{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:23:24Z","timestamp":1760243004102,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2015,5,13]],"date-time":"2015-05-13T00:00:00Z","timestamp":1431475200000},"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>Spin dynamics on networks allows us to understand how a global consensus emerges out of individual opinions. Here, we are interested in the effect of heterogeneity in the initial geographic distribution of a competing opinion on the competitiveness of its own opinion. Accordingly, in this work, we studied the effect of spatial heterogeneity on the majority rule dynamics using a three-state spin model, in which one state is neutral. Monte  Carlo simulations were performed on square lattices divided into square blocks (cells). Accordingly, one competing opinion was distributed uniformly among cells, whereas the spatial distribution of the rival opinion was varied from the uniform to heterogeneous, with the median-to-mean ratio in the range from 1 to 0. When the size of discussion group is odd, the uncommitted agents disappear completely after  3.30 \u00b1 0.05 update cycles, and then the system evolves in a two-state regime with complementary spatial distributions of two competing opinions. Even so, the initial heterogeneity in the spatial distribution of one of the competing opinions causes a decrease of this opinion competitiveness. That is, the opinion with initially heterogeneous spatial distribution has less probability to win, than the opinion with the initially uniform spatial distribution, even when the initial concentrations of both opinions are equal. We found that although the time to consensus , the opinion\u2019s recession rate is determined during the first 3.3 update cycles. On the other hand, we found that the initial heterogeneity of the opinion spatial distribution assists the formation of quasi-stable regions, in which this opinion is dominant. The results of Monte  Carlo simulations are discussed with regard to the electoral competition of political parties.<\/jats:p>","DOI":"10.3390\/e17053160","type":"journal-article","created":{"date-parts":[[2015,5,13]],"date-time":"2015-05-13T12:42:07Z","timestamp":1431520927000},"page":"3160-3171","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Effect of Heterogeneity in Initial Geographic Distribution on Opinions\u2019 Competitiveness"],"prefix":"10.3390","volume":"17","author":[{"given":"Alexander","family":"Balankin","sequence":"first","affiliation":[{"name":"Grupo \"Mec\u00e1nica Fractal\", ESIME-Zacatenco, Instituto Polit\u00e9cnico Nacional, M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Miguel","family":"Mart\u00ednez Cruz","sequence":"additional","affiliation":[{"name":"Grupo \"Mec\u00e1nica Fractal\", ESIME-Zacatenco, Instituto Polit\u00e9cnico Nacional, M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Felipe","family":"Gayosso Mart\u00ednez","sequence":"additional","affiliation":[{"name":"TEMPLE S.A de C.V., M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Claudia","family":"Mart\u00ednez-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Grupo \"Mec\u00e1nica Fractal\", ESIME-Zacatenco, Instituto Polit\u00e9cnico Nacional, M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leobardo","family":"Morales Ruiz","sequence":"additional","affiliation":[{"name":"TEMPLE S.A de C.V., M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Juli\u00e1n","family":"Pati\u00f1o Ortiz","sequence":"additional","affiliation":[{"name":"Grupo \"Mec\u00e1nica Fractal\", ESIME-Zacatenco, Instituto Polit\u00e9cnico Nacional, M\u00e9xico D.F. 07738, M\u00e9xico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,5,13]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Galam, S. 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