Abstract
We investigate the three-state majority rule model in a coevolving network with intensive average degree using Monte Carlo simulations. The key parameter investigated is the degree of homophily (heterophily), which is the probability p (\(q = 1 - p\)) of a given person being affected by others with the same (different) opinion. For a system with a uniformly random initial state, so that each person has an equal chance of selecting one of the three opinions, based on extensive Monte Carlo simulations, we found that there are three distinct phases: (1) When the population has an intermediate homophilic tendency, it reaches the consensus state very fast. (2) When the system has a moderate to large heterophilic tendency (a small value of p), the time to consensus (or convergence time) can be significantly longer. (3) When the system has a high homophilic tendency (a large value of p), the population can remain in a polarization state for a long time. We defined the convergence time for the system of voters to reach consensus operationally, and obtained a distribution function for the convergence time through Monte Carlo simulations. We observed that the average convergence time in a three-state opinion formation process is generally faster than in the same model of voting dynamics when only two states are available to voters, given that in the beginning of the simulations, different opinions are uniformly spread in the population. The implications in diversity are discussed.
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References
Benczik IJ, Benczik SZ, Schmittmann B, Zia RKP (2008) Lack of consensus in social systems. Europhys Lett 82(4):48006 (2008). doi:10.1209/0295-5075/82/48006. http://iopscience.iop.org/0295-5075/82/4/48006
Benczik IJ, Benczik SZ, Schmittmann B, Zia RKP (2009) Opinion dynamics on an adaptive random network. Phys Rev E 79(4):046104. doi:10.1103/PhysRevE.79.046104. http://link.aps.org/doi/10.1103/PhysRevE.79.046104
Castellano C, Fortunato S, Loreto V (2009) Statistical physics of social dynamics. Rev Mod Phys 81(2):591–646. doi:10.1103/RevModPhys.81.591. http://link.aps.org/doi/10.1103/RevModPhys.81.591
Chakrabarti BK, Chakraborti A, Chatterjee A (2007) Econophysics and sociophysics: trends and perspectives. Wiley-VCH, Weinheim
Fu X, Szeto KY, Cheung WK (2004) Phase transition of two-dimensional Ising model on random point patterns. Phys Rev E 70(5):056123. doi:10.1103/PhysRevE.70.056123. http://link.aps.org/doi/10.1103/PhysRevE.70.056123
Galam S (2002) Minority opinion spreading in random geometry. Eur Phys J B Condens Matter Complex Syst 25(4):403–406. doi:10.1140/epjb/e20020045. http://www.springerlink.com/content/mbteay82yjxatrpt/abstract/
Holley RA, Liggett TM (1975) Ergodic theorems for weakly interacting infinite systems and the voter model. Ann Probab 3(4):643–663. http://www.jstor.org/stable/2959329 [ArticleType: research-article/full publication date: Aug., 1975/Copyright 1975 Institute of Mathematical Statistics]
Holme P, Newman MEJ (2006) Nonequilibrium phase transition in the coevolution of networks and opinions. Phys Rev E 74(5):056108. doi:10.1103/PhysRevE.74.056108. http://link.aps.org/doi/10.1103/PhysRevE.74.056108
Katarzyna S, Sznajd J (2000) Opinion evolution in closed community. Int J Mod Phys C 11(6):1157–1165
Kuperman M, Zanette D (2002) Stochastic resonance in a model of opinion formation on small-world networks. Eur Phys J B Condens Matter Complex Syst 26(3):387–391. doi:10.1140/epjb/e20020104.http://www.springerlink.com/content/12wu6yqlk66u103u/abstract/
Nardini C, Kozma B, Barrat A (2008) Who’s talking first? Consensus or lack thereof in coevolving opinion formation models. Phys Rev Lett 100(15):158701. doi:10.1103/PhysRevLett.100.158701.http://link.aps.org/doi/10.1103/PhysRevLett.100.158701
Schmittmann B, Mukhopadhyay A (2010) Opinion formation on adaptive networks with intensive average degree. Phys Rev E 82(6):066104. doi:10.1103/PhysRevE.82.066104. http://link.aps.org/doi/10.1103/PhysRevE.82.066104
Sood V, Redner S (2005) Voter model on heterogeneous graphs. Phys Rev Lett 94(17):178701. doi:10.1103/PhysRevLett.94.178701.http://link.aps.org/doi/10.1103/PhysRevLett.94.178701
Stauffer D, De Oliveira SM, de Oliveira PM, Martins JS (2006) Biology, sociology, geology by computational physicists. Monogr Ser Nonlinear Sci Complex 1:i276
Weidlich W (1971) The statistical description of polarization phenomena in society. Br J Math Stat Psychol 24(2):251–266. doi:10.1111/j.2044-8317.1971.tb00470.x, 10.1111/j.2044-8317.1971.tb00470.x. http://onlinelibrary.wiley.com/doi/10.1111/j.2044-8317.1971.tb00470.x/abstract
Fu Xiu-Jun, Szeto KY (2009) Competition of multi-agent systems: analysis of a three-company econophysics model. Chin Phys Lett 26(9):098901. doi:10.1088/0256-307X/26/9/098901. http://iopscience.iop.org/0256-307X/26/9/098901
Acknowledgements
The authors thank Mathis Antony for his helpful discussions. K.Y. Szeto acknowledges the support of Grant FSGRF 13SC25.
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Wu, D., Szeto, K.Y. (2014). Three-State Opinion Formation Model on Adaptive Networks and Time to Consensus. In: Chen, SH., Terano, T., Yamamoto, R., Tai, CC. (eds) Advances in Computational Social Science. Agent-Based Social Systems, vol 11. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54847-8_6
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DOI: https://doi.org/10.1007/978-4-431-54847-8_6
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