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 $k \ge 0$, 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 winning agent then increments its $k$--value by one. We study the dynamics of the system in the entire $0 \le p \le 1$ range and compare with the case $p=1/2$, which corresponds to the standard voter model. When $0 \le p < 1/2$, the system approaches exponentially fast to the consensus state of the initial majority opinion. The mean consensus time $\tau$ grows logarithmically with the number of agents $N$, and it is greatly decreased relative to the linear behaviour $\tau \sim N$ found in the voter model. When $1/2 < p \le 1$, the system initially relaxes to a state with an even coexistence of opinions, but eventually reaches consensus by finite-size fluctuations. The approach to coexistence is monotonic for $1/2 < p < 0.8$, while for $0.8 \le p \le 1$ 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 $b$ increases with $p$. Also, $\tau$ increases respect to the 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 $\tau \sim N^\beta$, with $\beta \simeq 1.45$.