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Diffusing opinions in bounded confidence processes

We study the effects of diffusing opinions on the Deffuant et al. model for continuous opinion dynamics. Individuals are given the opportunity to change their opinion, with a given probability, to a randomly selected opinion inside an interval centered around the present opinion. We show that diffusion induces an order-disorder transition. In the disordered state the opinion distribution tends to be uniform, while for the ordered state a set of well defined opinion clusters are formed, although with some opinion spread inside them. If the diffusion jumps are not large, clusters coalesce, so that weak diffusion favors opinion consensus. A master equation for the process described above is presented. We find that the master equation and the Monte-Carlo simulations do not always agree due to finite-size induced fluctuations. Using a linear stability analysis we can derive approximate conditions for the transition between opinion clusters and the disordered state. The linear stability analysis is compared with Monte Carlo simulations. Novel interesting phenomena are analyzed.


Introduction
In a community, opinions evolve due to affinities and conflicts between mutually interacting individuals. These interactions lead to collective states, where either a major-ity of individuals adopt a similar opinion (consensus) or a number of opinion groups (clusters) arise. In many situations, the dynamics of this complex collective behavior goes beyond specific individual attributes and seems to be well characterized by quantities like statistical distri-arXiv:1004.1102v2 [physics.soc-ph] 7 Sep 2010 butions and averages. This explains why, in the last years, the understanding of opinion formation using tools and techniques borrowed from nonlinear and statistical physics has become a topic of interest for physicists. In particular, several models have been developed to reproduce the basic elements that drive the processes of opinion evolution [1]. These models can be classified in two broad groups: on the one hand, discrete, Ising-type, models where opinions can only adopt a finite set of integer values [2,3]; on the other, continuous opinion models where opinions can vary continuously in a finite interval [4,5,6,7,8,9,10].
In the context of continuous opinion dynamics a model introduced by Deffuant and collaborators has received much attention recently [6]. In this model individuals meet in random pairwise encounters and then converge to a common opinion only if their respective opinions prior to the encounter differ less than some given amount, in a kind of bounded confidence mechanism [11,12]. After some transient evolution, this leads to final states in which either full consensus is reached or the population splits in a finite number of clusters such that all individuals in one cluster share the same opinion. However, we believe that such dynamics misses the fact that the behavior of an individual in a society does not depend only on the influence of people with similar opinions. There are many additional factors, both personal and external, which also cause opinion changes. These factors may introduce some degree of discrepancy within otherwise well defined opinion clusters and should be modeled somehow. In a recent paper [13] we analyzed the effects induced by a modification of the model in which we allow individuals to change their opinion, with a given probability, to a randomly chosen value in the whole opinion space. One possible interpretation of this modification (which, from the technical point of view, can be understood as some sort of noise acting upon the dynamics) is that individuals keep at all times a basal opinion to which they return from time to time, no matter what the environment tells them to do, modelling in a crude way the intrinsic free-will of human decisions. We have shown [13] that this modification of the dynamical rules can induce new and interesting phenomena, such as noise-induced bistability, and that the model displays interesting and non-trivial finite-size effects that need to be considered in some detail.
In this paper, we continue our analysis of the uncertainty factors present in human decisions by considering a modification of the model in which the opinion of an individual can jump to a new value chosen randomly inside an interval of finite width centered around the current opinion. In other words, in addition to the bounded confidence mechanism, the opinions execute a random-walk motion or diffusion in opinion space. When the typical size of the jumps is sufficiently large we expect this model to behave as the previous free-will model discussed in [13], but when it is small a distinct behavior will be obtained. is introduced to reflect that individuals interact, discuss, and modify their opinions: at time-step n two individuals, say i and j, are randomly chosen; if their opinions satisfy |x i n − x j n | < , they converge to the common value: for the initial opinions, the typical realization is that for ≥ 0.5 the system evolves to a state of consensus where all individuals share the same opinion and that, decreasing , the population splits into opinion clusters separated by distances larger than [5,14].
The new ingredient we add to the dynamics is that individuals can perform random jumps in their opinions.
More specifically the dynamical rules are modified as fol- Nevertheless, for mathematical simplicity periodic boundary conditions will be considered in some particular cases as properly mentioned.

