A voter model with time dependent flip rates

Abstract

We introduce time variation in the flip rates of the voter model. This type of generalizationmay be applied to other diffusion-like models in which interaction rates at the microscopiclevel may change with time, for example in models of language change, allowing therepresentation of changes in speakers’ learning rates over their lifetime. The mean timetaken to reach consensus varies in a nontrivial way with the rate of change of theflip rates, varying between bounds given by the mean consensus times for statichomogeneous (the original voter model) and static heterogeneous flip rates. Byconsidering the mean time between interactions for each agent, we derive excellentestimates of the mean consensus times and exit probabilities for any timescale of fliprate variation. The scaling of consensus times with population size on complexnetworks is correctly predicted, and is as would be expected for the ordinaryvoter model. Heterogeneity in the initial distribution of opinions has a strongeffect, considerably reducing the mean time to consensus, while increasing theprobability of survival of the opinion which initially occupies the most slowly changingagents. The mean times taken to reach consensus for different states are verydifferent. An opinion originally held by the fastest changing agents has a smallerchance of succeeding, and takes much longer to do so than an evenly distributedopinion.