{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T22:24:12Z","timestamp":1776205452970,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2019,10,30]],"date-time":"2019-10-30T00:00:00Z","timestamp":1572393600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2019,10,30]],"date-time":"2019-10-30T00:00:00Z","timestamp":1572393600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007601","name":"Horizon 2020","doi-asserted-by":"publisher","award":["620"],"award-info":[{"award-number":["620"]}],"id":[{"id":"10.13039\/501100007601","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Dyn Games Appl"],"published-print":{"date-parts":[[2020,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We provide a detailed study of the threshold model, where both conformist and anti-conformist agents coexist. Our study bears essentially on the convergence of the opinion dynamics in the society of agents, i.e., finding absorbing classes, cycles, etc. Also, we are interested in the existence of cascade effects, as this may constitute an undesirable phenomenon in collective behavior. We divide our study into two parts. In the first one, we basically study the threshold model supposing a fixed complete network, where every one is connected to every one, like in the seminal work of Granovetter. We study the case of a uniform distribution of the threshold, of a Gaussian distribution, and finally give a result for arbitrary distributions, supposing there is one type of anti-conformist. In a second part, we suppose that the neighborhood of an agent is random, drawn at each time step from a distribution. We distinguish the case where the degree (number of links) of an agent is fixed, and where there is an arbitrary degree distribution. We show the existence of cascades and that for most societies, the opinion converges to a chaotic situation.<\/jats:p>","DOI":"10.1007\/s13235-019-00332-0","type":"journal-article","created":{"date-parts":[[2019,10,31]],"date-time":"2019-10-31T17:18:41Z","timestamp":1572542321000},"page":"444-477","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":23,"title":["Anti-conformism in the Threshold Model of Collective Behavior"],"prefix":"10.1007","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3283-1496","authenticated-orcid":false,"given":"Michel","family":"Grabisch","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fen","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2019,10,30]]},"reference":[{"key":"332_CR1","first-page":"142","volume-title":"Contributions to mathematical psychology","author":"R Abelson","year":"1964","unstructured":"Abelson R (1964) Mathematical models of the distribution of attitudes under controversy. In: Frederiksen N, Guliksen H (eds) Contributions to mathematical psychology. Holt, Rinehart & Winston, Inc, New York, pp 142\u2013160"},{"key":"332_CR2","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s13235-010-0004-1","volume":"1","author":"D Acemoglu","year":"2011","unstructured":"Acemoglu D, Ozdaglar A (2011) Opinion dynamics and learning in social networks. Dyn Games Appl 1:3\u201349","journal-title":"Dyn Games Appl"},{"key":"332_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s001820400178","volume":"33","author":"Y Bramoull\u00e9","year":"2004","unstructured":"Bramoull\u00e9 Y, L\u00f3pez-Pintado D, Goyal S, Vega-Redondo F (2004) Network formation and anti-coordination games. Int J Game Theory 33:1\u201319","journal-title":"Int J Game Theory"},{"key":"332_CR4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-51865-7","volume-title":"Mob control: models of threshold collective behavior","author":"V Breer","year":"2017","unstructured":"Breer V, Novikov D, Rogatkin A (2017) Mob control: models of threshold collective behavior. Springer, Berlin"},{"key":"332_CR5","doi-asserted-by":"publisher","first-page":"e49441","DOI":"10.1371\/journal.pone.0049441","volume":"8","author":"Z Cao","year":"2013","unstructured":"Cao Z, Gao H, Qu X, Yang M, Yang X (2013) Fashion, cooperation, and social interactions. PLoS ONE 8:e49441","journal-title":"PLoS ONE"},{"key":"332_CR6","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1103\/RevModPhys.81.591","volume":"81","author":"C Castellano","year":"2009","unstructured":"Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81:591\u2013646","journal-title":"Rev Mod Phys"},{"key":"332_CR7","first-page":"181","volume":"63","author":"J French Jr","year":"1956","unstructured":"French J Jr (1956) A formal theory of social power. Physchol Rev 63:181\u2013194","journal-title":"Physchol Rev"},{"key":"332_CR8","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1080\/0022250X.1990.9990069","volume":"15","author":"N Friedkin","year":"1990","unstructured":"Friedkin N, Johnsen E (1990) Social influences and opinion. J Math Sociol 15:193\u2013205","journal-title":"J Math Sociol"},{"key":"332_CR9","first-page":"403","volume":"25","author":"S Galam","year":"2002","unstructured":"Galam S (2002) Minority opinion spreading in random geometry. Eur Phys J B 25:403\u2013406","journal-title":"Eur Phys J B"},{"key":"332_CR10","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1016\/j.physa.2003.10.041","volume":"333","author":"S Galam","year":"2004","unstructured":"Galam S (2004) Contrarian deterministic effects on opinion dynamics: the hung elections scenario. Physica A 333:453\u2013460","journal-title":"Physica A"},{"key":"332_CR11","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4614-2032-3","volume-title":"Sociophysics\u2014a physicist\u2019s modeling of psycho-political phenomena","author":"S Galam","year":"2012","unstructured":"Galam