Title:

Opinion dynamics : from local interactions to global trends

We study the dynamics of consensus formation in a finite twostate voter model in which each agent has a confidence in their current opinion, k. The evolution of the distribution of kvalues and the opinion change rules are coupled together to allow the opinion dynamics to dynamically develop heterogeneity of agent states in a simple way. We introduce two models. In both models pairs of agents with different opinions interact, which means kvalues are compared and, with probability p, the agent with the lower value adopts the spin of the one with the higher value. In our nonconserved confidence model the agent with the higher kvalue increments their confidence by one. In our conserved confidence model, additionally, the agent changing opinion reduces their kvalue by one, so total confidence is conserved. The only parameter in both models is the probability p. We study the nonconserved model on the complete graph and compare the consensus time with the case p = 1/2 in which the opinion dynamics are decoupled from the kvalues and are equivalent to standard voter model dynamics. When 1/2 < p < 1, agents with higher k{values are more persuasive and the consensus time is increased relative to the standard voter model although it still scales linearly with the number of agents, N. When p = 1, the consensus time scales as Nα with α = 1.4. When 0 < p < 1/2, agents with higher kvalues are less persuasive and the consensus time is greatly decreased relative to the standard voter model and appears to be logarithmic in N. We provide some partial explanations for these observations using a meanfield model of the dynamics and a lowdimensional heuristic model which tracks only the sizes and mean kvalues of each group. We also study the conserved model on the complete graph. When 1/2 < p < 1 this model also has consensus time that scales linearly with N and when p = 1 it scales as Nα with α = 1.4. However when 0 < p < 1/2 this conserved confidence model does not behave in the same way as the nonconserved model and the consensus time scales linearly with N. We compare the meanfield model dynamics with those of the nonconserved confidence model to partially explain model behaviour differences. We find consensus times for the nonconserved confidence model on low dimension regular lattices and compare with the complete graph. When 1/2 < p < 1, the consensus for the model on a 2d lattice is slower than the fully connected model, but in 3d results suggest that it is comparable. When p = 1, the population is prevented from reaching consensus by stable locally coordinated confidence arrangements. When 0 < p < 1/2, the consensus time is slow compared to the fully connected model and we notice spatial structures in the simulations. Also we implement a modification to the Axelrod model to introduce heterogeneity in the opinion space. We implement a bias in which opinions (features) are updated and separately a bias in the weight of influence of opinions on whether interaction occurs. Despite affecting the dynamics of the opinions and the time to absorbing state, we find the state, consensus or coexistence, the model typically reaches is robust under the effect of these two types of heterogeneity.
