Z","timestamp":1637452800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["61802316"],"award-info":[{"award-number":["61802316"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation of Xinjiang","award":["2021D01A65"],"award-info":[{"award-number":["2021D01A65"]}]},{"name":"Scientific Research and Education Project of Xinjiang Institute of Engineering","award":["2020xgy332302"],"award-info":[{"award-number":["2020xgy332302"]}]},{"name":"National innovation and entrepreneurship training program for College Students","award":["202110994006"],"award-info":[{"award-number":["202110994006"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, several multi-layer-coupled star-composed networks with similar symmetrical structures are defined by using the theory of graph operation. The supra-Laplacian matrix of the corresponding multi-layer networks is obtained according to the master stability equation (MSF). Two important indexes that reflect the synchronizability of these kinds of networks are derived in the case of bounded and unbounded synchronized regions. The relationships among the synchronizability, the number of layers, the length of the paths, the branchings, and the interlayer and intralayer coupling strengths in the two cases are studied. At the same time, the simulation experiments are carried out with the MATLAB software, and the simulated images of the two symmetrical structure networks\u2019 synchronizability are compared. Finally, the factors affecting the synchronizability of multi-layer-coupled star-composed networks are found. On this basis, optimization schemes are given to improve the synchronizability of multi-layer-coupled star-composed networks and the influences of the number of central nodes on the networks\u2019 synchronizability are further studied.<\/jats:p>","DOI":"10.3390\/sym13112224","type":"journal-article","created":{"date-parts":[[2021,11,21]],"date-time":"2021-11-21T21:00:50Z","timestamp":1637528450000},"page":"2224","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Synchronizability of Multi-Layer-Coupled Star-Composed Networks"],"prefix":"10.3390","volume":"13","author":[{"given":"Haiping","family":"Gao","sequence":"first","affiliation":[{"name":"Department of Basic Science, Xinjiang Institute of Light Industry Technology, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xianyong","family":"Li","sequence":"additional","affiliation":[{"name":"School of Computer and Software Engineering, Xihua University, Chengdu 610039, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xing","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"128701","DOI":"10.1103\/PhysRevLett.111.128701","article-title":"Dynamical interplay between awareness and epidemic spreading in multiplex networks","volume":"111","author":"Granell","year":"2013","journal-title":"Phys. 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