{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,3]],"date-time":"2026-09-03T00:44:03Z","timestamp":1788396243779,"version":"build-2803163510"},"reference-count":47,"publisher":"AIP Publishing","issue":"3","content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,3,1]]},"abstract":"<jats:p>Opinion evolution in social networks represents a complex dynamical process, particularly when accounting for the interplay between expressed and private opinions\u2014a dynamic that captures the tension between social conformity and individual authenticity. In this paper, we examine the evolution of these two opinion types in signed networks using the expressed and private opinion (EPO) model, in which each individual carries distinct expressed and private opinions. By applying the lifting technique to the EPO model, we rigorously address the influence of negative edges, while the path-dependence theory is employed to model transitions between consecutive topics. Through analytical study, we derive sufficient conditions to eliminate discrepancies between expressed and private opinions across a sequence of topics, ensuring convergence to zero consensus, bipartite consensus, or bounded opinion stability. Notably, iterative topic discussions lead to asymptotic alignment between private and expressed opinions, with all trajectories confined within a well-defined range. Furthermore, for structurally unbalanced networks, we establish a sufficient condition guaranteeing bipartite consensus. Numerical simulations validate all theoretical findings.<\/jats:p>","DOI":"10.1063\/5.0322595","type":"journal-article","created":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T17:41:16Z","timestamp":1773423676000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":2,"title":["Expressed and private opinion dynamics in signed networks based on path-dependence framework"],"prefix":"10.1063","volume":"36","author":[{"given":"Yutong Christina","family":"Wang","sequence":"first","affiliation":[{"name":"Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, Hong Kong 1 ,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuzhi","family":"Kong","sequence":"additional","affiliation":[{"name":"College of Computer Science and Software Engineering, Shenzhen University 2 , Shenzhen 518060,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongfeng","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Wuhan University 3 , Wuhan 430072,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-9654-2298","authenticated-orcid":false,"given":"Zixi","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Journalism and Communication, Wuhan University 4 , Wuhan 430072,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5065-6460","authenticated-orcid":false,"given":"Xiaoqun","family":"Wu","sequence":"additional","affiliation":[{"name":"College of Computer Science and Software Engineering, Shenzhen University 2 , Shenzhen 518060,","place":["China"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,3,13]]},"reference":[{"key":"2026031313411437900_c1","doi-asserted-by":"publisher","first-page":"186","DOI":"10.1038\/s41567-019-0739-6","article-title":"Negative representation and instability in democratic elections","volume":"16","year":"2020","journal-title":"Nat. Phys."},{"key":"2026031313411437900_c2","doi-asserted-by":"publisher","first-page":"1442","DOI":"10.1007\/s10955-020-02506-2","article-title":"Threshold behavior of democratic opinion dynamics","volume":"178","year":"2020","journal-title":"J. Stat. Phys."},{"key":"2026031313411437900_c3","doi-asserted-by":"publisher","first-page":"2391","DOI":"10.1007\/s00530-022-00966-y","article-title":"Combating multimodal fake news on social media: Methods, datasets, and future perspective","volume":"28","year":"2022","journal-title":"Multimedia Syst."},{"key":"2026031313411437900_c4","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.3982\/TE3552","article-title":"The wisdom of the crowd in dynamic economies","volume":"15","year":"2020","journal-title":"Theor. Econ."},{"key":"2026031313411437900_c5","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1016\/j.arcontrol.2014.09.002","article-title":"Synchronization in complex networks and its application\u2013a survey of recent advances and challenges","volume":"38","year":"2014","journal-title":"Annu. Rev. Control"},{"key":"2026031313411437900_c6","doi-asserted-by":"publisher","first-page":"6304","DOI":"10.1109\/TSMC.2020.2966712","article-title":"Consensus reaching and strategic manipulation in group decision making with trust relationships","volume":"51","year":"2021","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"2026031313411437900_c7","doi-asserted-by":"publisher","first-page":"2394","DOI":"10.1109\/TSMC.2017.2758338","article-title":"Opinion dynamics-based group recommender