{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:49:57Z","timestamp":1760244597166,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T00:00:00Z","timestamp":1672012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Science Foundation of Xinjiang (NSFXJ)","award":["2019D01B10","62003289"],"award-info":[{"award-number":["2019D01B10","62003289"]}]},{"name":"National Natural Science Foundation of Peoples Republic of China (NSFC)","award":["2019D01B10","62003289"],"award-info":[{"award-number":["2019D01B10","62003289"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This work studies the first-order coherence of noisy multi-agent networks with multi-layered structures. The coherence, which is a sort of performance index of networks, can be seen as a sort of measurement for a system\u2019s robustness. Graph operations are applied to design the novel multi-layered networks, and a graph spectrum approach, along with analysis methods, is applied to derive the mathematical expression of the coherence, and the corresponding asymptotic results on the performance index have been obtained. In addition, the coherence of these non-isomorphic multi-layered networks with three different graph parameters are compared and analyzed. We find that, when the cardinalities of the vertex sets of corresponding counterpart layers are the same, the multi-layered topology class with a balanced, complete, multi-partite structure has the best robustness of all the considered networks, if the sufficient conditions for the node-related parameters hold. Finally, simulations are given to verify the asymptotic results.<\/jats:p>","DOI":"10.3390\/e25010040","type":"journal-article","created":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T04:45:54Z","timestamp":1672116354000},"page":"40","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Consensus Performance of Multi-Layered MASs with Various Graph Parameters\u2014From the Perspective of Cardinalities of Vertex Sets"],"prefix":"10.3390","volume":"25","author":[{"given":"Da","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Mathematics and Physics, Xinjiang Institute of Engineering, Urumqi 830023, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhiyong","family":"Yu","sequence":"additional","affiliation":[{"name":"College of Mathematics and System Science, Xinjiang University, Urumqi 830017, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1520","DOI":"10.1109\/TAC.2004.834113","article-title":"Consensus problems in Networks of Agents With Switching Topology and Time-Delays","volume":"49","author":"Saber","year":"2004","journal-title":"IEEE Trans. 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