{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T13:57:29Z","timestamp":1781272649024,"version":"3.54.1"},"reference-count":64,"publisher":"AIP Publishing","issue":"6","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62071248"],"award-info":[{"award-number":["62071248"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62201284"],"award-info":[{"award-number":["62201284"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Postgraduate Research &amp; Practice Innovation Program of Jiangsu Province","award":["KYCX25_1240"],"award-info":[{"award-number":["KYCX25_1240"]}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,6,1]]},"abstract":"<jats:p>With the proliferation of online social networks, rumors can spread rapidly, amplifying public-opinion risks and threatening social stability. To address this issue, we develop a six-state quantum-like rumor propagation model on a static social network by coupling ternary beliefs with binary behaviors. Unlike traditional discrete-state models, the proposed model uses a density operator to characterize individual states and their temporal evolution, allowing multiple belief-behavior tendencies to coexist. Decoherence models coherence loss induced by environmental noise, while a time-dependent multi-channel relaxation mechanism captures late-stage suppression and drives the dynamics from diffusion dominance to inhibition dominance. We further show that the resulting single-node evolution map is completely positive and trace-preserving, ensuring a valid probabilistic interpretation. Simulations show that forwarding-related states rise rapidly in the early stage and then approach a stable regime dominated by disbelief and non-forwarding as decoherence and dissipation increase. At the belief level, neutral groups differentiate quickly and gradually evolve toward disbelief dominance, while belief and neutral states persist as low-level long tails. At the behavioral level, a bimodal forwarding pattern emerges, and later forwarders decay into a long tail. Parameter sensitivity analysis shows that rumor coupling, coherence, and relaxation jointly shape spreading intensity, peak structure, and long-tail behavior. An illustrative comparison with a real rumor sequence and two classical baseline models, namely the susceptible-infected-recovered (SIR) model and the susceptible-exposed-infected-recovered (SEIR) model, further shows that the proposed model can capture early-stage bimodal spreading and middle-to-late-stage long-tail decay in a representative case while offering an interpretable framework for analyzing rumor propagation and suppression mechanisms.<\/jats:p>","DOI":"10.1063\/5.0335015","type":"journal-article","created":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T13:06:24Z","timestamp":1781269584000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["A quantum-like rumor propagation model captures bimodal spreading dynamics in social networks"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0004-3742-7849","authenticated-orcid":false,"given":"Haojie","family":"Hou","sequence":"first","affiliation":[{"name":"School of Science, Nanjing University of Posts and Telecommunications , Nanjing 210023,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5369-3133","authenticated-orcid":false,"given":"Youguo","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Science, Nanjing University of Posts and Telecommunications , Nanjing 210023,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3649-3893","authenticated-orcid":false,"given":"Qiqing","family":"Zhai","sequence":"additional","affiliation":[{"name":"School of Science, Nanjing University of Posts and Telecommunications , Nanjing 210023,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,6,12]]},"reference":[{"key":"2026061209061654300_c1","doi-asserted-by":"publisher","first-page":"daaf023","DOI":"10.1093\/heapro\/daaf023","article-title":"Social media and the spread of misinformation: Infectious and a threat to public health","volume":"40","year":"2025","journal-title":"Health Promot. Int."},{"key":"2026061209061654300_c2","doi-asserted-by":"publisher","first-page":"102780","DOI":"10.1016\/j.ijinfomgt.2024.102780","article-title":"Social media trust: Fighting misinformation in the time of crisis","volume":"77","year":"2024","journal-title":"Int. J. Inf. Manage."},{"key":"2026061209061654300_c3","doi-asserted-by":"publisher","first-page":"108546","DOI":"10.1016\/j.chb.2024.108546","article-title":"Large-scale analysis of online social data on the long-term sentiment and content dynamics of online (mis)information","volume":"165","year":"2025","journal-title":"Comput. Hum. Behav."