systems","volume":"48","year":"2017","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"2026031313411437900_c8","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":"2026031313411437900_c9","doi-asserted-by":"publisher","first-page":"112544","DOI":"10.1016\/j.automatica.2025.112544","article-title":"Design of asymmetric Lyapunov functions for global stability in dynamic surface control","volume":"182","year":"2025","journal-title":"Automatica"},{"key":"2026031313411437900_c10","doi-asserted-by":"publisher","first-page":"4061","DOI":"10.1109\/TAC.2025.3528344","article-title":"Performance-guaranteed finite-time tracking of strict-feedback systems with unknown control directions: A novel switching mechanism","volume":"70","year":"2025","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026031313411437900_c11","doi-asserted-by":"publisher","DOI":"10.1109\/TSMC.2026.3663317","article-title":"Predefined-time exact tracking for nonstrict feedback nonlinear systems: A novel switching feedback gain method","year":"2026","journal-title":"IEEE Trans. Syst., Man Cybern. Syst."},{"key":"2026031313411437900_c12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2024.01.005","article-title":"Synchronization in multiplex networks","volume":"1060","year":"2024","journal-title":"Phys. Rep."},{"key":"2026031313411437900_c13","doi-asserted-by":"publisher","first-page":"073109","DOI":"10.1063\/5.0211026","article-title":"Steering opinion dynamics through control of social networks","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026031313411437900_c14","doi-asserted-by":"publisher","first-page":"2165","DOI":"10.1109\/TCYB.2025.3547772","article-title":"An improved topology identification method of complex dynamical networks","volume":"55","year":"2025","journal-title":"IEEE Trans. Cybern."},{"key":"2026031313411437900_c15","doi-asserted-by":"publisher","first-page":"5184","DOI":"10.1038\/s41467-024-49278-x","article-title":"Reconstructing higher-order interactions in coupled dynamical systems","volume":"15","year":"2024","journal-title":"Nat. Commun."},{"key":"2026031313411437900_c16","doi-asserted-by":"publisher","first-page":"073156","DOI":"10.1063\/5.0210741","article-title":"Stepwise reconstruction of higher-order networks from dynamics","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026031313411437900_c17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.physrep.2021.10.005","article-title":"Social physics","volume":"948","year":"2022","journal-title":"Phys. Rep."},{"key":"2026031313411437900_c18","doi-asserted-by":"publisher","first-page":"118","DOI":"10.1080\/01621459.1974.10480137","article-title":"Reaching a consensus","volume":"69","year":"1974","journal-title":"J. Am. Stat. Assoc."},{"key":"2026031313411437900_c19","doi-asserted-by":"publisher","first-page":"655","DOI":"10.1109\/TAC.2005.846556","article-title":"Consensus seeking in multiagent systems under dynamically changing interaction topologies","volume":"50","year":"2005","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026031313411437900_c20","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.mathsocsci.2024.01.009","article-title":"Dynamics of opinion polarization in a population","volume":"128","year":"2024","journal-title":"Math. Soc. Sci."},{"key":"2026031313411437900_c21","volume-title":"Opinion Dynamics and the Evolution of Social Power in Social Networks","year":"2019"},{"key":"2026031313411437900_c22"},{"key":"2026031313411437900_c23","doi-asserted-by":"publisher","first-page":"053127","DOI":"10.1063\/5.0086796","article-title":"Reconstructing community structure of online social network via user opinions","volume":"32","year":"2022","journal-title":"Chaos"},{"key":"2026031313411437900_c24","doi-asserted-by":"publisher","first-page":"063131","DOI":"10.1063\/5.0276020","article-title":"A bounded confidence model to predict how group work affects student math anxiety","volume":"35","year":"2025","journal-title":"Chaos"},{"key":"2026031313411437900_c25","doi-asserted-by":"publisher","first-page":"296","DOI":"10.1109\/TSMC.2022.3170745","article-title":"Mixed opinion dynamics based on DeGroot model and Hegselmann-Krause model in social networks","volume":"53","year":"2023","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"2026031313411437900_c26","doi-asserted-by":"publisher","first-page":"11920","DOI":"10.1016\/j.ifacol.2017.08.2310","article-title":"Consensus stability in the Hegselmann-Krause model with coopetition and cooperosity","volume":"50","year":"2017","journal-title":"IFAC-PapersOnLine"},{"key":"2026031313411437900_c27","doi-asserted-by":"publisher","first-page":"5302","DOI":"10.1002\/mma.3902","article-title":"Krause\u2019s model of opinion dynamics on isolated time scales","volume":"39","year":"2016","journal-title":"Math. Methods Appl. Sci."