The Role of Noise and Initial Conditions in the Asymptotic Solution of a Bounded Confidence, Continuous-Opinion Model

Abstract

We study a model for continuous-opinion dynamics under bounded confidence. In particular, we analyze the importance of the initial distribution of opinions in determining the asymptotic configuration. Thus, we sketch the structure of attractors of the dynamical system, by means of the numerical computation of the time evolution of the agents density. We show that, for a given bound of confidence, a consensus can be encouraged or prevented by certain initial conditions. Furthermore, a noisy perturbation is added to the system with the purpose of modeling the free will of the agents. As a consequence, the importance of the initial condition is partially replaced by that of the statistical distribution of the noise. Nevertheless, we still find evidence of the influence of the initial state upon the final configuration for a short range of the bound of confidence parameter.

Appendix: Lyapunov Function for the DW et al. Model

In order to write a Lyapunov function for the DW et al. process, we need to find a function , where x(t) is a vector whose components are the agents opinions at time t, which satisfies

(A.1)

(A.2)

where (A.1) simply means that it is a positive-definite function and (A.2) that it cannot increase with time.

We will first write a positive-definite function and then prove that it is decreasing in time for the model interaction rules. Let us write a function which only depends on the distances between the agents opinions as

(A.3)

where we only sum over all terms which are different. Let us then focus on one interaction, i.e., on the changes occurred to the positions of only two agents in the opinion space, say agents i ₁ and j ₁. Thus, we may divide the Lyapunov function into two terms, one dependent and the other independent of i ₁ and j ₁, and let us call A to the independent part for simplicity:

(A.4)

Each sum contains N−2 terms, being N the number of agents. Now we use the new opinions $x_{i_{1}}(t+1)$ and $x_{j_{1}}(t+1)$ that agents i ₁ and j ₁ hold after the interaction. It is important to notice that this interaction does only take place in case the opinions of the agents are nearer than the bound of confidence ε. However, this does not affect our analysis, as we are only interested in effective interactions, those which actually take place. The Lyapunov function after the interaction is

(A.5)

Replacing the new values $x_{i_{1}}(t+1)$ and $x_{j_{1}}(t+1)$ as given by application of the rule Eq. (1) and subtracting, we get the variation of the Lyapunov as which, after some algebra, reads:

(A.6)

Therefore, we see that the Lyapunov function is strictly decreasing in time when any interaction takes place, and it stays constant when no interaction occurs.