Constrained opinion dynamics: freezing and slow evolution

We study opinion formation in a population of leftists, centrists and rightist. In an interaction between neighbouring agents, a centrist and a leftist can become both centrists or leftists (and similarly for a centrist and a rightist), while leftists and rightists do not affect each other. The evolution is controlled by the initial density of centrists ρ0. For any spatial dimension the system reaches a centrist consensus with probability ρ0, while with probability 1 − ρ0 the final state is either an extremist consensus, or a frozen population of leftists and rightists. In one dimension, we determine the opinion evolution by mapping the system onto a spin-1 Ising model with zero-temperature Glauber kinetics. The approach to the final state is governed by a t−ψ long-time tail, with ψ → 2ρ0/π as ρ0 → 0. In the one-dimensional frozen state, the length distribution of single-opinion domains has an algebraic small-size tail x−2(1−ψ) and the average domain length is L2ψ , where L is the length of the system. PACS numbers: 64.60.My, 05.40.−a, 05.50.+q, 75.40.Gb One of the basic issues in opinion dynamics is to understand the conditions under which consensus or diversity is reached from an initial population of individuals (agents) with different opinions. Models for such evolution are typically based on each agent freely adopting a new state in response to the opinions in a local neighbourhood [1]. The attribute of incompatibility—in which agents with sufficiently disparate opinions do not interact— has recently been found to prevent ultimate consensus from being reached [2, 3]. Related phenomenology arises in the Axelrod model [4, 5], a simple model that accounts for the formation and evolution of cultural domains. The goal of the present letter is to investigate the role of the incompatibility constraint on opinion formation within a minimal model for which quantitative results can be obtained. We find that the incompatibility restriction has a profound effect on the dynamics and on the nature of the final state. 0305-4470/03/030061+08$30.00 © 2003 IOP Publishing Ltd Printed in the UK L61 L62 Letter to the Editor Figure 1. Typical frozen final state in our opinion dynamics model on a 100 × 100 square lattice for ρ0 = 0.1. The two extreme opinions are represented by black and white squares. We consider a ternary system in which each agent can adopt the opinions of leftist, centrist and rightist. The agents populate a lattice and in a single microscopic event an agent adopts the opinion of a randomly-chosen neighbour, but with the crucial proviso that leftists and rightists are considered to be so incompatible that they do not interact. Note that while a leftist cannot directly become a rightist (and vice versa) in a single step, the evolution leftist ⇒ centrist ⇒ rightist is possible. Our model is similar to the classical voter model [6] and also turns out to be isomorphic to the two-trait two-state Axelrod model [4, 5]. Due to the incompatibility constraint in our model, the final opinion outcome can be either consensus or a frozen mixture of extremists with no centrists. Figure 1 shows a typical frozen state on the square lattice (with periodic boundary conditions). Note the existence of nested enclaves of opposite opinions and the clearly visible clustering. We can exploit the connection between the voter model and our opinion dynamics model to infer the final state of the latter for any spatial dimension. If we temporarily disregard the difference between leftists and rightists, the resulting binary system of centrists and extremists reduces to the voter model for which one of the two absorbing states of all centrists or all extremists is eventually reached. In the context of the ternary opinion system, the latter case can mean either a consensus of extremists or a frozen mixed state of leftists and rightists as depicted in figure 1. Because of the underlying voter model dynamics, the average density of each species is globally conserved in any spatial dimension. That is, 〈ρi(t)〉 = ρi(t = 0), where i refers to one of the states (+, 0,−) and the angle brackets denote an average over all realizations of the dynamics and over all initial states that are compatible with the specified density. Thus with probability P0 = ρ0 the final state consists of all centrists, while with probability 1 − ρ0 the final state consists of no centrists. In the latter case, there can be either a consensus of + (this occurs with probability P+), consensus of − (probability P−), or a frozen mixed state (probability P+−). Figure 2 shows these final-state probabilities as a function of the initial densities. Because of the conservation of the global density in the model, this behaviour is valid for any spatial dimension. In principle, the final-state probabilities could depend on the system size, but this dependence turns out to be extremely weak. To understand the approach to the final state, we now focus on the case of one dimension. Here our opinion dynamics model is equivalent to a constrained spin-1 Ising chain that is endowed with single-spin flip zero-temperature Glauber kinetics [7], with leftist, centrist and Letter to the Editor L63 0.0 0.2 0.4 0.6 0.8 1.0 ρ0 0.0 0.2 0.4 0.6 0.8 1.0