{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T19:52:19Z","timestamp":1775850739331,"version":"3.50.1"},"reference-count":55,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T00:00:00Z","timestamp":1555545600000},"content-version":"vor","delay-in-days":107,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"NCN, Poland","award":["2016\/21\/B\/HS6\/01256"],"award-info":[{"award-number":["2016\/21\/B\/HS6\/01256"]}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Complexity"],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We study two variants of the modified Watts threshold model with a noise (with nonconformity, in the terminology of social psychology) on a complete graph. Within the first version, a noise is introduced via so\u2010called independence, whereas in the second version anticonformity plays the role of a noise, which destroys the order. The modified Watts threshold model, studied here, is homogeneous and possesses an up\u2010down symmetry, which makes it similar to other binary opinion models with a single\u2010flip dynamics, such as the majority\u2010vote and the <jats:italic>q<\/jats:italic>\u2010voter models. Because within the majority\u2010vote model with independence only continuous phase transitions are observed, whereas within the <jats:italic>q<\/jats:italic>\u2010voter model with independence also discontinuous phase transitions are possible, we ask the question about the factor, which could be responsible for discontinuity of the order parameter. We investigate the model via the mean\u2010field approach, which gives the exact result in the case of a complete graph, as well as via Monte Carlo simulations. Additionally, we provide a heuristic reasoning, which explains observed phenomena. We show that indeed if the threshold <jats:italic>r<\/jats:italic> = 0.5, which corresponds to the majority\u2010vote model, an order\u2010disorder transition is continuous. Moreover, results obtained for both versions of the model (one with independence and the second one with anticonformity) give the same results, only rescaled by the factor of 2. However, for <jats:italic>r<\/jats:italic> &gt; 0.5 the jump of the order parameter and the hysteresis is observed for the model with independence, and both versions of the model give qualitatively different results.<\/jats:p>","DOI":"10.1155\/2019\/5150825","type":"journal-article","created":{"date-parts":[[2019,4,18]],"date-time":"2019-04-18T23:43:23Z","timestamp":1555631003000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["Homogeneous Symmetrical Threshold Model with Nonconformity: Independence versus Anticonformity"],"prefix":"10.1155","volume":"2019","author":[{"given":"Bart\u0142omiej","family":"Nowak","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1851-8508","authenticated-orcid":false,"given":"Katarzyna","family":"Sznajd-Weron","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2019,4,18]]},"reference":[{"key":"e_1_2_8_1_2","doi-asserted-by":"publisher","DOI":"10.1103\/revmodphys.81.591"},{"key":"e_1_2_8_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-2032-3"},{"key":"e_1_2_8_3_2","volume-title":"Sociophysics: An Introduction","author":"Sen P.","year":"2013"},{"key":"e_1_2_8_4_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-25658-0_17"},{"key":"e_1_2_8_5_2","doi-asserted-by":"publisher","DOI":"10.1142\/s0219525900000078"},{"key":"e_1_2_8_6_2","article-title":"Opinion dynamics and bounded confidence: models, analysis and simulation","volume":"5","author":"Hegselmann R.","year":"2002","journal-title":"Journal of Artificial Societies and Social Simulation"},{"key":"e_1_2_8_7_2","doi-asserted-by":"publisher","DOI":"10.3390\/e19070371"},{"key":"e_1_2_8_8_2","doi-asserted-by":"publisher","DOI":"10.1088\/1742-5468\/2014\/03\/P03007"},{"key":"e_1_2_8_9_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.rser.2016.04.063"},{"key":"e_1_2_8_10_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2018.03.086"},{"key":"e_1_2_8_11_2","doi-asserted-by":"publisher","DOI":"10.1007\/s11135-007-9072-8"},{"key":"e_1_2_8_12_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0217979217420152"},{"key":"e_1_2_8_13_2","doi-asserted-by":"publisher","DOI":"10.3389\/fphy.2018.00024"},{"key":"e_1_2_8_14_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8542-4"},{"key":"e_1_2_8_15_2","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/24\/15\/033"},{"key":"e_1_2_8_16_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01060069"},{"key":"e_1_2_8_17_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01027314"},{"key":"e_1_2_8_18_2","doi-asserted-by":"publisher","DOI":"10.1142\/s0129183100000936"},{"key":"e_1_2_8_19_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.082090499"},{"key":"e_1_2_8_20_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.80.041129"},{"key":"e_1_2_8_21_2","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-013-0701-4"},{"key":"e_1_2_8_22_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.97.052106"},{"key":"e_1_2_8_23_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevX.3.021004"},{"key":"e_1_2_8_24_2","doi-asserted-by":"publisher","DOI":"10.1086\/226707"},{"key":"e_1_2_8_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2018.03.077"},{"key":"e_1_2_8_26_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.86.011105"},{"key":"e_1_2_8_27_2","doi-asserted-by":"publisher","DOI":"10.1086\/518527"},{"key":"e_1_2_8_28_2","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0209620"},{"key":"e_1_2_8_29_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.physa.2016.01.013"},{"key":"e_1_2_8_30_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.4990038"},{"key":"e_1_2_8_31_2","unstructured":"JedrzejewskiA.andSznajd-WeronK. 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