Monte Carlo simulations
We present in this subsection the main results obtained from Monte Carlo simulations for a system with a finite population of N individuals and adsorbing boundary conditions. As in previous studies, we assume that the ini- ξ i (t) are independent random jumps of zero mean and variance σ 2 = γ 2 /3. Reordering, and taking squares and mean value one obtains where we have used that the expected value of N 1 is mN .
This implies that the cluster's center of mass experiences a random walk with an effective diffusion coefficient D cm equal to the single-individual's opinion one divided by the number of individuals in the cluster D cm = D/N . Fig. 1 illustrates cluster random walks for three different population sizes. For isolated clusters (i.e. separated by distances larger than ) the arguments above are only valid when boundary effects are unimportant, which means that clusters remain sufficiently far from the interval extremes (at distances larger than γ) so that jumping particles can not reach the boundaries. For larger γ or particles will feel the boundaries more easily and we expect cluster mobility to be much reduced by boundary effects.
From Monte Carlo simulations, it is also seen that cluster formation always occurs for small values of , γ, and m.
But, in contrast with the diffusionless model, coarsening is observed for small γ. Figure 2 shows successive merging of clusters, occurring after the collisions which arise because of the diffusive wandering of clusters (by collision we mean that clusters become closer than , so that they interact).
We expect that in this regime, at very long times, a state containing a single cluster would be the final regime.
The behavior is different at large values of γ: in addition to the expected reduced wandering of the clusters, which tend to stay at relatively well defined positions, we see that there is no tendency to reducing the number of clusters (See Figs. 3a and c), so that a pattern of opinions is established. The pattern of clusters is approximately periodic when 1. This is similar to the behavior of the model in [13]. In fact, for γ of the order of system size the jumps are of global size so that there should be no difference between both models. As in [13], a remarkable fact occurs in which one can find regions of bistability: inside these regions, the inherent fluctuations arising from the finite number of particles take the system from one state to another and back. The sort of transitions between steady states is observed for instance, in Monte Carlo simulations at = 0.316. Figure 3b shows multiple jumps between a state of two big opinion cluster and another state of a big cluster with two smaller ones near the edges of the opinion interval.

Master equation approach
In this subsection we analyze the master equation description of the process introduced above. Following standard arguments (see for example [13]) one finds the master equation for the probability density function P (x, t) of an The term proportional to (1 − m) is the one coming from the original rules of the Deffuant et al. model [5,14], whereas the one proportional to m describes the random jumps. The function G(x, t) is, for the case of adsorbing boundary conditions: For small values of γ the boundary effects become less important, and the second case in (5) applies in the majority of cases. In addition, for γ small enough, this term can be approximated as so that the diffusion term, i.e. the term proportional to m in the right-hand side of Eq. (4) becomes of the form , where one finds again the diffusion coefficient We have solved numerically the master equation (4)  , and m. We have found that for small values of γ only one opinion cluster, centered around x = 0.5, remains for almost any value of the parameter (see Fig. 4a). We interpret this as a consequence of the dynamics discussed in section 2.1, where we showed that although several clusters were initially formed, collisions reduced their number and a single one was expected to survive at long times.
We stress that the single cluster in Fig. 4a  For large values of γ the situation is rather different: as in the model with free-will [13], for small several opinion clusters form, which do not coalesce. Asymptotic states as a function of are displayed in Fig. 4b. Different cluster bifurcations, as in the original Deffuant model [5] and in the case of free-will [13] are seen.
An important feature is that, for all values of γ considered, the clusters become less defined below a critical value of (for large γ one can always observe however the two large clusters at x = 0, 1 arising from the adsorbing boundary conditions). This point will be further addressed in section 3.
where the over-bar denotes a temporal average in steady  In Fig. 6(a)  or less homogeneously populated. Fig. 6(b) shows a similar behavior in the case γ = 0.4, but in this case the number of individuals whose opinion is adsorbed by the extremes is larger. Therefore, G M saturates farther away from its maximum possible value 1. We will define the transition from order to disorder as the location m c of the absolute maximum value of G M . In the next subsection we will explain this transition via a simple linear stability analysis that turns out to be very accurate, in particular, for small values of γ.

A linear stability analysis
Although the transition to cluster formation is a nonlinear process, one can still derive approximate analytical conditions for the existence of cluster formation as a function To analyze the stability of the homogeneous solution P (x) = 1 we write P (x, t) = 1 + A q e iqx+λqt , where q is the wave number of the perturbation, λ q its growth rate and A q the amplitude. Introducing this ansatz in Eq. (8) we find the dispersion relation We note that, for small γ the second term, propor-  ical expressions can be obtained expanding λ q in powers of q: In the case in which the q 4 term remains negative (which occurs if the mγ 4 term remains smaller than the one containing ), the change of sign of the q 2 term iden- as the value of below which an unstructured configuration remains stable. Alternatively, for fixed we find m c = 2 3 2 3 +γ 2 , the critical value above which clusters will disappear. Within this approximation the expression for the fastest growing mode near the orderdisorder transition is: We stress that all these expressions following Eq. (10) are only valid for the case in which the appearance of positive values of λ q when varying a parameter occurs first at values of q close to zero, corresponding to a longwavelength instability. As seen in Fig. 7 this is not always the case for large values of γ, for which the transitions have to be obtained numerically. In fact, we note that the change in behavior observed in the simulations from a tendency to coarsening to the establishment of a robust periodic pattern, occurring for increasing γ, seems to be correlated with the change in the character of the linear instability, from long-wavelength to finite wavelength, this last one being the one occurring in the free-will model in [13]. A more detailed discussion of the relationship between the linear dynamics and the nonlinear long-time states is however beyond the scope of this paper. Comparison of the predicted order-disorder transition with Monte Carlo simulations is performed in the bottom panels of