CONSENSUS FORMATION OF TWO-LEVEL OPINION DYNAMICS

doi:10.1016/S0252-9602(14)60067-9

Abstract

Abstract:

Opinion dynamics have received significant attention in recent years. This pa-per proposes a bounded confidence opinion model for a group of agents with two different confidence levels. Each agent in the population is endowed with a confidence interval around her opinion with radius αd or (1 − α)d, where α∈ (0, 1/2] represents the differentiation of
confidence levels. We analytically derived the critical confidence bound d_c= 1/(4α) for the two-level opinion dynamics on Z. A single opinion cluster is formed with probability 1 above this critical value regardless of the ratio p of agents with high/low confidence. Extensive numerical simulations are performed to illustrate our theoretical results. Noticed is a clear impact of p on the collective behavior: more agents with high confidence lead to harder agreement. It is also experimentally revealed that the sharpness of the threshold d_c increases with α but does not depend on p.

Key words: social dynamics, bounded confidence, phase transition, Monte Carlo simulation

CLC Number:

60K35

Yilun SHANG. CONSENSUS FORMATION OF TWO-LEVEL OPINION DYNAMICS[J].Acta mathematica scientia,Series B, 2014, 34(4): 1029-1040.

Trendmd

References

[1] Castellano C, Fortunato S, Loreto V. Statistical physics of social dynamics. Rev Mod Phys, 2009, 81: 591–646

[2] Lorenz J. Continuous opinion dynamics under bounded confidence: a survey. Int J Mod Phys C, 2007, 18: 1819–1838

[3] Galam S. Sociophysics: a review of Galam models. Int J Mod Phys C, 2008, 19: 409–440

[4] Clifford P, Sudbury A. A model for spatial conflict. Biometrika, 1973, 60: 581–588

[5] Galam S. Minority opinion spreading in random geometry. Eur Phys J B, 2002, 25: 403–406

[6] Sznajd-Weron K, Sznajd J. Opinion evolution in closed community. Int J Mod Phys C, 2000, 11: 1157–1165

[7] Galam S. Local dynamics vs. social mechanisms: a unifying frame. Europhys Lett, 2005, 70: 705–711

[8] Deffuant G, Neau D, Amblard F, Weisbuch G. Mixing beliefs among interacting agents. Adv Complex Syst, 2000, 3: 87–98

[9] Hegselmann R, Krause U. Opinion dynamics and bounded confidence: models, analysis and simulation. J Artif Soc Soc Simulat, 2002, 5: article no 2

[10] Ben-Naim E, Krapivsky P, Redner S. Bifurcations and patterns in compromise processes. Physica D, 2003, 183: 190–204

[11] Lorenz J. Consensus strikes back in the Hegselmann-Krause model of continuous opinion dynamics under
bounded confidence. J Artif Soc Soc Simulat, 2006, 9: article no 8

[12] Lorenz J. Heterogeneous bounds of confidence: meet, discuss and find consensus! Complexity, 2010, 4: 43–52

[13] Kou G, Zhao Y, Peng Y, Shi Y. Multi-level opinion dynamics under bounded confidence. PLoS ONE, 2012, 7: e43507

[14] Shang Y. A note on the 2-connectivity in one-dimensional ad hoc networks. Sci China: Inf Sci, 2011, 54: 123–128

[15] Deffuant G, Amblard F, Weisbuch G, Faure T. How can extremism prevail? a study based on the relative agreement interaction model. J Artif Soc Soc Simulat, 2002, 5: article no 1

[16] Weisbuch G, Deffuant G, Amblard F. Persuasion dynamics. Physica A, 2005, 353: 555–575

[17] Kurmyshev E, Ju´arez H, Gonz´alez-Silva R. Dynamics of bounded confidence opinion in heterogeneous social
networks: concord againts partial antagonism. Physica A, 2011, 390: 2945–2955

[18] Godehardt E, Jaworski J. Two models of random intersection graphs for classification//Schwaiger M, Opitz O, eds. Exploratory Data Analysis in Empirical Research. Berlin: Springer, 2002: 68–81

[19] Shang Y. Joint probability generating function for degrees of active/passive random intersection graphs. Front Math China, 2012, 7: 117–124

[20] Shang Y. L1 group consensus of multi-agent systems with switching topologies and stochastic inputs. Phys Lett A, 2013, 377: 1582–1586

[21] Shang Y. Consensus recovery from intentional attacks in directed nonlinear multi-agent systems. Int J Nonlinear Sci Numer Simul, 2013, 14: 355–361

[22] Helbing D, Balietti S. Agent-based modeling//Helbing D, ed. Social Self-Organization: Agent-Based Simulations
and Experiments to Study Emergent Social Behavior. Berlin: Springer, 2012: 25–70

[23] H¨aggstr¨om O. A pairwise averaging procedure with application to consensus formation in the Deffuant model. Acta Appl Math, 2012, 119: 185–201

[24] Lanchier N. The critical value of the Deffuant model equals one half. ALEA Lat Am J Probab Math Stat, 2012, 9: 383–402

[25] Shang Y. Deffuant model with general opinion distributions: first impression and critical confidence bound.
Complexity, 2013, 19: 38–49

[26] Fortunato S. Universality of the threshold for complete consensus for the opinion dynamics of Deffuant et al. Int J Mod Phys C, 2004, 15: 1301–1307

[27] Lorenz J, Urbig D. About the power to enforce and prevent consensus by manipulating communication rules. Adv Complex Syst, 2007, 10: 251–269

[28] Weisbuch G, Deffuant G, Amblard F, Nadal J -P. Meet, discuss and segregate! Complexity, 2002, 7: 55–63

[29] Durrett R. Probability: Theory and Examples. Belmont: Duxbury Press, 1996

[30] Golub B, Jackson M O. How homophily affects the speed of learning and best-response dynamics. Quarterly
Journal of Economics, 2012, 127: 1287–1338

[31] Currarini S, Jackson M O, Pin P. An economic model of friendship: homophily, minorities and segregation.
Econometrica, 2009, 77: 1003–1045