Voter Model with Time dependent Flip-rates

TL;DR

This paper extends the Voter Model by incorporating time-dependent flip-rates, analyzing how changing interaction rates influence consensus times and opinion survival, with implications for language change and other dynamic systems.

Contribution

It introduces a novel generalization of the Voter Model with time-varying flip-rates and provides accurate estimates for consensus times and probabilities across different scenarios.

Findings

Consensus time varies nontrivially with flip-rate change rate.

Heterogeneity reduces mean consensus time and affects opinion survival.

Scaling predictions match complex network simulations.

Abstract

We introduce time variation in the flip-rates of the Voter Model. This type of generalisation is relevant to models of ageing in language change, allowing the representation of changes in speakers' learning rates over their lifetime and may be applied to any other similar model in which interaction rates at the microscopic level change with time. The mean time taken to reach consensus varies in a nontrivial way with the rate of change of the flip-rates, varying between bounds given by the mean consensus times for static homogeneous (the original Voter Model) and static heterogeneous flip-rates. By considering the mean time between interactions for each agent, we derive excellent estimates of the mean consensus times and exit probabilities for any time scale of flip-rate variation. The scaling of consensus times with population size on complex networks is correctly predicted, and is as would be expected for the ordinary voter model. Heterogeneity in the initial distribution of opinions has a strong effect, considerably reducing the mean time to consensus, while increasing the probability of survival of the opinion which initially occupies the most slowly changing agents. The mean times to reach consensus for different states are very different. An opinion originally held by the fastest changing agents has a smaller chance to succeed, and takes much longer to do so than an evenly distributed opinion.