TY - JOUR

T1 - Binary and multivariate stochastic models of consensus formation

AU - Miguel, Maxi San

AU - Eguíluz, Victor M.

AU - Toral, Raul

AU - Klemm, Konstantin

N1 - Funding Information:
Figure 7 shows simulation results for a d = 2 square lattice: Figure 7. Simulation results. The normalized order parameter we observe a transition from multicultural to consensus <Smax>/N as a function of the effective noise rate r for states controlled by an effective noise rate r = r(1 – 1/q). different values of q in a d = 2 square lattice of size N = 50 50 The factor (1 – 1/q) takes into account the probability that and F = 2 shows a noise-induced transition.25 the single feature perturbation doesn’t change the trait’s value. This is a noise-induced transition because the control parameter is a noise property. In addition, the transition has n interesting open question for future developments universal scaling properties with respect to q: we find the A is to go beyond the static networks of interaction same result for different values of q and a consensus state for considered here, allowing for a co-evolution of the net-any value of q as r goes to zero. Therefore, cultural drift de-work and agent states. Other computer simulations of so-stroys the transition controlled by q that we find in the ab-cial dynamics have already started to implement this sence of exogenous perturbations (r = 0). In this sense, noise general idea of co-evolution.26 here is an essential parameter that completely changes the type of transition the system exhibits. Acknowledgments An additional important point is the character of the states We acknowledge the collaboration of Krzysztof Suchecki in found at both sides of the noise-induced transition. The dis-the original studies of the voter model dynamics. We also ordered multicultural state found for large r is no longer a acknowledge financial support from the Ministerio de Edu- frozen configuration—rather, it exhibits disordered noise-sus-cación y Ciencia (Spain) through project CONOCE2 tained dynamics. On the other hand, the consensus or ordered (FIS2004-00953). state found for small r is metastable. Once it reaches one of the equivalent qF cultural states, the system doesn’t stay there for- ever, but eventually a fluctuation takes it from this state to an- other one of the equivalent qF states, as Figure 8 shows. Why does the noise rate cause a transition? Here, we have a competition between two time scales: the time scale at which noise acts (1/r) and the relaxation time of perturbations T. For a small noise rate r, there is time to relax, and the system de- cays to a consensus state, but for a large noise rate, stochastic perturbations accumulate and multicultural disorder builds up. We then expect the transition to occur for rT ~ 1. We can calculate the relaxation time T of perturbations as an exit time in a random walk.23,25 A mean-field approximation gives it as the time needed to reach consensus in a finite system follow- ing the voter model dynamics; for a d = 2 square lattice, this is T ~ N ln N.11,25 The noise-induced transition occurs for a sys- tem-size-dependent value of r, but curves such as the ones plotted in Figure 8 for different values of N collapse into a sin- gle curve when plotted versus rN ln N.25 The general result is that in very large systems’ limits, disordered multicultural states prevail at any noise rate. Thus, cultural drift causes global polarization in large systems, but as a state with noise- sustained dynamics rather than a frozen configuration of spa- tially coexisting equivalent cultures.
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2005/11

Y1 - 2005/11

N2 - Binary and multivariate stochastic models of consensus formation are discussed. The voter model, which is the simplest model of collective behavior, is defined by a set of voters who have two opinions or spins at a network's nodes. The elementary dynamical step consists of randomly choosing a node and assigning it the opinion or spin value of one of its nearest neighbors, also chosen at random. This opinion-formation mechanism reflects the agents' complete lack of self-confidence and could be appropriate for describing processes of opinion formation in certain groups in which imitation is prevalent. Axelrod Model, in which order-disorder transition becomes system-size dependent, is also discussed.

AB - Binary and multivariate stochastic models of consensus formation are discussed. The voter model, which is the simplest model of collective behavior, is defined by a set of voters who have two opinions or spins at a network's nodes. The elementary dynamical step consists of randomly choosing a node and assigning it the opinion or spin value of one of its nearest neighbors, also chosen at random. This opinion-formation mechanism reflects the agents' complete lack of self-confidence and could be appropriate for describing processes of opinion formation in certain groups in which imitation is prevalent. Axelrod Model, in which order-disorder transition becomes system-size dependent, is also discussed.

UR - http://www.scopus.com/inward/record.url?scp=28244484657&partnerID=8YFLogxK

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U2 - 10.1109/MCSE.2005.114

DO - 10.1109/MCSE.2005.114

M3 - Review article

AN - SCOPUS:28244484657

VL - 7

SP - 67

EP - 73

JO - Computing in Science and Engineering

JF - Computing in Science and Engineering

SN - 1521-9615

IS - 6

ER -