},{"key":"2026031313411437900_c28","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1109\/TBDATA.2017.2657624","article-title":"Dynamics of public opinions in an online and offline social network","volume":"7","year":"2017","journal-title":"IEEE Trans. Big Data"},{"key":"2026031313411437900_c29","doi-asserted-by":"publisher","first-page":"106240","DOI":"10.1016\/j.knosys.2020.106240","article-title":"Consensus reaching for social network group decision making by considering leadership and bounded confidence","volume":"204","year":"2020","journal-title":"Knowl. Based Syst."},{"key":"2026031313411437900_c30","first-page":"1","article-title":"Social influence networks and opinion change","volume":"16","year":"1999","journal-title":"Adv. Group Process."},{"key":"2026031313411437900_c31","doi-asserted-by":"publisher","first-page":"063116","DOI":"10.1063\/5.0085222","article-title":"How zealots affect the energy cost for controlling complex social networks","volume":"32","year":"2022","journal-title":"Chaos"},{"key":"2026031313411437900_c32","doi-asserted-by":"publisher","first-page":"e0122963","DOI":"10.1371\/journal.pone.0122963","article-title":"Association between an internet-based measure of area racism and black mortality","volume":"10","year":"2015","journal-title":"PLoS One"},{"key":"2026031313411437900_c33","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1038\/scientificamerican1155-31","article-title":"Opinions and social pressure","volume":"193","year":"1955","journal-title":"Sci. Am."},{"key":"2026031313411437900_c34","doi-asserted-by":"publisher","first-page":"103123","DOI":"10.1063\/5.0090831","article-title":"Cooperative evolution with opinion formation in a complex social environment","volume":"32","year":"2022","journal-title":"Chaos"},{"key":"2026031313411437900_c35","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1016\/j.automatica.2019.05.056","article-title":"An influence network model to study discrepancies in expressed and private opinions","volume":"107","year":"2019","journal-title":"Automatica"},{"key":"2026031313411437900_c36","doi-asserted-by":"publisher","first-page":"935","DOI":"10.1109\/TAC.2012.2214451","article-title":"Consensus problems on networks with antagonistic interactions","volume":"58","year":"2012","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026031313411437900_c37","doi-asserted-by":"publisher","first-page":"495","DOI":"10.1109\/TNSE.2021.3075595","article-title":"Leader-follower opinion dynamics of signed social networks with asynchronous trust\/distrust level evolution","volume":"9","year":"2022","journal-title":"IEEE Trans. Netw. Sci. Eng."},{"key":"2026031313411437900_c38","doi-asserted-by":"publisher","first-page":"46","DOI":"10.1109\/TCNS.2015.2424662","article-title":"Structural balance and opinion separation in trust\u2013mistrust social networks","volume":"3","year":"2015","journal-title":"IEEE Trans. Control Netw. Syst."},{"key":"2026031313411437900_c39"},{"key":"2026031313411437900_c40","doi-asserted-by":"publisher","first-page":"3339","DOI":"10.1109\/TAC.2018.2882198","article-title":"Polarizability, consensusability, and neutralizability of opinion dynamics on coopetitive networks","volume":"64","year":"2019","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026031313411437900_c41","doi-asserted-by":"publisher","first-page":"2270","DOI":"10.1109\/TAC.2016.2613905","article-title":"Novel multidimensional models of opinion dynamics in social networks","volume":"62","year":"2017","journal-title":"IEEE Trans. Autom. Control"},{"key":"2026031313411437900_c42","doi-asserted-by":"publisher","first-page":"112279","DOI":"10.1016\/j.automatica.2025.112279","article-title":"Coevolutionary dynamics of multidimensional opinions over coopetitive influence networks","volume":"177","year":"2025","journal-title":"Automatica"},{"key":"2026031313411437900_c43","doi-asserted-by":"publisher","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","year":"2018","journal-title":"Automatica"},{"key":"2026031313411437900_c44","volume-title":"Matrix Iterative Analysis","year":"2009"},{"key":"2026031313411437900_c45"},{"key":"2026031313411437900_c46","volume-title":"Linear System Theory","year":"1996"},{"key":"2026031313411437900_c47","doi-asserted-by":"publisher","first-page":"111840","DOI":"10.1016\/j.automatica.2024.111840","article-title":"Opinion dynamics analysis for stubborn individuals in cooperation\u2013competition networks based on path-dependence framework","volume":"169","year":"2024","journal-title":"Automatica"}],"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\/5.0322595\/20940164\/033126_1_5.0322595.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0322595\/20940164\/033126_1_5.0322595.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T17:41:17Z","timestamp":1773423677000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/3\/033126\/3383610\/Expressed-and-private-opinion-dynamics-in-signed"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,1]]},"references-count":47,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0322595","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,3]]},"published":{"date-parts":[[2026,3,1]]},"article-number":"033126"}}