A voter model with time dependent flip rates

Abstract

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 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.

Fast fixation with a generic network structure

“…Here we find that the fixation time can be reduced relative to the case of uniform influence (H ij = const) even on homogeneous networks. This result contrasts with those of [27,28], in which variation in the willingness to change state (our α parameter) causes a slower onset of fixation, a result we also obtained here.…”

“…(14) in this case, we see that it is the smallest values of α i which contribute most to r. In fact we find that r ∼ 1/α /N 2 . This result is similar to that found in [27,28] where different agents in the network could change state with different rates: this is one way to interpret variation of the α parameter in the present work.…”

Neutral Evolution: A Null Model for Language Dynamics

“…One would be correct in levelling Castellano et al 's criticism of being "justified mainly by vague plausibility arguments, with no direct connection to measurable facts" at this model. However, we launch our defence of neutral evolution as a null model for language dynamics with the observation that very many models in which replication takes place at a rate that does not depend on the replicator's structure are also described by the diffusion equation (1).…”

Reduction from non-Markovian to Markovian dynamics: the case of aging in the noisy-voter model

“…As we mentioned before, the equations for the time evolution of n ± i |n are Eqs. (11,12) with x = n/N regarded as a parameter. The stationary average values are then equivalent to Eqs.…”