{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:29:08Z","timestamp":1760059748810,"version":"build-2065373602"},"reference-count":64,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T00:00:00Z","timestamp":1751587200000},"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":["72025405","72421002","92467302","72401289"],"award-info":[{"award-number":["72025405","72421002","92467302","72401289"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["BDCC"],"abstract":"<jats:p>The evolution of public opinion on social media affects societal security and stability. To effectively control the societal impact of public opinion evolution, it is essential to study its underlying mechanisms. Public opinion evolution on social media primarily involves two processes: information dissemination and opinion interaction. However, existing studies overlook the bidirectional coupling relationship between these two processes, with limitations such as weak coupling and insufficient consideration of individual heterogeneity. To address this, we propose the SIS-FJ model with a bidirectional coupling mechanism, which combines the strengths of the SIS (Susceptible\u2013Infected\u2013Susceptible) model in information dissemination and the FJ (Friedkin\u2013Johnsen) model in opinion interaction. Specifically, the SIS model is used to describe information dissemination, while the FJ model is used to describe opinion interaction. In the computation of infection and recovery rates of the SIS model, we introduce the opinion differences between individuals and their observable neighbors from the FJ model. In the computation of opinion values in the FJ model, we introduce the node states from the SIS model, thus achieving bidirectional coupling between the two models. Moreover, the model considers individual heterogeneity from multiple aspects, including infection rate, recovery rate, and individual susceptibility. Through simulation experiments, we investigate the effects of initial opinion distribution, individual susceptibility, and network structure on public opinion evolution. Interestingly, neither initial opinion distribution, individual susceptibility, nor network structure exerts a significant influence on the proportion of disseminating and non-disseminating individuals at termination. Furthermore, we optimize the model by adjusting the functions for infection and recovery rates.<\/jats:p>","DOI":"10.3390\/bdcc9070180","type":"journal-article","created":{"date-parts":[[2025,7,4]],"date-time":"2025-07-04T11:20:11Z","timestamp":1751628011000},"page":"180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Modeling and Simulation of Public Opinion Evolution Based on the SIS-FJ Model with a Bidirectional Coupling Mechanism"],"prefix":"10.3390","volume":"9","author":[{"given":"Wenxuan","family":"Fu","sequence":"first","affiliation":[{"name":"College of Systems Engineering, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Renqi","family":"Zhu","sequence":"additional","affiliation":[{"name":"College of Systems Engineering, National University of Defense Technology, Changsha 410073, China"}]},{"given":"Bo","family":"Li","sequence":"additional","affiliation":[{"name":"College of Systems Engineering, National University of Defense Technology, Changsha 410073, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3547-6493","authenticated-orcid":false,"given":"Xin","family":"Lu","sequence":"additional","affiliation":[{"name":"College of Systems Engineering, National University of Defense Technology, Changsha 410073, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-5870-928X","authenticated-orcid":false,"given":"Xiang","family":"Lin","sequence":"additional","affiliation":[{"name":"College of Systems Engineering, National University of Defense Technology, Changsha 410073, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"104192","DOI":"10.1016\/j.engappai.2021.104192","article-title":"HK\u2013SEIR Model of Public Opinion Evolution Based on Communication Factors","volume":"100","author":"Li","year":"2021","journal-title":"Eng. Appl. Artif. Intell."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3404","DOI":"10.1007\/s13132-024-02059-9","article-title":"Analyzing the Complexity of Public Opinion Evolution on Weibo: A Super Network Model","volume":"16","author":"Li","year":"2024","journal-title":"J. Knowl. Econ."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"123591","DOI":"10.1016\/j.eswa.2024.123591","article-title":"Multiple Criteria Decision Modeling in Social Networks: An Application to Identify Opinion Leaders in Online Public Opinion Events","volume":"249","author":"Sun","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_4","first-page":"4791527","article-title":"Modeling Public Opinion Polarization in Group Behavior by Integrating SIRS-Based Information Diffusion Process","volume":"2020","author":"Chen","year":"2020","journal-title":"Complexity"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.bushor.2009.09.003","article-title":"Users of the World, Unite! The Challenges and Opportunities of Social Media","volume":"53","author":"Kaplan","year":"2010","journal-title":"Bus. Horiz."