Abstract
The voter model is widely used to model opinion dynamics in society. In this paper, we propose three modifications to incorporate heterogeneity into the model. We address the corresponding oversimplifications of the conventional voter model which are unrealistic. We first consider the voter model with popularity bias. The influence of each node on its neighbors depends on its degree. We find the consensus probabilities and expected consensus times for each of the states. We also find the fixation probability, which is the probability that a single node whose state differs from every other node imposes its state on the entire system. In addition, we find the expected fixation time. Then two other extensions to the model are proposed and the motivations behind them are discussed. The first one is confidence, where in addition to the states of neighbors, nodes take their own state into account at each update. We repeat the calculations for the augmented model and investigate the effects of adding confidence to the model. The second proposed extension is irreversibility, where one of the states is given the property that once nodes adopt it, they cannot switch back. This is motivated by applications where, agents take an irreversible action such as seeing a movie, purchasing a music album online, or buying a new product. The dynamics of densities, fixation times and consensus times are obtained.
Similar content being viewed by others
References
D. Kempe, J. Kleinberg, E. Tardos, in Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining (ACM, 2003), pp. 137–146
L. Backstrom, D. Huttenlocher, J. Kleinberg, X. Lan, in Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining (ACM, 2006), pp. 44–54
G. Kossinets, J. Kleinberg, D. Watts, in Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining (ACM, 2008), pp. 435–443
J. Leskovec, L. Backstrom, R. Kumar, A. Tomkins, in Proceedings of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining (ACM, 2008), pp. 462–470
A. Tahbaz-Salehi, A. Sandroni, A. Jadbabaie, in Proceedings of the 48th IEEE Conference on Decision and Control, 2009 held jointly with the 2009 28th Chinese Control Conference, Shanghai, 2009, pp. 1513–1519
A. Mirtabatabaei, F. Bullo, SIAM J. Control Opt. 50, 2763 (2012)
D. Acemoglu, A. Ozdaglar, Dyn. Games Appl. 1, 3 (2011)
D. Acemoglu, A. Ozdaglar, A. ParandehGheibi, Games Econ. Behav. 70, 194 (2010)
R.A. Blythe, A.J. McKane, J. Stat. Mech. 2007, P07018 (2007)
F. Brauer, C. Castillo-Châavez, Mathematical models in population biology and epidemiology (Springer, New York, 2012)
C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2009)
S. Galam, Int. J. Mod. Phys. C 19, 409 (2008)
K. Sznajd-Weron, Acta Phys. Pol. B 36, 2537 (2005)
R. Olfati-Saber, J.A. Fax, R.M. Murray, Proc. IEEE 95, 215 (2007)
A. Jadbabaie, J. Lin, A.S. Morse, IEEE Trans. Automatic Control 48, 988 (2003)
D. Acemoglu, G. Como, F. Fagnani, A. Ozdaglar, in Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), 2011 (IEEE, 2011), pp. 2347–2352
G. Shi, M. Johansson, K.H. Johansson, IEEE J. Selected Areas Commun. 31, 1061 (2013)
T.M. Liggett, Interacting Particle Systems (Springer, New York, 1985)
M.E. Yildiz, R. Pagliari, A. Ozdaglar, A. Scaglione, in Inf. Th. App. Workshop (ITA) (IEEE, 2010), pp. 1–7
