{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T01:25:55Z","timestamp":1776475555766,"version":"3.51.2"},"reference-count":54,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,7,19]],"date-time":"2023-07-19T00:00:00Z","timestamp":1689724800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Heterogeneity among interacting units plays an important role in numerous biological and man-made complex systems. While the impacts of heterogeneity on synchronization, in terms of structural mismatch of the layers in multiplex networks, has been studied thoroughly, its influence on intralayer synchronization, in terms of parameter mismatch among the layers, has not been adequately investigated. Here, we study the intralayer synchrony in multiplex networks, where the layers are different from one other, due to parameter mismatch in their local dynamics. In such a multiplex network, the intralayer coupling strength for the emergence of intralayer synchronization decreases upon the introduction of impurity among the layers, which is caused by a parameter mismatch in their local dynamics. Furthermore, the area of occurrence of intralayer synchronization also widens with increasing mismatch. We analytically derive a condition under which the intralayer synchronous solution exists, and we even sustain its stability. We also prove that, in spite of the mismatch among the layers, all the layers of the multiplex network synchronize simultaneously. Our results indicate that a multiplex network with mismatched layers can induce synchrony more easily than a multiplex network with identical layers.<\/jats:p>","DOI":"10.3390\/e25071083","type":"journal-article","created":{"date-parts":[[2023,7,19]],"date-time":"2023-07-19T21:20:14Z","timestamp":1689801614000},"page":"1083","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Synchronization Induced by Layer Mismatch in Multiplex Networks"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3846-2601","authenticated-orcid":false,"given":"Md Sayeed","family":"Anwar","sequence":"first","affiliation":[{"name":"Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sarbendu","family":"Rakshit","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, University of California, Riverside, CA 92521, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5926-4276","authenticated-orcid":false,"given":"J\u00fcrgen","family":"Kurths","sequence":"additional","affiliation":[{"name":"Potsdam Institute for Climate Impact Research, Telegraphenberg A 31, 14473 Potsdam, Germany"},{"name":"Department of Physics, Humboldt University Berlin, 12489 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4832-5210","authenticated-orcid":false,"given":"Dibakar","family":"Ghosh","sequence":"additional","affiliation":[{"name":"Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,7,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Newman, M. (2018). Networks, Oxford University Press.","DOI":"10.1093\/oso\/9780198805090.001.0001"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Latora, V., Nicosia, V., and Russo, G. (2017). Complex Networks: Principles, Methods and Applications, Cambridge University Press.","DOI":"10.1017\/9781316216002"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1137\/S003614450342480","article-title":"The structure and function of complex networks","volume":"45","author":"Newman","year":"2003","journal-title":"SIAM Rev."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"268","DOI":"10.1038\/35065725","article-title":"Exploring complex networks","volume":"410","author":"Strogatz","year":"2001","journal-title":"Nature"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1093\/comnet\/cnu016","article-title":"Multilayer networks","volume":"2","author":"Arenas","year":"2014","journal-title":"J. Complex Netw."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Bianconi, G. (2018). Multilayer Networks: Structure and Function, Oxford University Press.","DOI":"10.1093\/oso\/9780198753919.001.0001"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2014.07.001","article-title":"The structure and dynamics of multilayer networks","volume":"544","author":"Boccaletti","year":"2014","journal-title":"Phys. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1038\/srep01344","article-title":"Emergence of network features from multiplexity","volume":"3","author":"Cardillo","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"E680","DOI":"10.1073\/pnas.1110586109","article-title":"Suppressing cascades of load in interdependent networks","volume":"109","author":"Brummitt","year":"2012","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"13636","DOI":"10.1073\/pnas.1004008107","article-title":"Multirelational organization of large-scale social networks in an online world","volume":"107","author":"Szell","year":"2010","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"0101","DOI":"10.1038\/s41559-017-0101","article-title":"The multilayer nature of ecological networks","volume":"1","author":"Pilosof","year":"2017","journal-title":"Nat. Ecol. Evol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"028701","DOI":"10.1103\/PhysRevLett.110.028701","article-title":"Diffusion