},{"key":"2026061209061654300_c4","doi-asserted-by":"publisher","first-page":"2397","DOI":"10.1109\/TCSS.2023.3277465","article-title":"Rumor propagation control with anti-rumor mechanism and intermittent control strategies","volume":"11","year":"2024","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"key":"2026061209061654300_c5","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1038\/2041118a0","article-title":"Epidemics and rumours","volume":"204","year":"1964","journal-title":"Nature"},{"key":"2026061209061654300_c6","doi-asserted-by":"publisher","first-page":"457","DOI":"10.1016\/j.physa.2006.07.017","article-title":"Theory of rumour spreading in complex social networks","volume":"374","year":"2007","journal-title":"Physica A"},{"key":"2026061209061654300_c7","doi-asserted-by":"publisher","first-page":"2103","DOI":"10.1016\/j.physleta.2012.05.021","article-title":"An information diffusion model based on retweeting mechanism for online social media","volume":"376","year":"2012","journal-title":"Phys. Lett. A"},{"key":"2026061209061654300_c8","doi-asserted-by":"publisher","first-page":"995","DOI":"10.1016\/j.physa.2012.09.030","article-title":"SIR rumor spreading model in the new media age","volume":"392","year":"2013","journal-title":"Physica A"},{"key":"2026061209061654300_c9","doi-asserted-by":"publisher","first-page":"015002","DOI":"10.1088\/1572-9494\/ab4ef6","article-title":"SIS model of rumor spreading in social network with time delay and nonlinear functions","volume":"72","year":"2020","journal-title":"Commun. Theor. Phys."},{"key":"2026061209061654300_c10","doi-asserted-by":"publisher","first-page":"127887","DOI":"10.1016\/j.amc.2023.127887","article-title":"Dynamical analysis of hyper-SIR rumor spreading model","volume":"446","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"2026061209061654300_c11","doi-asserted-by":"publisher","first-page":"043104","DOI":"10.1063\/5.0195918","article-title":"A multilayer network model of interaction between rumor propagation and media influence","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026061209061654300_c12","doi-asserted-by":"publisher","first-page":"12419","DOI":"10.1007\/s11071-024-10817-2","article-title":"Synergistic control of negative information diffusion in improved semi-randomized epidemic networks","volume":"113","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"2026061209061654300_c13","doi-asserted-by":"publisher","first-page":"39590","DOI":"10.1038\/s41598-025-22166-0","article-title":"A dynamics-based model of multi-state rumor propagation with triple identities","volume":"15","year":"2025","journal-title":"Sci. Rep."},{"key":"2026061209061654300_c14","doi-asserted-by":"publisher","first-page":"103105","DOI":"10.1063\/5.0225731","article-title":"Misinformation spreading on activity-driven networks with heterogeneous spreading rates","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026061209061654300_c15","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1145\/3040230.3040233","article-title":"Competitive rumor spread in social networks","volume":"44","year":"2017","journal-title":"ACM SIGMETRICS Perform. Eval. Rev."},{"key":"2026061209061654300_c16","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1109\/TNSE.2017.2783190","article-title":"An efficient randomized algorithm for rumor blocking in online social networks","volume":"7","year":"2020","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"2026061209061654300_c17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3701738","article-title":"Adaptive rumor suppression on social networks: A multi-round hybrid approach","volume":"19","year":"2025","journal-title":"ACM Trans. Knowl. Discov. Data"},{"key":"2026061209061654300_c18","doi-asserted-by":"publisher","first-page":"105265","DOI":"10.1016\/j.knosys.2019.105265","article-title":"Influence maximization based on the realistic independent cascade model","volume":"191","year":"2020","journal-title":"Knowl.-Based Syst."},{"key":"2026061209061654300_c19","doi-asserted-by":"publisher","first-page":"083127","DOI":"10.1063\/5.0011658","article-title":"A generalized linear threshold model for an improved description of the spreading dynamics","volume":"30","year":"2020","journal-title":"Chaos"},{"key":"2026061209061654300_c20","doi-asserted-by":"publisher","first-page":"083101","DOI":"10.1063\/5.0283611","article-title":"Tale of one emergent game: Opinion formation in dynamical undirected hypergraphs","volume":"35","year":"2025","journal-title":"Chaos"},{"key":"2026061209061654300_c21","doi-asserted-by":"publisher","first-page":"130837","DOI":"10.1016\/j.physa.2025.130837","article-title":"Analysis of three-party quantum game strategy in rumor spreading","volume":"675","year":"2025","journal-title":"Physica A"},{"key":"2026061209061654300_c22","doi-asserted-by":"publisher","first-page":"073117","DOI":"10.1063\/5.0084178","article-title":"Game-theoretical approach for opinion dynamics on social networks","volume":"32","year":"2022","journal-title":"Chaos"},{"key":"2026061209061654300_c23","doi-asserted-by":"publisher","first-page":"033131","DOI":"10.1063\/5.0259050","article-title":"Rumor propagation in the framework of evolutionary game analysis","volume":"35","year":"2025","journal-title":"Chaos"},{"key":"2026061209061654300_c24","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1007\/s40745-025-00606-y","article-title":"Rumor governance under uncertain conditions: An evolutionary game theory analysis","volume":"12","year":"2025","journal-title":"Ann. Data Sci."