Fitness voter model: damped oscillations and anomalous consensus

TL;DR

This paper introduces a heterogeneous voter model with fitness parameters, revealing how opinion dynamics and fitness evolution lead to damped oscillations, anomalous consensus times, and different relaxation behaviors depending on the probability p.

Contribution

It presents a novel opinion formation model incorporating fitness, analyzing its dynamics across all p values, and discovering new phenomena like damped oscillations and anomalous scaling of consensus times.

Findings

For p<0.5, the system rapidly reaches consensus with logarithmic consensus time growth.

For p>0.5, the system exhibits damped oscillations around coexistence before consensus.

At p=1, the relaxation to coexistence follows a power law, and consensus time scales superlinearly with N.

Abstract

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 $\beta \simeq 1.45$.