S (2012) Sociophysics\u2014a physicist\u2019s modeling of psycho-political phenomena. Springer, Berlin"},{"key":"332_CR12","first-page":"187","volume":"30","author":"E Goles","year":"1980","unstructured":"Goles E, Olivos J (1980) Periodic behavior of generalized threshold functions. Discrete Appl Math 30:187\u2013189","journal-title":"Discrete Appl Math"},{"key":"332_CR13","doi-asserted-by":"publisher","first-page":"103773","DOI":"10.1016\/j.jedc.2019.103773","volume":"109","author":"M Grabisch","year":"2019","unstructured":"Grabisch M, Poindron A, Rusinowska A (2019) A model of anonymous influence with anti-conformist agrnts. J Econ Dyn Control 109:103773. https:\/\/doi.org\/10.1016\/j.jedc.2019.103773","journal-title":"J Econ Dyn Control"},{"key":"332_CR14","doi-asserted-by":"publisher","first-page":"1420","DOI":"10.1086\/226707","volume":"83","author":"M Granovetter","year":"1978","unstructured":"Granovetter M (1978) Threshold models of collective behavior. Am J Sociol 83:1420\u20131443","journal-title":"Am J Sociol"},{"key":"332_CR15","doi-asserted-by":"publisher","first-page":"241","DOI":"10.1006\/aama.1998.0599","volume":"21","author":"J Gravner","year":"1998","unstructured":"Gravner J, Griffeath D (1998) Cellular automaton growth on $$\\mathbb{Z}^2$$: theorems, examples and problems. Adv Appl Math 21:241\u2013304","journal-title":"Adv Appl Math"},{"key":"332_CR16","doi-asserted-by":"publisher","DOI":"10.1515\/9781400833993","volume-title":"Social and economic networks","author":"M Jackson","year":"2008","unstructured":"Jackson M (2008) Social and economic networks. Princeton University Press, Princeton"},{"issue":"2","key":"332_CR17","doi-asserted-by":"publisher","first-page":"022313","DOI":"10.1103\/PhysRevE.99.022313","volume":"99","author":"JS Juul","year":"2019","unstructured":"Juul JS, Porter MA (2019) Hipsters on networks: how a minority group of individuals can lead to an antiestablishment majority. Phys Rev E 99(2):022313","journal-title":"Phys Rev E"},{"issue":"1","key":"332_CR18","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1142\/S0219198909002145","volume":"11","author":"D L\u00f3pez-Pintado","year":"2009","unstructured":"L\u00f3pez-Pintado D (2009) Network formation, cost-sharing and anti-coordination. Int Game Theory Rev 11(1):53\u201376","journal-title":"Int Game Theory Rev"},{"key":"332_CR19","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1111\/1467-937X.00121","volume":"67","author":"S Morris","year":"2000","unstructured":"Morris S (2000) Contagion. Rev Econ Stud 67:57\u201378","journal-title":"Rev Econ Stud"},{"key":"332_CR20","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1214\/14-PS230","volume":"14","author":"E Mossel","year":"2017","unstructured":"Mossel E, Tamuz O (2017) Opinion exchange dynamics. Probab Surv 14:155\u2013204","journal-title":"Probab Surv"},{"key":"332_CR21","doi-asserted-by":"crossref","unstructured":"Nowak, N., Sznajd-Weron, K.: Homogeneous symmetrical threshold model with nonconformity: independence versus anticonformity. Complexity pp 1\u201314, Article ID 5150825 (2019)","DOI":"10.1155\/2019\/5150825"},{"key":"332_CR22","doi-asserted-by":"publisher","first-page":"174","DOI":"10.1007\/s10955-013-0701-4","volume":"151","author":"P Nyczka","year":"2013","unstructured":"Nyczka P, Sznajd-Weron K (2013) Anticonformity or independence?\u2014Insights from statistical physics. J Stat Phys 151:174\u2013202","journal-title":"J Stat Phys"},{"key":"332_CR23","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/BF01737559","volume":"2","author":"RW Rosenthal","year":"1973","unstructured":"Rosenthal RW (1973) A class of games possessing pure-strategy nash equilibria. Int J Game Theory 2:65\u201367","journal-title":"Int J Game Theory"},{"key":"332_CR24","volume-title":"Micromotives and macrobehaviour","author":"T Schelling","year":"1978","unstructured":"Schelling T (1978) Micromotives and macrobehaviour. Norton, New York"},{"key":"332_CR25","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1177\/001872676802100202","volume":"21","author":"M Taylor","year":"1968","unstructured":"Taylor M (1968) Towards a mathematical theory of influence and attitude change. Hum Relat 21:121\u2013139","journal-title":"Hum Relat"},{"key":"332_CR26","unstructured":"Touboul J (2014) The hipster effect: when anticonformists all look the same. arXiv preprint arXiv:1410.8001"},{"issue":"9","key":"332_CR27","doi-asserted-by":"publisher","first-page":"5766","DOI":"10.1073\/pnas.082090499","volume":"99","author":"DJ Watts","year":"2002","unstructured":"Watts DJ (2002) A simple model of global cascades on random networks. Proc Natl Acad Sci USA 99(9):5766\u20135771","journal-title":"Proc Natl Acad Sci USA"}],"container-title":["Dynamic Games and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-019-00332-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s13235-019-00332-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s13235-019-00332-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T10:07:12Z","timestamp":1695377232000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s13235-019-00332-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,10,30]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,6]]}},"alternative-id":["332"],"URL":"https:\/\/doi.org\/10.1007\/s13235-019-00332-0","relation":{},"ISSN":["2153-0785","2153-0793"],"issn-type":[{"value":"2153-0785","type":"print"},{"value":"2153-0793","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,10,30]]},"assertion":[{"value":"30 October 2019","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}