},{"key":"2026061209061654300_c25","doi-asserted-by":"publisher","first-page":"3625","DOI":"10.1109\/TNSE.2025.3563360","article-title":"A rumor dissemination control model based on evolutionary game and multiple user states","volume":"12","year":"2025","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"2026061209061654300_c26","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1038\/s42005-025-02181-3","article-title":"Adaptive rumor propagation and activity contagion in higher-order networks","volume":"8","year":"2025","journal-title":"Commun. Phys."},{"key":"2026061209061654300_c27","doi-asserted-by":"publisher","first-page":"2441","DOI":"10.1038\/s41562-025-02373-5","article-title":"Higher-order interactions shape collective human behaviour","volume":"9","year":"2025","journal-title":"Nat. Hum. Behav."},{"key":"2026061209061654300_c28","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1038\/s42005-024-01666-x","article-title":"Higher-order non-Markovian social contagions in simplicial complexes","volume":"7","year":"2024","journal-title":"Commun. Phys."},{"key":"2026061209061654300_c29","doi-asserted-by":"publisher","first-page":"7391","DOI":"10.1038\/s41467-024-50918-5","article-title":"The temporal dynamics of group interactions in higher-order social networks","volume":"15","year":"2024","journal-title":"Nat. Commun."},{"key":"2026061209061654300_c30","doi-asserted-by":"publisher","first-page":"146","DOI":"10.1038\/s42254-025-00916-3","article-title":"Collective dynamics on higher-order networks","volume":"8","year":"2026","journal-title":"Nat. Rev. Phys."},{"key":"2026061209061654300_c31","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1109\/TSMC.2025.3624366","article-title":"Reinforcement learning can be a double-edged sword for cooperation on higher-order networks","volume":"56","year":"2026","journal-title":"IEEE Trans. Syst. Man Cybern.: Syst."},{"key":"2026061209061654300_c32","doi-asserted-by":"publisher","first-page":"114172","DOI":"10.1016\/j.chaos.2023.114172","article-title":"Conformity versus credibility: A coupled rumor-belief model","volume":"176","year":"2023","journal-title":"Chaos Soliton. Fract."},{"key":"2026061209061654300_c33","doi-asserted-by":"publisher","first-page":"13220","DOI":"10.1038\/s41598-025-96006-6","article-title":"Research on rumor and anti rumor propagation models based on quantum superposition states","volume":"15","year":"2025","journal-title":"Sci. Rep."},{"key":"2026061209061654300_c34","volume-title":"Social Laser: Application of Quantum Information and Field Theories to Modeling of Social Processes","year":"2020"},{"key":"2026061209061654300_c35","doi-asserted-by":"publisher","first-page":"8566","DOI":"10.1038\/s41598-022-12327-w","article-title":"Mean-field theory of social laser","volume":"12","year":"2022","journal-title":"Sci. Rep."},{"key":"2026061209061654300_c36","doi-asserted-by":"publisher","first-page":"e0321718","DOI":"10.1371\/journal.pone.0321718","article-title":"An opinion evolution model for online social networks considering higher-order interactions","volume":"20","year":"2025","journal-title":"PLoS One"},{"key":"2026061209061654300_c37","doi-asserted-by":"publisher","first-page":"2450028","DOI":"10.1142\/S0129183124500281","article-title":"Rumor propagation model with neutral state","volume":"35","year":"2024","journal-title":"Int. J. Mod. Phys. C"},{"key":"2026061209061654300_c38","doi-asserted-by":"publisher","first-page":"331","DOI":"10.1016\/j.physa.2018.02.096","article-title":"Rumor spreading model with the different attitudes towards rumors","volume":"502","year":"2018","journal-title":"Physica A"},{"key":"2026061209061654300_c39","volume-title":"Quantum Models of Cognition and Decision: Principles and Applications","year":"2025","edition":"2nd"},{"key":"2026061209061654300_c40","doi-asserted-by":"publisher","first-page":"77","DOI":"10.3389\/fphy.2015.00077","article-title":"Quantum-like modeling of cognition","volume":"3","year":"2015","journal-title":"Front. Phys."},{"key":"2026061209061654300_c41","doi-asserted-by":"publisher","first-page":"886","DOI":"10.3390\/e25060886","article-title":"Open systems, quantum probability, and logic for quantum-like modeling in biology, cognition, and decision-making","volume":"25","year":"2023","journal-title":"Entropy"},{"key":"2026061209061654300_c42","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.mathsocsci.2015.02.004","article-title":"A survey of quantum-like approaches to decision making and cognition","volume":"75","year":"2015","journal-title":"Math. Soc. Sci."