},{"key":"ref_6","first-page":"51","article-title":"The Study on the Formation Mechanism and Guiding Strategies of Group Polarization in Social Media Platforms","volume":"8","author":"Lu","year":"2021","journal-title":"Inf. Stud. Theory Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"165","DOI":"10.2307\/249394","article-title":"Polarization and Persuasive Argumentation: A Study of Decision Making in Group Settings","volume":"22","author":"Vinze","year":"1998","journal-title":"MIS Q."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1038\/s41562-019-0541-6","article-title":"The Wisdom of Polarized Crowds","volume":"3","author":"Shi","year":"2019","journal-title":"Nat. Hum. Behav."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3316809","article-title":"Polarization and Fake News","volume":"13","author":"Vicario","year":"2019","journal-title":"ACM Trans. Web"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"554","DOI":"10.1073\/pnas.1517441113","article-title":"The Spreading of Misinformation Online","volume":"113","author":"Bessi","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Tucker, J., Guess, A., Barbera, P., Vaccari, C., Siegel, A., Sanovich, S., Stukal, D., and Nyhan, B. (2018). Social Media, Political Polarization, and Political Disinformation: A Review of the Scientific Literature. SSRN Electron. J.","DOI":"10.2139\/ssrn.3144139"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"113287","DOI":"10.1016\/j.chaos.2023.113287","article-title":"Controversy-Seeking Fuels Rumor-Telling Activity in Polarized Opinion Networks","volume":"169","author":"Maia","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1002\/hbe2.202","article-title":"Political Polarization Drives Online Conversations about COVID-19 in the United States","volume":"2","author":"Jiang","year":"2020","journal-title":"Hum. Behav. Emerg. Technol."},{"key":"ref_14","unstructured":"Sunstein, C.R. (2001). Republic.com, Princeton University Press."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1146\/annurev-polisci-051117-073034","article-title":"The Origins and Consequences of Affective Polarization in the United States","volume":"22","author":"Iyengar","year":"2019","journal-title":"Annu. Rev. Political Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"663","DOI":"10.1093\/poq\/nfab029","article-title":"How Affective Polarization Undermines Support for Democratic Norms","volume":"85","author":"Kingzette","year":"2021","journal-title":"Public Opin. Q."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1131","DOI":"10.1016\/j.ejor.2020.05.025","article-title":"Parallel Evolution and Response Decision Method for Public Sentiment Based on System Dynamics","volume":"287","author":"Xie","year":"2020","journal-title":"Eur. J. Oper. Res."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1202","DOI":"10.1016\/j.ins.2022.07.021","article-title":"Empirical Analysis and Modelling Social Network User Interaction Behavior and Time Characteristics Based on Selection Preference","volume":"608","author":"Luo","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3379443","article-title":"Capturing Dynamics of Information Diffusion in SNS: A Survey of Methodology and Techniques","volume":"55","author":"Li","year":"2021","journal-title":"ACM Comput. Surv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"112959","DOI":"10.1016\/j.chaos.2022.112959","article-title":"Online Public Opinion Dissemination Model and Simulation under Media Intervention from Different Perspectives","volume":"166","author":"Geng","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"7595","DOI":"10.1109\/TCSS.2024.3438212","article-title":"Research on Opinion Diffusion and Evolution Model in Complex Network","volume":"11","author":"Liu","year":"2024","journal-title":"IEEE Trans. Comput. Soc. Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"113095","DOI":"10.1016\/j.chaos.2022.113095","article-title":"Strategies for Reducing Polarization in Social Networks","volume":"167","author":"Wu","year":"2023","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3563388","article-title":"Fighting False Information from Propagation Process: A Survey","volume":"55","author":"Sun","year":"2023","journal-title":"ACM Comput. Surv."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.inffus.2017.11.009","article-title":"A Survey on the Fusion Process in Opinion Dynamics","volume":"43","author":"Dong","year":"2018","journal-title":"Inf. Fusion"},{"key":"ref_25","first-page":"94","article-title":"Contributions to the Mathematical Theory of Epidemics. III. Further Studies of the Problem of Endemicity","volume":"141","author":"Kermack","year":"1933","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/S0378-8733(96)00298-5","article-title":"Social Positions in Influence Networks","volume":"19","author":"Friedkin","year":"1997","journal-title":"Soc. Netw."