{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T02:49:00Z","timestamp":1778899740040,"version":"3.51.4"},"reference-count":54,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T00:00:00Z","timestamp":1680739200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Political polarization is perceived as a threat to democracies. Using the Galam model of opinion dynamics deployed in a five-dimensional parameter space, I show that polarization is the byproduct of an essential hallmark of a vibrant democratic society, namely open and informal discussions among agents. Indeed, within a homogeneous social community with floaters, the dynamics lead gradually toward unanimity (zero entropy). Polarization can eventually appear as the juxtaposition of non-mixing social groups sharing different prejudices about the issue at stake. On the other hand, the inclusion of contrarian agents produces a polarization within a community that mixes when their proportion x is beyond a critical value xc=16\u22480.167 for discussing groups of size three and four. Similarly, the presence of stubborn agents also produces a polarization of a community that mixes when the proportion of stubborn agents is greater than some critical value. For equal proportions of stubborn agents a along each opinion, ac=29\u22480.22 for group size four against ac=14=0.25 for group size three. However, the evaluation of the proportion of individual opinion shifts at the attractor 12 and indicates that the polarization produced by contrarians is fluid with a good deal of agents who keep shifting between the two opposed blocks (high entropy). That favors a coexistence of opposite opinions in a divided community. In contrast, the polarization created by stubborn agents is found to be frozen with very few individuals shifting opinion between the two opinions (low entropy). That yields a basis for the emergence of hate between the frozen opposed blocks.<\/jats:p>","DOI":"10.3390\/e25040622","type":"journal-article","created":{"date-parts":[[2023,4,6]],"date-time":"2023-04-06T05:21:23Z","timestamp":1680758483000},"page":"622","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":24,"title":["Unanimity, Coexistence, and Rigidity: Three Sides of Polarization"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9003-5420","authenticated-orcid":false,"given":"Serge","family":"Galam","sequence":"first","affiliation":[{"name":"CEVIPOF\u2014Centre for Political Research, Sciences Po and CNRS, 1 Place Saint Thomas d\u2019Aquin, 75007 Paris, France"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"024125","DOI":"10.1103\/PhysRevE.105.024125","article-title":"Transitions between polarization and radicalization in a temporal bilayer echo-chamber model","volume":"105","author":"Gajewski","year":"2022","journal-title":"Phys. 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