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 k0, 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 01 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<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 appears to grow logarithmically with the number of agents N, and it is greatly decreased relative to the linear behavior found in the standard voter model. When 1/2<p1, 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₎0. 8, while for 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, 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 N^, with 1. 45.