{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:07:39Z","timestamp":1760238459987,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,14]],"date-time":"2020-08-14T00:00:00Z","timestamp":1597363200000},"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":["61673320 and 61976130"],"award-info":[{"award-number":["61673320 and 61976130"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100012551","name":"Applied Basic Research Program of Sichuan Province","doi-asserted-by":"publisher","award":["2020YJ0270"],"award-info":[{"award-number":["2020YJ0270"]}],"id":[{"id":"10.13039\/100012551","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["2682018CX59"],"award-info":[{"award-number":["2682018CX59"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Seeking truth is an important objective of agents in social groups. Opinion leaders in social groups may help or hinder the other agents on seeking the truth by symmetric nature. This paper studies the impact of opinion leaders by considering four characteristics of opinion leaders\u2014reputation, stubbornness, appeal, and extremeness\u2014on the truth-seeking behavior of agents based on a bounded confidence model. Simulations show that increasing the appeal of the leader whose opinion is opposite to the truth has a straightforward impact, i.e., it normally prevents the agents from finding the truth. On the other hand, it also makes the agents who start out close to the truth move away from the truth by increasing the group bound of confidence, if there is an opinion leader opposite to the truth. The results demonstrate that the opinion of the leader is important in affecting the normal agents to reach the truth. Furthermore, for some cases, small variations of the parameters defining the agents\u2019 characteristics can lead to large scale changes in the social group.<\/jats:p>","DOI":"10.3390\/sym12081362","type":"journal-article","created":{"date-parts":[[2020,8,14]],"date-time":"2020-08-14T13:00:18Z","timestamp":1597410018000},"page":"1362","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["How Opinion Leaders Affect Others on Seeking Truth in a Bounded Confidence Model"],"prefix":"10.3390","volume":"12","author":[{"given":"Shuwei","family":"Chen","sequence":"first","affiliation":[{"name":"School of Mathematics, Southwest Jiaotong University, Chengdu 610031, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David H.","family":"Glass","sequence":"additional","affiliation":[{"name":"School of Computing, Ulster University, Newtownabbey BT37 0QB, Northern Ireland, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark","family":"McCartney","sequence":"additional","affiliation":[{"name":"School of Computing, Ulster University, Newtownabbey BT37 0QB, Northern Ireland, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1037\/h0046123","article-title":"A formal theory of social power","volume":"63","author":"French","year":"1956","journal-title":"Psychol. Rev."},{"key":"ref_2","unstructured":"Cartwright, D. (1956). A criterion for unanimity in French\u2019s theory of social power. Studies in Social Power, Institute for Social Research."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1142\/S0129183105008126","article-title":"Vector opinion dynamics in a bounded confidence consensus Model","volume":"16","author":"Fortunato","year":"2005","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1142\/S0129183105007388","article-title":"Multidimensional consensus model on a Barabasi-Albert network","volume":"16","author":"Jacobmeier","year":"2005","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1819","DOI":"10.1142\/S0129183107011789","article-title":"Continuous opinion dynamics under bounded confidence: A survey","volume":"18","author":"Lorenz","year":"2007","journal-title":"Int. J. Mod. Phys. C"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s13235-010-0004-1","article-title":"Opinion dynamics and learning in social networks","volume":"1","author":"Acemoglu","year":"2011","journal-title":"Dyn. Games Appl."},{"key":"ref_7","unstructured":"Helbing, D. (2008). Fostering consensus in multidimensional continuous opinion dynamics under bounded confidence. Managing Complexity, Springer."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1177\/1470594X11416766","article-title":"Social network structure and the achievement of consensus","volume":"11","author":"Zollman","year":"2012","journal-title":"Politics Philos. Econ."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4938","DOI":"10.1038\/srep04938","article-title":"Opinion dynamics on interacting networks: Media competition and social influence","volume":"4","author":"Quattrociocchi","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/j.physa.2014.10.089","article-title":"Mass media and heterogeneous bounds of confidence in continuous opinion dynamics","volume":"420","author":"Pineda","year":"2015","journal-title":"Physica A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1137\/130913250","article-title":"Opinion dynamics and the evolution of social power in influence networks","volume":"57","author":"Jia","year":"2015","journal-title":"SIAM Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"050301","DOI":"10.1103\/PhysRevE.100.050301","article-title":"Nonuniversal opinion dynamics driven by opposing external influences","volume":"100","author":"Bhat","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Elaydi, S., Ladas, G., Popenda, J., and Rakowski, J. (2000). A discrete nonlinear and non-autonomous model of consensus formation. Communications in Difference Equations, Gordon and Breach Publ.","DOI":"10.1201\/b16999"},{"key":"ref_14","first-page":"1","article-title":"Opinion dynamics and bounded confidence: Models, analysis, and simulations","volume":"5","author":"Hegselmann","year":"2002","journal-title":"J. Artif. Soc. Soc. Simul."