Fitness voter model: Damped oscillations and anomalous consensus.

We study the dynamics of opinion formation in a heterogeneous voter model on a complete graph, in which each agent is endowed with an integer fitness parameter k≥0, in addition to its + or - opinion state. The evolution of the distribution of k-values and the opinion dynamics are coupled together, so as to allow the system to dynamically develop heterogeneity and memory in a simple way. When two agents with different opinions interact, their k-values are compared, and with probability p the agent with the lower value adopts the opinion of the one with the higher value, while with probability 1-p the opposite happens. The agent that keeps its opinion (winning agent) increments its k-value by one. We study the dynamics of the system in the entire 0≤p≤1 range and compare with the case p=1/2, in which opinions are decoupled from the k-values and the dynamics is equivalent to that of the standard voter model. When 0≤p<1/2, agents with higher k-values are less persuasive, and the system approaches exponentially fast to the consensus state of the initial majority opinion. The mean consensus time τ appears to grow logarithmically with the number of agents N, and it is greatly decreased relative to the linear behavior τ∼N found in the standard voter model. When 1/2<p≤1, agents with higher k-values are more persuasive, and the system initially relaxes to a state with an even coexistence of opinions, but eventually reaches consensus by finite-size fluctuations. The approach to the coexistence state is monotonic for 1/2<p<p_{o}≃0.8, while for p_{o}≤p≤1 there are damped oscillations around the coexistence value. The final approach to coexistence is approximately a power law t^{-b(p)} in both regimes, where the exponent b increases with p. Also, τ increases respect to the standard voter model, although it still scales linearly with N. The p=1 case is special, with a relaxation to coexistence that scales as t^{-2.73} and a consensus time that scales as τ∼N^{β}, with β≃1.45.