Abstract
We present a detailed investigation of the behavior of the nonlinear q-voter model for opinion dynamics. At the mean-field level we derive analytically, for any value of the number q of agents involved in the elementary update, the phase diagram, the exit probability and the consensus time at the transition point. The mean-field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks. We finally discuss the case of random regular networks and compare analytical results with simulations.
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Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover, New York (1972)
Albert, R., Barabási, A.L.: Statistical mechanics of complex networks. Rev. Mod. Phys. 74, 47–97 (2002)
Al Hammal, O., Chaté, H., Dornic, I., Muñoz, M.A.: Langevin description of critical phenomena with two symmetric absorbing states. Phys. Rev. Lett. 94, 230601 (2005)
Barrat, A., Barthélemy, M., Vespignani, A.: Dynamical Processes on Complex Networks. Cambridge University Press, Cambridge (2008)
Blythe, R.A.: Ordering in voter models on networks: exact reduction to a single-coordinate diffusion. J. Phys. A 43, 385003 (2010)
Blythe, R.A., McKane, A.J.: Stochastic models of evolution in genetics, ecology and linguistics. J. Stat. Mech. 2007, P07018 (2007)
Boguñá, M., Pastor-Satorras, R.: Epidemic spreading in correlated complex networks. Phys. Rev. E 66, 047104 (2002)
Bray, A.J.: Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459 (1994)
Canet, L., Chaté, H., Delamotte, B., Dornic, I., Muñoz, M.A.: Nonperturbative fixed point in a nonequilibrium phase transition. Phys. Rev. Lett. 95, 100601 (2005)
Castellano, C.: Effect of network topology on the ordering dynamics of voter models. AIP Conf. Proc. 779, 114 (2005)
Castellano, C., Pastor-Satorras, R.: Universal and nonuniversal features of the generalized voter class for ordering dynamics in two dimensions. Phys. Rev. E 86, 051123 (2012)
Castellano, C., Fortunato, S., Loreto, V.: Statistical physics of social dynamics. Rev. Mod. Phys. 81, 591–646 (2009)
Castellano, C., Muñoz, M.A., Pastor-Satorras, R.: Nonlinear q-voter model. Phys. Rev. E 80, 041129 (2009)
Clifford, P., Sudbury, A.: A model for spatial conflict. Biometrika 60, 581–588 (1973)
de Oliveira, M., Mendes, J., Santos, M.: Nonequilibrium spin models with Ising universal behaviour. J. Phys. A 26, 2317–2324 (1993)
Dornic, I., Chaté, H., Chave, J., Hinrichsen, H.: Critical coarsening without surface tension: the universality class of the voter model. Phys. Rev. Lett. 87, 045701 (2001)
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Critical phenomena in complex networks. Rev. Mod. Phys. 80, 1275–1335 (2008)
Dorogovtsev, S.N., Mendes, J.F.F.: Evolution of networks. Adv. Phys. 51, 1079–1187 (2002)
Drouffe, J.M., Godrèche, C.: Phase ordering and persistence in a class of stochastic processes interpolating between the Ising and voter models. J. Phys. A 32, 249–261 (1999)
Gardiner, C.W.: Handbook of Stochastic Methods, 2nd edn. Springer, Berlin (1985)
Holley, R.A., Liggett, T.M.: Ergodic theorems for weakly interacting infinite systems and voter model. Ann. Probab. 3, 643–663 (1975)
Krapivsky, P., Redner, S., Ben-Naim, E.: A Kinetic View of Statistical Physics. Cambridge University Press, Cambridge (2010)
Liggett, T.M.: Stochastic Interacting Particle Systems: Contact, Voter, and Exclusion Processes. Springer, New York (1999)
Molofsky, J., Durrett, R., Dushoff, J., Griffeath, D., Levin, S.: Local frequency dependence and global coexistence. Theor. Popul. Biol. 55, 270–282 (1999)
Moretti, P., Liu, S.Y., Baronchelli, A.: Pastor-Satorras, R.: Heterogenous mean-field analysis of a generalized voter-like model on networks. Eur. Phys. J. B 85, 88 (2012)
Newman, M.E.J.: Networks: An Introduction. Oxford University Press, Oxford (2010)
Pugliese, E., Castellano, C.: Heterogeneous pair approximation for voter models on networks. Europhys. Lett. 88, 58004 (2009)
Sood, V., Redner, S.: Voter model on heterogeneous graphs. Phys. Rev. Lett. 94, 178701 (2005)
Sood, V., Antal, T., Redner, S.: Voter models on heterogeneous networks. Phys. Rev. E 77, 041121 (2008)
Suchecki, K., Eguíluz, V.M., Miguel, M.S.: Conservation laws for the voter model in complex networks. Europhys. Lett. 69, 228–234 (2005)
Vázquez, F., López, C.: Systems with two symmetric absorbing states: Relating the micro scopic dynamics with the macroscopic behavior. Phys. Rev. E 78, 061127 (2008)
Acknowledgements
R.P.-S. acknowledges financial support from the Spanish MEC, under project No. FIS2010-21781-C02-01; the Junta de Andalucía, under project No. P09-FQM4682; ICREA Academia, funded by the Generalitat de Catalunya; partial support by the NSF under Grant No. PHY1066293, and the hospitality of the Aspen Center for Physics, CO, USA, where part of this work was performed. P.M. acknowledges financial support from Junta de Andalucía project P09-FQM4682 and MICINN–FEDER project FIS2009-08451. S.Y.L. acknowledges the support of the 973 Program of China (No. 2012CB720500) and the National High Technology R&D Program of China (2012AA041102).
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Moretti, P., Liu, S., Castellano, C. et al. Mean-Field Analysis of the q-Voter Model on Networks. J Stat Phys 151, 113–130 (2013). https://doi.org/10.1007/s10955-013-0704-1
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DOI: https://doi.org/10.1007/s10955-013-0704-1