Some new results on one-dimensional outflow dynamics.
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.
Social and economic systems Fluctuation phenomena, random processes, noise, and Brownian motion Probability theory, stochastic processes, and statistics
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  and developed later in the Sznajd  and Majority rule  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, 5, 6, 7, 8]). Here we deal with a more mathematical problem, namely finding the analytical formula for the probability of reaching consensus on opinion as a function of the initial fraction of opinion . This quantity is commonly called exit probability [9, 10]. In fact, we follow the method used in  for the Majority-rule model.
The one dimensional outflow dynamics is defined as follows: if pair of neighboring spins the the two neighbors of the pair followed its direction, i.e. and ; 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 . Within the mean field approach the mean relaxation time as a function of the initial fraction 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  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 to find at random an agent sharing opinion at time as a function of . Among several models, he considered the one dimensional rule, which we investigate in this paper, i.e.: if pair of neighboring spins 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 :
Iterating above formula the exit probability 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 , but not in one dimension . Step-like function for exit probability has been found also in .
In the paper  real space renormalization approach has been proposed to calculate the probability of reaching consensus on opinion as a function of the initial fraction p of opinion . They have found in case of two sites convincing others the following analytical formulas:
in the case of growing network (growing hierarchical or the Barabasi-Albert scale-free network)
in the case of fixed network they have found as the step function observed also in computer simulations on the square lattice.
It is seen on Fig. 1 that RG results for growing networks agree much better with simulations that RG results for fixed network which are exactly the same as obtained using GSPF by Galam.
In this paper we present analytical results obtained using Kirkwood approximation  following the method used in for the Majority-rule model and we obtain perfect agreement with computer simulations. Moreover, we consider two types of non-random initial conditions and we show how analytical formulas for exit probabilities change for such cases.
2 Approximate solution in 1D
We consider linear chain with sites. Each site can be in two
We use the following notation:
state of the system.
state of the spin at site if the system is in state .
state which differs from by flipping spin at site . Therefore .
We introduce here slight modifications with respect to original outflow rule: choose pair of neighbors and if they both are in the same state, then adjust one (instead of two) of its neighbors (chosen randomly on left or right with equal probability ) to the common state. Because this way at most one spin is flipped in one step while in original formulation two can be flipped simultaneously, the time must be rescaled by factor . We measure the time so that the speed of all processes remains constant when , so normally one update takes time . Here, instead, we consider also the factor , so single update takes time . Our modification eliminates some correlations due to simultaneous flip of spins at distance . However, if we look at later stages of the evolution , where typically the domains are larger than , simultaneous flips occurs very rarely. Therefore, we do not expect any substantial difference. Indeed, computer simulations confirm our expectations - only time has to be rescaled (see Fig. 2).
On the other hand, the modification simplifies the analytical treatment. Indeed, the update rule can be equivalently formulated as follows:
Choose randomly a spin and side ( for right, for left.) The updated state is if , otherwise .
Within such a formulation the probability that in one update the flip occurs:
These flip probabilities are then inserted into master equation:
Now, we make the limit which also implies the continuous time limit, as . We also note that most of the transition probabilities are zero, since only one spin flip is allowed in one step. Finally we end with
where the transition rates are trivially related to transition probabilities (3), . (The sum is now over infinite set of sites.) For completeness we repeat the formula for transition rates:
It is hopeless to solve the master equation as it is. Instead, we write evolution equations for some correlation functions derived from it. We define:
Only two equations are relevant for us. The first is:
and the second:
These two become closed set of equations, if we apply the approximations described in the next section. Before going to it, it is perhaps instructive to show the intermediate results which lead to equations (8), (9), and analogically to others, for more complicated correlation functions.
So, for example, for the lowest correlation function - the average of one spin - we have
and for the next one in the level of complexity
The pattern is transparent. When computing the correlation function of spins at sites , , , …, on the RHS we have sum of terms, in which we average the product of spins at sites , , , …with transition rate (which is constructed from the spin configuration according to (6)) for flip at positions , , , …. As a formula, this sentence means
2.1 Kirkwood approximation
The first one is the usual Kirkwood approximation, or decoupling, which is used in various contexts and accordingly it assumes different names. For example in the classical quantum many-body theory of electrons and phonons in solids, it is nothing else than the Hatree-Fock approximation (but contrary to this theory, which may be improved systematically using diagrammatic techniques, here the systematic expansions are not developed). We use the name Kirkwood approximation, following the work .