dynamics on multiplex networks","volume":"110","author":"Gomez","year":"2013","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"032804","DOI":"10.1103\/PhysRevE.89.032804","article-title":"Structural measures for multiplex networks","volume":"89","author":"Battiston","year":"2014","journal-title":"Phys. Rev. E"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1140\/epjb\/e2015-50742-1","article-title":"Towards real-world complexity: An introduction to multiplex networks","volume":"88","author":"Lee","year":"2015","journal-title":"Eur. Phys. J. B"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Barrat, A., Barthelemy, M., and Vespignani, A. (2008). Dynamical Processes on Complex Networks, Cambridge University Press.","DOI":"10.1017\/CBO9780511791383"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","article-title":"Critical phenomena in complex networks","volume":"80","author":"Dorogovtsev","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"138302","DOI":"10.1103\/PhysRevLett.118.138302","article-title":"Collective phenomena emerging from the interactions between dynamical processes in multiplex networks","volume":"118","author":"Nicosia","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Pikovsky, A., Kurths, J., and Rosenblum, M. (2001). Synchronization: A Universal Concept in Nonlinear Sciences, Cambridge University Press.","DOI":"10.1017\/CBO9780511755743"},{"key":"ref_21","first-page":"172","article-title":"Sync: The emerging science of spontaneous order","volume":"41","author":"Strogatz","year":"2004","journal-title":"Phys. Today"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"821","DOI":"10.1103\/PhysRevLett.64.821","article-title":"Synchronization in chaotic systems","volume":"64","author":"Pecora","year":"1990","journal-title":"Phys. Rev. Lett."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"034101","DOI":"10.1103\/PhysRevLett.98.034101","article-title":"Paths to synchronization on complex networks","volume":"98","author":"Moreno","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"4191","DOI":"10.1109\/TNNLS.2020.3017171","article-title":"Synchronization of switched discrete-time neural networks via quantized output control with actuator fault","volume":"32","author":"Yang","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Wang, H., Yang, X., Xiang, Z., Tang, R., and Ning, Q. (2022). Synchronization of switched neural networks via attacked mode-dependent event-triggered control and its application in image encryption. IEEE Trans. Cybern., 1\u201310.","DOI":"10.1109\/TCYB.2022.3227021"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"4545","DOI":"10.1109\/TCYB.2022.3208156","article-title":"Novel LKF Method on H\u221e Synchronization of Switched Time-Delay Systems","volume":"53","author":"Qi","year":"2023","journal-title":"IEEE Trans. Cybern."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"105436","DOI":"10.1016\/j.sysconle.2022.105436","article-title":"Monotone decreasing LKF method for secure consensus of second-order mass with delay and switching topology","volume":"172","author":"Zhang","year":"2023","journal-title":"Syst. Control Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3179","DOI":"10.1038\/s41467-020-16343-0","article-title":"Symmetries and cluster synchronization in multilayer networks","volume":"11","author":"Pecora","year":"2020","journal-title":"Nat. Commun."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"30002","DOI":"10.1209\/0295-5075\/113\/30002","article-title":"Cluster synchronization in multiplex networks","volume":"113","author":"Jalan","year":"2016","journal-title":"Europhys. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"038701","DOI":"10.1103\/PhysRevLett.114.038701","article-title":"Explosive synchronization in adaptive and multilayer networks","volume":"114","author":"Zhang","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"073109","DOI":"10.1063\/1.4993836","article-title":"Chimera states in a multilayer network of coupled and uncoupled neurons","volume":"27","author":"Majhi","year":"2017","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"20010","DOI":"10.1209\/0295-5075\/110\/20010","article-title":"Intra-layer synchronization in multiplex networks","volume":"110","author":"Gambuzza","year":"2015","journal-title":"EPL Europhys. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"024303","DOI":"10.1103\/PhysRevE.105.024303","article-title":"Stability analysis of intralayer synchronization in time-varying multilayer networks with generic coupling functions","volume":"105","author":"Anwar","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"45475","DOI":"10.1038\/srep45475","article-title":"Inter-layer synchronization in non-identical multi-layer networks","volume":"7","author":"Leyva","year":"2017","journal-title":"Sci. Rep."