{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T08:16:50Z","timestamp":1772612210567,"version":"3.50.1"},"reference-count":45,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T00:00:00Z","timestamp":1772409600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"crossref","award":["2023D01A78"],"award-info":[{"award-number":["2023D01A78"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"crossref","award":["202501A65"],"award-info":[{"award-number":["202501A65"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["12361110"],"award-info":[{"award-number":["12361110"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Autonomous Region Level College Students\u2019 Innovation and Entrepreneurship Training Program","award":["S202410994017"],"award-info":[{"award-number":["S202410994017"]}]},{"name":"Xinjiang University Scientific Research","award":["XJEDU2024P080"],"award-info":[{"award-number":["XJEDU2024P080"]}]},{"name":"The Foundation of Xinjiang Institute of Engineering","award":["2024xgy092605"],"award-info":[{"award-number":["2024xgy092605"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study analyzes network coherence in two-layer and three-layer networks with positive and negative inter-layer correlation patterns. Using algebraic graph theory, we construct the Laplacian matrices for different correlated graph structures and compute their Laplacian spectra. The impact of correlation patterns on network coherence is investigated for two-layer and three-layer structures, and numerical evaluations are performed. The results show that negative-correlation patterns yield better network coherence than positive ones. This work provides fundamental insights into the structural robustness of layered networks and offers theoretical guidance for the design of robust networked systems.<\/jats:p>","DOI":"10.3390\/axioms15030182","type":"journal-article","created":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T14:06:56Z","timestamp":1772460416000},"page":"182","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Structural Robustness of Consensus Models with Layered Correlated Graphs"],"prefix":"10.3390","volume":"15","author":[{"given":"Zhi","family":"Song","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jian","family":"Zhu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Da","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xing","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhongming","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Energy Engineering, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1343","DOI":"10.1016\/j.automatica.2007.09.019","article-title":"Leader\u2013follower formation control of nonholonomic mobile robots with input constraints","volume":"44","author":"Consolini","year":"2008","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"162","DOI":"10.1109\/JSYST.2014.2318054","article-title":"Autonomous formation flight of multi-UAVs based on leader\u2013follower approach","volume":"10","author":"Kim","year":"2016","journal-title":"IEEE Syst. 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