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1093\/jigpal\/jzp059","article-title":"Extending the Hegselmann\u2013Krause model I","volume":"18","author":"Douven","year":"2010","journal-title":"Log. J. IGPL"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"558","DOI":"10.1016\/j.physa.2014.10.045","article-title":"Opinion dynamics of modified Hegselmann\u2013Krause model in a group-based population with heterogeneous bounded confidence","volume":"419","author":"Fu","year":"2015","journal-title":"Physica A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/j.ins.2017.02.052","article-title":"Managing consensus based on leadership in opinion dynamics","volume":"397","author":"Dong","year":"2017","journal-title":"Inf. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1200","DOI":"10.1109\/TAC.2017.2742139","article-title":"Control via leadership of opinion dynamics with state and time-dependent interactions","volume":"63","author":"Dietrich","year":"2018","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.ins.2017.10.031","article-title":"Understanding influence power of opinion leaders in e-commerce networks: An opinion dynamics theory perspective","volume":"426","author":"Zhao","year":"2018","journal-title":"Inf. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1086\/266687","article-title":"The two-step flow of communication: An up-to-date report on a hypothesis","volume":"21","author":"Katz","year":"1957","journal-title":"Public Opin. Q."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"153","DOI":"10.24846\/v23i2y201403","article-title":"Bounded confidence-based opinion formation for opinion leaders and opinion followers on social networks","volume":"23","author":"Zhao","year":"2014","journal-title":"Stud. Inform. Control"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Lachowicz, M., Leszczy\u0144ski, H., and Pu\u017aniakowska-Ga\u0142uch, E. (2019). Diffusive and anti-diffusive behavior for Kinetic Models of opinion dynamics. Symmetry, 11.","DOI":"10.3390\/sym11081024"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1007\/s00146-017-0784-6","article-title":"Two-dimensional opinion dynamics in social networks with conflicting beliefs","volume":"34","author":"Chen","year":"2019","journal-title":"AI Soc."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"426","DOI":"10.1016\/j.physa.2015.12.107","article-title":"Characteristics of successful opinion leaders in a bounded confidence model","volume":"449","author":"Chen","year":"2016","journal-title":"Physica A"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Lima, F.W.S., and Plascak, J.A. (2019). Kinetic models of discrete opinion dynamics on directed Barab\u00e1si\u2013Albert networks. Entropy, 21.","DOI":"10.3390\/e21100942"},{"key":"ref_26","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_27","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1016\/j.physa.2005.01.054","article-title":"Persuasion dynamics","volume":"353","author":"Weisbuch","year":"2005","journal-title":"Physica A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1140\/epjb\/e2006-00131-0","article-title":"Compromise and synchronization in opinion dynamics","volume":"50","author":"Pluchino","year":"2006","journal-title":"Eur. Phys. J. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"145","DOI":"10.3366\/E1742360009000616","article-title":"Extending the Hegselmann\u2013Krause model III: From Single Beliefs to Complex Belief States","volume":"6","author":"Riegler","year":"2009","journal-title":"Episteme"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kou, G., Zhao, Y., Peng, Y., and Shi, Y. (2012). Multi-level opinion dynamics under bounded confidence. PLoS ONE, 7.","DOI":"10.1371\/journal.pone.0043507"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"2968","DOI":"10.1038\/srep02968","article-title":"Opinion dynamics with similarity-based random neighbours","volume":"3","author":"Liu","year":"2013","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.physa.2014.10.090","article-title":"Opinion dynamics in networks with common-neighbours-based connections","volume":"421","author":"Wang","year":"2015","journal-title":"Physica A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/2538508","article-title":"Binary opinion dynamics with stubborn agents","volume":"1","author":"Yildiz","year":"2013","journal-title":"ACM Trans. Econ. Comput."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/j.automatica.2018.06.041","article-title":"Opinion dynamics in social networks with stubborn agents: An issue-based perspective","volume":"96","author":"Tian","year":"2018","journal-title":"Automatica"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"10032","DOI":"10.1038\/srep10032","article-title":"How does public opinion become extreme?","volume":"5","author":"Ramos","year":"2015","journal-title":"Sci. Rep."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"3339","DOI":"10.1109\/TAC.2018.2879599","article-title":"Polarizability, consensusability, and neutralizability of opinion dynamics on coopetitive networks","volume":"64","author":"Liu","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"477","DOI":"10.3934\/nhm.2015.10.477","article-title":"Opinion dynamics under the influence of radical groups, charismatic leaders, and other constant signals: A simple unifying model","volume":"10","author":"Hegselmann","year":"2015","journal-title":"Netw. Heterog. Media"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Glass, C.A., and Glass, D.H. (2020). Opinion dynamics of social learning with a conflicting source. Physica A, under review.","DOI":"10.1016\/j.physa.2020.125480"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1362\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:01:14Z","timestamp":1760176874000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/8\/1362"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,8,14]]},"references-count":38,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,8]]}},"alternative-id":["sym12081362"],"URL":"https:\/\/doi.org\/10.3390\/sym12081362","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,8,14]]}}}