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"065304","DOI":"10.1063\/1.4952967","article-title":"Inter-layer synchronization in multiplex networks of identical layers","volume":"26","author":"Leyva","year":"2016","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"033125","DOI":"10.1063\/5.0074641","article-title":"Intralayer and interlayer synchronization in multiplex network with higher-order interactions","volume":"32","author":"Anwar","year":"2022","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Anwar, M.S., Ghosh, D., and Frolov, N. (2021). Relay synchronization in a weighted triplex network. Mathematics, 9.","DOI":"10.3390\/math9172135"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"8629","DOI":"10.1038\/s41598-018-26945-w","article-title":"Relay synchronization in multiplex networks","volume":"8","author":"Leyva","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"051104","DOI":"10.1063\/5.0008341","article-title":"Effect of topology upon relay synchronization in triplex neuronal networks","volume":"30","author":"Drauschke","year":"2020","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"034314","DOI":"10.1103\/PhysRevE.106.034314","article-title":"Stability of synchronization in simplicial complexes with multiple interaction layers","volume":"106","author":"Anwar","year":"2022","journal-title":"Phys. Rev. E"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"681","DOI":"10.1088\/1361-6544\/ac3c2f","article-title":"Relay interlayer synchronisation: Invariance and stability conditions","volume":"35","author":"Rakshit","year":"2022","journal-title":"Nonlinearity"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"074101","DOI":"10.1103\/PhysRevLett.98.074101","article-title":"Wave mediated synchronization of nonuniform oscillatory media","volume":"98","author":"Kheowan","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2353","DOI":"10.1103\/PhysRevE.55.2353","article-title":"Phase synchronization effects in a lattice of nonidentical R\u00f6ssler oscillators","volume":"55","author":"Osipov","year":"1997","journal-title":"Phys. Rev. E"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"037114","DOI":"10.1063\/1.2979693","article-title":"Synchronization in interacting populations of heterogeneous oscillators with time-varying coupling","volume":"18","author":"So","year":"2008","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"012203","DOI":"10.1103\/PhysRevE.94.012203","article-title":"Synchronization in heterogeneous FitzHugh-Nagumo networks with hierarchical architecture","volume":"94","author":"Plotnikov","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"036107","DOI":"10.1103\/PhysRevE.77.036107","article-title":"Synchronization in networks of networks: The onset of coherent collective behavior in systems of interacting populations of heterogeneous oscillators","volume":"77","author":"Barreto","year":"2008","journal-title":"Phys. Rev. E"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"60011","DOI":"10.1209\/0295-5075\/85\/60011","article-title":"Master stability functions for coupled nearly identical dynamical systems","volume":"85","author":"Sun","year":"2009","journal-title":"EPL Europhys. Lett."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"107127","DOI":"10.1016\/j.cnsns.2023.107127","article-title":"Creation of heterogeneity or defects in a memristive neural network under energy flow","volume":"119","author":"Yang","year":"2023","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1007\/s11431-022-2188-2","article-title":"Formation of local heterogeneity under energy collection in neural networks","volume":"66","author":"Xie","year":"2023","journal-title":"Sci. China Technol. Sci."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"2109","DOI":"10.1103\/PhysRevLett.80.2109","article-title":"Master stability functions for synchronized coupled systems","volume":"80","author":"Pecora","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"012304","DOI":"10.1103\/PhysRevE.99.012304","article-title":"Master stability functions for complete, intralayer, and interlayer synchronization in multiplex networks of coupled R\u00f6ssler oscillators","volume":"99","author":"Tang","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1038\/s41467-021-21486-9","article-title":"Stability of synchronization in simplicial complexes","volume":"12","author":"Gambuzza","year":"2021","journal-title":"Nat. Commun."},{"key":"ref_53","unstructured":"Erd\u00f6s, P., and R\u00e9nyi, A. (2011). The Structure and Dynamics of Networks, Princeton University Press."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","article-title":"An equation for continuous chaos","volume":"57","year":"1976","journal-title":"Phys. Lett. A"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/7\/1083\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:14:43Z","timestamp":1760127283000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/7\/1083"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,19]]},"references-count":54,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,7]]}},"alternative-id":["e25071083"],"URL":"https:\/\/doi.org\/10.3390\/e25071083","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,7,19]]}}}