- Published 2008

In this paper we introduce modified version of one-dimensional outflow dynamics (known as a Sznajd model) which simplifies the analytical treatment. We show that simulations results of the original and modified rules are exactly the same for various initial conditions. We obtain the analytical formula for exit probability using Kirkwood approximation and we show that it agrees perfectly with computer simulations in case of random initial conditions. Moreover, we compare our results with earlier analytical calculations obtained from renormalization group and from general sequential probabilistic frame introduced by Galam. Using computer simulations we investigate the time evolution of several correlation functions to show if Kirkwood approximation can be justified. Surprisingly, it occurs that Kirkwood approximation gives correct results even for these initial conditions for which it cannot be easily justified. Introduction. – The outflow dynamics was introduced to describe the opinion change in the society. The idea was based on the fundamental social phenomenon called ”social validation”. By now, the opinion dynamics was studied by many authors, starting perhaps from the works by Galam [1] and developed later in the Sznajd [2] and Majority rule [3] models. The common feature of these models is that the complexity of real-world opinions is reduced to the minimum set of two options, + or −. However, in this paper we do not focus on social applications of the model (an interested reader may resort to reviews [4–8]). Here we deal with a more mathematical problem, namely finding the analytical formula for the probability P+(p) of reaching consensus on opinion + as a function of the initial fraction p of opinion +. This quantity is commonly called exit probability [9,10]. In fact, we follow the method used in [9] for the Majority-rule model. The one dimensional outflow dynamics is defined as follows: if pair of neighboring spins SiSi+1 = 1 the the two neighbors of the pair followed its direction, i.e. Si−1 → Si(= Si+1) and Si+2 → Si(= Si+1); in case of different opinions at the central pair, the two neighboring states are unchanged. Until now several analytical approaches have been proposed. One of the analytical approaches used for the outflow dynamics was based on the mean field idea [11]. Within the mean field approach the mean relaxation time 〈τ〉 as a function of the initial fraction p of opinion + was computed, as well as distribution of relaxation times. The exit probability found in this approach is the trivial step function, at odds with the known simulation results for 1D dynamics. Later, Galam in [12] presented a general sequential probabilistic frame (GSPF), which extended a series of earlier opinion dynamics models. Within his frame he was able to find analytic formulas for the probability p(t+1) to find at random an agent sharing opinion + at time t+1 as a function of p(t). Among several models, he considered the one dimensional rule, which we investigate in this paper, i.e.: if pair of neighboring spins SiSi+1 = 1 then the two neighbors of the pair followed its direction; in case of different opinions at the central pair, the two neighboring states are unchanged. For such a rule, within his GSPF calculation Galam has found the following formula [12]: p(t+ 1) = p(t) + 7 2 p(t)[1− p(t)] + 3p(t)[1− p(t)] + 1 2 p(t)[1 − p(t)] (1) Iterating above formula the exit probability P+ can be found as a step function (see Fig. 1). It is worth to notice that step-like function describes exit probability in the case of two dimensional outflow dynamics [13], but not in

@inproceedings{Slanina200812D,
title={1 2 D ec 2 00 7 epl draft Some new results on one - dimensional outflow dynamics},
author={Frantisek Slanina and K . Sznajd - Weron and Phuong La},
year={2008}
}