S.E. Asch, in Effects of group pressure upon the modification and distortion of judgment, edited by H. Guetzkow, leadership and men (Carnegie Press, Pittsburgh, 1951)
M. Sherif, The psychology of social norms (Harper, Oxford, 1936)
S. Redner, A guide to first-passage processes (Cambridge University Press, 2001)
V. Sood, T. Antal, S. Redner, Phys. Rev. E 77, 041121 (2008)
D. Volovik, M. Mobilia, S. Redner, Europhys. Lett. 85, 48033 (2009)
M. Bramson, D. Griffeath, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 53, 183 (1980)
J.T. Cox, D. Griffeath, Ann. Prob. 14, 347 (1986)
J.T. Cox, Ann. Prob. 17, 1333 (1989)
J.T. Cox, R. Durrett, Nonlinear voter models, in Random Walks, Brownian Motion and Interacting Particle Systems, edited by R. Durrett, H. Kesten (Birkhauser, Boston, 1991), pp. 189–202
T.M. Liggett, Ann. Prob. 22, 764 (1994)
M. Mobilia, A. Petersen, S. Redner, J Stat. Mech. 2007, P08029 (2007)
M. Mobilia, Phys. Rev. Lett. 91, 028701 (2003)
H.U. Stark, C.J. Tessone, F. Schweitzer, Phys. Rev. Lett. 101, 018701 (2008)
L. Dall’Asta, C. Castellano, Europhys. Lett. 77, 60005 (2007)
N. Crokidakis, C. Anteneodo, Phys. Rev. E 86, 061127 (2012)
S.R. Souza, S. Gonçalves, Phys. Rev. E 85, 056103 (2012)
P. Holme, M.E.J. Newman, Phys. Rev. E 74, 056108 (2006)
S.D. Yi, S.K. Baek, C.P. Zhu, B.J. Kim, Phys. Rev. E 87, 012806 (2013)
G. Zschaler, G.A. Böhme, M. Seißinger, C. Huepe, T. Gross, Phys. Rev. E 85, 046107 (2012)
C.M. Schneider-Mizell, L.M. Sander, J. Stat. Phys. 136, 59 (2009)
J.P. Gleeson, S. Melnik, J.A. Ward, M.A. Porter, P.J. Mucha, Phys. Rev. E 85, 026106 (2012)
A. Baronchelli, C. Castellano, R. Pastor-Satorras, Phys. Rev. E 83, 066117 (2011)
F.M. Bass, J. Marketing Res. 11, 1 (1974)
F.M. Bass, W.W. Talarzyk, J. Marketing Res. 9, 93 (1972)
V. Mahajan, E. Muller, F.M. Bass, J. Marketing 54, 1 (1990)
E.M. Rogers, Diffusion of Innovation (The Free Press, New York, 2003)
R. Peres, E. Muller, V. Mahajan, Int. J. Res. Marketing 27, 91 (2010)
S. Morris, Rev. Econ. Studies 67, 57 (2000)
J. Coleman, E. Katz, H. Menzel, Sociometry 20, 253 (1957)
L. Kuandykov, M. Sokolov, Decision Support System 48, 531 (2010)
S.A. Delre, W. Jager, T.H. Bijmolt, M.A. Janssen, J. Prod. Innov. Manag. 27, 267 (2010)
H. Amini, M. Draief, M. Lelarge, Network Control and Optimization, Lecture Notes in Computer Science (Springer, 2009), Vol. 5894, p. 17
S.A. Delre, W. Jager, M.A. Janssen, Comput. Math. Organ. Theory 13, 185 (2007)
M. Molloy, B. Reed, Random Struct. Alg. 6, 161 (1995)
M. Molloy, B. Reed, Combinatorics Prob. Comput. 7, 295 (1998)
M. Catanzaro, M. Boguñá, R. Pastor-Satorras, Phys. Rev. E 71, 027103 (2005)
A.L. Barabási, R. Albert, H. Jeong, Physica A 272, 173 (1999)
P. Krapivsky, S. Redner, J. Phys. A 35, 9517 (2002)
G.J. Baxter, R.A. Blythe, W. Croft, A.J. McKane, Phys. Rev. E 73, 046118 (2006)
I.D. Chase, Am. Soci. Rev. 45, 905 (1980)
I.D. Chase, Behaviour 80, 218 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fotouhi, B., Rabbat, M. Voter model with arbitrary degree dependence: clout, confidence and irreversibility. Eur. Phys. J. B 87, 55 (2014). https://doi.org/10.1140/epjb/e2014-41088-3
Received:
Revised:
Published:
DOI: https://doi.org/10.1140/epjb/e2014-41088-3