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"864","DOI":"10.1086\/591197","article-title":"Modeling Infectious Diseases in Humans and Animals: Modeling Infectious Diseases in Humans and Animals","volume":"47","author":"Keeling","year":"2008","journal-title":"Clin. Infect. Dis."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Centola, D. (2018). How Behavior Spreads: The Science of Complex Contagions, Princeton University Press.","DOI":"10.23943\/9781400890095"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2019.07.001","article-title":"Coevolution Spreading in Complex Networks","volume":"820","author":"Wang","year":"2019","journal-title":"Phys. Rep."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF00171519","article-title":"Population Models for Diseases with No Recovery","volume":"28","author":"Pugliese","year":"1990","journal-title":"J. Math. Biol."},{"key":"ref_31","first-page":"700","article-title":"A Contribution to the Mathematical Theory of Epidemics","volume":"115","author":"Kermack","year":"1927","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1016\/j.techfore.2016.12.003","article-title":"Forecasting of Potential Impacts of Disruptive Technology in Promising Technological Areas: Elaborating the SIRS Epidemic Model in RFID Technology","volume":"117","author":"Cheng","year":"2017","journal-title":"Technol. Forecast. Soc. Change"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/S2468-2667(20)30073-6","article-title":"The Effect of Control Strategies to Reduce Social Mixing on Outcomes of the COVID-19 Epidemic in Wuhan, China: A Modelling Study","volume":"5","author":"Prem","year":"2020","journal-title":"Lancet Public Health"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"797","DOI":"10.2307\/2118364","article-title":"A Simple Model of Herd Behavior","volume":"107","author":"Banerjee","year":"1992","journal-title":"Q. J. Econ."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1420","DOI":"10.1086\/226707","article-title":"Threshold Models of Collective Behavior","volume":"83","author":"Granovetter","year":"1978","journal-title":"Am. J. Sociol."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"129630","DOI":"10.1016\/j.physa.2024.129630","article-title":"Coupled Dynamics of Information Propagation and Emotion Influence: Emerging Emotion Clusters for Public Health Emergency Messages on the Chinese Sina Microblog","volume":"639","author":"Yin","year":"2024","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"127533","DOI":"10.1109\/ACCESS.2019.2939150","article-title":"SETQR Propagation Model for Social Networks","volume":"7","author":"Zhang","year":"2019","journal-title":"IEEE Access"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"e36658","DOI":"10.1016\/j.heliyon.2024.e36658","article-title":"Social Network Influence Based on SHIR and SLPR Propagation Models","volume":"10","author":"Liu","year":"2024","journal-title":"Heliyon"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2250014","DOI":"10.1142\/S0129183122500140","article-title":"SCIR Rumor Propagation Model with the Chord Mechanism in Social Networks","volume":"33","author":"Zhou","year":"2021","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Govindankutty, S., and Gopalan Padinjappurathu, S. (2022). SEDIS\u2014A Rumor Propagation Model for Social Networks by Incorporating the Human Nature of Selection. Systems, 11.","DOI":"10.3390\/systems11010012"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"2350003","DOI":"10.1142\/S0129183123500031","article-title":"ISRT Rumor Spreading Model with Different Influence Mechanisms in Social Networks","volume":"34","author":"Wang","year":"2022","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.inffus.2021.06.004","article-title":"Influence Identification of Opinion Leaders in Social Networks: An Agent-Based Simulation on Competing Advertisements","volume":"76","author":"Chen","year":"2021","journal-title":"Inf. Fusion"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"123462","DOI":"10.1016\/j.physa.2019.123462","article-title":"Controlling Opinions in Deffuant Model by Reconfiguring the Network Topology","volume":"544","author":"Bashari","year":"2019","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/j.jebo.2021.04.013","article-title":"Convergence and Divergence in Dynamic Voting with Inequality","volume":"187","author":"Galanis","year":"2021","journal-title":"J. Econ. Behav. Organ."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"121292","DOI":"10.1016\/j.eswa.2023.121292","article-title":"Opinion Formation Analysis for Expressed and Private Opinions (EPOs) Models: Reasoning Private Opinions from Behaviors in Group Decision-Making Systems","volume":"236","author":"Dong","year":"2024","journal-title":"Expert Syst. Appl."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"643","DOI":"10.1214\/aop\/1176996306","article-title":"Ergodic Theorems for Weakly Interacting Infinite Systems and the Voter Model","volume":"3","author":"Holley","year":"1975","journal-title":"Ann. Probab."