In our case, the Kirkwood approximation amounts to
in Eq. (9) and
in Eq. (8). While the latter assumption (14) enables us to relate the equation (8) directly to (9) and therefore to solve it as soon as we have the solution of (9), the approximation (13) does not yet make of (9) a closed equation. The point is that there is also the correlation at distance , the function . So, we make also an additional approximation, which is also made in . We suppose that only weakly depends on distance , or else, that the decay of the correlations is relatively slow. If the spins are correlated to certain extent on distance (the neighbors) , they are correlated to essentially the same extent also on distance (next-next neighbors). This is also justified if the domains are large enough, i. e. at later stages of the evolution. So, we assume
In the figure 3 we present a sample (not averaged) time evolution of several correlation functions. Indeed, our assumptions can be justified at later stages of the evolution, although the assumption (13) agrees perfectly with simulation results from very beginning. The second given by (14) agrees with simulations also quite well. Only the assumption (15) that the decay of the correlations is relatively slow is valid only at later stages of the evolution.
where and satisfy the equations (dot denotes the time-derivative
The solution is straightforward. We assume initial conditions and . First we solve the second equation from the set (17). This gives
and inserting that into the first of the set (17) we have
The most important result is the asymptotics
How to interpret this finding? The average is the average magnetization. In other terms, it determines the probability that a randomly chosen spin will have state at time . This probability is . Therefore, is the initial magnetization. When we go to the limit , we know that ultimately the homogeneous state is reached. The asymptotic magnetization therefore says what is the probability that the final state will have all spins . It is . So, (20) means that
If the initial state is completely uncorrelated, i. e. we set the spins at random, with the only condition that average magnetization is , we have and
Finally, we express this result in terms of the probability to have a randomly chosen spin spin in state at the beginning and the probability that all spins are in state at the end. We have
Computer simulations for random initial conditions, in which assumption can be done shows perfect agreement with analytical formula (23). In the next section we show how results will change in case of correlated initial conditions.
3 Correlated initial conditions
Here we consider two examples of correlated initial conditions with fraction of up-spins:
Ordered initial state that consists of two clusters: -length of up-spins and -length of down-spins, for example in case of :
Correlated, completely homogeneous, initial state, i.e. for , for example in case of :
In both cases it is easy to calculate exactly correlation function . In the first case of ordered initial conditions we obtain:
Thus, from equation (21):
Computer simulations shows that indeed for such a initial conditions (see Fig. 4).
As we see Kirkwood approximation surprisingly gives correct results also in this case. However, if we look at figure 5 we see that Kirkwood approximation given equations by (13) and (14) cannot be justified by computer simulations.
We have checked also the mean relaxation time in case of ordered initial conditions (Fig. 6). It occurs that analogously like for random initial conditions the mean relaxation time scales with the system size as (see Figs. 2 and 6). The same scaling has been found in the voter model [16, 17, 18]. However, contrary to the random initial conditions for which bell-shaped curve is observed, here the mean relaxation times is well described by simple parabola:
For the second correlated initial conditions, which are completely homogeneous we observed in computer simulations that exit probability is step like, i.e.
In this case two-spins correlation function can be also calculated easily. For we obtain:
Thus, from equation (21):
which again agrees perfectly with computer simulations, although Kirkwood approximation cannot be easily justified.
Very interesting results is obtained for homogeneous initial condition if we measure the mean relaxation time (see Fig. 7). Computer simulation shows that for (and , respectively) the mean relaxation time does not depend on the system size and for depends linearly on system size, i.e. .
We introduced modified version of one-dimensional outflow dynamics in which we choose pair of neighbours and if they both are in the same state, then adjust one (in original version both) of its neighbours (chosen randomly on left or right with equal probability ) to the common state. We checked in computer simulations that accordingly to our expectations results in the case of modified rule are the same as in the case of original outflow dynamics, only the time must be rescaled by factor . Modified version simplified the analytical treatment and allowed to derive the master equation. Following the method proposed in  we wrote evolution equations for some correlation functions and used the Kirkwood approximation. This approach allowed us to derive the analytical formula for final magnetization (21). In fact, just before finishing this paper the same result was published by Lambiotte and Redner as a special case in the work  where a model interpolating the voter, Majority-rule (or Sznajd) and so-called vacillating voter dynamics was investigated, using also the Kirkwood approximation.
In the case of random initial conditions Kirkwood approximation can be justified looking at time evolution of simulated correlation functions. In this case our analytical results can be simplified to eq. (23) and agrees perfectly with simulations on contrary to earlier approaches [12, 15]. We have checked also how the Kirkwood approximation works in the case of two types of correlated initial conditions. Although in both cases the Kirkwood approximation cannot be easily justified, surprisingly we obtained perfect agreement with computer simulations.
Acknowledgements.This work was supported by the MŠMT of the Czech Republic, grant no. 1P04OCP10.001, and by the Research Program CTS MSM 0021620845. Katarzyna Sznajd-Weron gratefully acknowledges the financial support in period 2007-2009 of the Polish Ministery of Science and Higher Education through the scientific grant no. N N202 0194 33
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