},{"key":"2026061209061654300_c43","volume-title":"The Theory of Open Quantum Systems","year":"2002"},{"key":"2026061209061654300_c44","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1016\/j.jmp.2016.07.005","article-title":"Advanced tools and concepts for quantum cognition: A tutorial","volume":"78","year":"2017","journal-title":"J. Math. Psychol."},{"key":"2026061209061654300_c45","doi-asserted-by":"publisher","first-page":"990","DOI":"10.3390\/e22090990","article-title":"Application of quantum\u2014Markov open system models to human cognition and decision","volume":"22","year":"2020","journal-title":"Entropy"},{"key":"2026061209061654300_c46","doi-asserted-by":"publisher","first-page":"20160387","DOI":"10.1098\/rsta.2016.0387","article-title":"Quantum-like dynamics applied to cognition: A consideration of available options","volume":"375","year":"2017","journal-title":"Phil. Trans. R. Soc. A"},{"key":"2026061209061654300_c47","doi-asserted-by":"publisher","first-page":"025106","DOI":"10.1063\/1.5115323","article-title":"A short introduction to the Lindblad master equation","volume":"10","year":"2020","journal-title":"AIP Adv."},{"key":"2026061209061654300_c48","doi-asserted-by":"publisher","first-page":"042307","DOI":"10.1103\/PhysRevA.78.042307","article-title":"Preparation of entangled states by quantum Markov processes","volume":"78","year":"2008","journal-title":"Phys. Rev. A: At. Mol. Opt. Phys."},{"key":"2026061209061654300_c49","volume-title":"Quantum Computation and Quantum Information","year":"2010"},{"key":"2026061209061654300_c50","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1063\/1.522979","article-title":"Completely positive dynamical semigroups of N-level systems","volume":"17","year":"1976","journal-title":"J. Math. Phys."},{"key":"2026061209061654300_c51","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1007\/BF01608499","article-title":"On the generators of quantum dynamical semigroups","volume":"48","year":"1976","journal-title":"Commun. Math. Phys."},{"key":"2026061209061654300_c52","volume-title":"Ordinary Differential Equations and Dynamical Systems","year":"2012"},{"key":"2026061209061654300_c53","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1126\/science.286.5439.509","article-title":"Emergence of scaling in random networks","volume":"286","year":"1999","journal-title":"Science"},{"key":"2026061209061654300_c54","doi-asserted-by":"publisher","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of \u2018small-world\u2019 networks","volume":"393","year":"1998","journal-title":"Nature"},{"key":"2026061209061654300_c55","volume-title":"Advances in Neural Information Processing Systems","year":"2012"},{"key":"2026061209061654300_c56","unstructured":"C.\n            McClain\n          , R.Widjaya, G.Rivero, and A.Smith, \u201cThe behaviors and attitudes of U.S. adults on Twitter,\u201d Pew Research Center (2021). Available at: https:\/\/www.pewresearch.org\/internet\/2021\/11\/15\/the-behaviors-and-attitudes-of-u-s-adults-on-twitter\/."},{"key":"2026061209061654300_c57"},{"key":"2026061209061654300_c58"},{"key":"2026061209061654300_c59"},{"key":"2026061209061654300_c60","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0363-8111(02)00107-8","article-title":"Corporate rumor activity, belief and accuracy","volume":"28","year":"2002","journal-title":"Public Relat. Rev."},{"key":"2026061209061654300_c61","doi-asserted-by":"publisher","first-page":"484","DOI":"10.1037\/0003-066X.46.5.484","article-title":"Inside rumor: A personal journey","volume":"46","year":"1991","journal-title":"Am. Psychol."},{"key":"2026061209061654300_c62","first-page":"1\u201327","article-title":"What makes news sharable on social media?","volume":"3","year":"2023","journal-title":"J. Quant. Descr. Digit. Media"},{"key":"2026061209061654300_c63"},{"key":"2026061209061654300_c64","volume-title":"The Theory of Quantum Information","year":"2018"}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0335015\/21038414\/063127_1_5.0335015.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0335015\/21038414\/063127_1_5.0335015.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T13:06:28Z","timestamp":1781269588000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/6\/063127\/3394642\/A-quantum-like-rumor-propagation-model-captures"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,6,1]]},"references-count":64,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2026,6,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0335015","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,6]]},"published":{"date-parts":[[2026,6,1]]},"article-number":"063127"}}