Fitness voter model: Damped oscillations and anomalous consensus

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 ≥ 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 .