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1140\/epjb\/e20020045","article-title":"Minority Opinion Spreading in Random Geometry","volume":"25","author":"Galam","year":"2002","journal-title":"Eur. Phys. J. B"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"1157","DOI":"10.1142\/S0129183100000936","article-title":"Opinion Evolution in Closed Community","volume":"11","author":"Sznajd","year":"2000","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1080\/01621459.1974.10480137","article-title":"Reaching a Consensus","volume":"69","author":"DeGroot","year":"1974","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1142\/S0219525900000078","article-title":"Mixing Beliefs among Interacting Agents","volume":"3","author":"Deffuant","year":"2000","journal-title":"Adv. Complex Syst."},{"key":"ref_51","first-page":"1","article-title":"Opinion Dynamics and Bounded Confidence Models, Analysis and Simulation","volume":"5","author":"Hegselmann","year":"2002","journal-title":"RePEc Res. Pap. Econ."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s10588-005-6282-2","article-title":"Uniformity, Bipolarization and Pluriformity Captured as Generic Stylized Behavior with an Agent-Based Simulation Model of Attitude Change","volume":"10","author":"Jager","year":"2005","journal-title":"Comput. Math. Organ. Theory"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1016\/j.ins.2020.01.052","article-title":"A Two-Step Communication Opinion Dynamics Model with Self-Persistence and Influence Index for Social Networks Based on the DeGroot Model","volume":"519","author":"Zhou","year":"2020","journal-title":"Inf. Sci."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"11896","DOI":"10.1016\/j.ifacol.2017.08.1424","article-title":"Opinion Evolution in Time-Varying Social Influence Networks with Prejudiced Agents","volume":"50","author":"Proskurnikov","year":"2017","journal-title":"IFAC-PapersOnLine"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"42118","DOI":"10.1103\/PhysRevE.95.042118","article-title":"Deffuant Model on a Ring with Repelling Mechanism and Circular Opinions","volume":"95","author":"Chen","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"102173","DOI":"10.1016\/j.inffus.2023.102173","article-title":"Analysis of the Influence of Trust in Opposing Opinions: An Inclusiveness-Degree Based Signed Deffuant\u2013Weisbush Model","volume":"104","author":"Jiang","year":"2023","journal-title":"Inf. Fusion"},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Xu, H., Cai, H., Wu, S., Ai, K., and Xu, M. (2020, January 11\u201314). HKML: A Novel Opinion Dynamics Hegselmann-Krause Model with Media Literacy. Proceedings of the IEEE International Conference on Systems, Man, and Cybernetics (SMC), Toronto, ON, Canada.","DOI":"10.1109\/SMC42975.2020.9283055"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"52001","DOI":"10.1209\/0295-5075\/acf39a","article-title":"Opinion Dynamics of Improved Hegselmann-Krause Model with Agent\u2019s Stubbornness and Stubborn Agents","volume":"143","author":"Wang","year":"2023","journal-title":"EPL (Europhys. Lett.)"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.patrec.2023.01.016","article-title":"Opinion-Climate-Based Hegselmann-Krause Dynamics","volume":"167","author":"Xu","year":"2023","journal-title":"Pattern Recognit. Lett."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1155\/2017\/1972532","article-title":"Opinion Dynamics of Social-Similarity-Based Hegselmann\u2013Krause Model","volume":"2017","author":"Chen","year":"2017","journal-title":"Complexity"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"4557","DOI":"10.1016\/j.aej.2021.10.014","article-title":"Modelling Network Public Opinion Polarization Based on SIR Model Considering Dynamic Network Structure","volume":"61","author":"Yuan","year":"2022","journal-title":"Alex. Eng. J."},{"key":"ref_62","first-page":"1820257","article-title":"Modeling and Analyzing Information Propagation Evolution Integrating Internal and External Influences","volume":"7","author":"Yin","year":"2023","journal-title":"Adv. Theory Simul."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"567","DOI":"10.1016\/j.physa.2017.05.034","article-title":"Topological Evolution of the Internet Public Opinion","volume":"486","author":"Lian","year":"2017","journal-title":"Phys. A Stat. Mech. Its Appl."},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.neucom.2022.06.111","article-title":"Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark","volume":"503","author":"Dubey","year":"2022","journal-title":"Neurocomputing"}],"container-title":["Big Data and Cognitive Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-2289\/9\/7\/180\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:04:30Z","timestamp":1760033070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-2289\/9\/7\/180"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,4]]},"references-count":64,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["bdcc9070180"],"URL":"https:\/\/doi.org\/10.3390\/bdcc9070180","relation":{},"ISSN":["2504-2289"],"issn-type":[{"type":"electronic","value":"2504-2289"}],"subject":[],"published":{"date-parts":[[2025,7,4]]}}}