Ordering dynamics in the voter model with aging

Abstract

We study theoretically and numerically the voter model with memory-dependent dynamics at the mean-field level. The “internal age”, related to the time an individual spends holding the same state, is added to the set of binary states of the population, such that the activation probability pi for attempting a change of state has an explicit dependence on the internal age i. We derive a closed set of integro-differential equations describing the time evolution of the fraction of individuals with a given state and internal age, and from it we obtain analytical results characterizing the behavior of the system close to the absorbing states. In general, different internal age-dependent activation probabilities pi have different effects on the dynamics. In the case of “aging”, i.e. pi being a decreasing function of its argument i, either the system reaches consensus or it gets trapped in a frozen state, depending on whether the limiting value p∞ is zero or positive, and on the rate at which pi approaches p∞. Moreover, when the system reaches consensus, the fraction x(t) of nodes not holding the consensus state may decay exponentially with time or as a power-law, depending again on the specific details of the functional form of pi. For the case of “anti-aging”, when the activation probability pi is an increasing function of i, the system always reaches a steady state with coexistence of opinions. Exact conditions for having one or another behavior, together with the equations and explicit expressions for the power-law exponents, are provided.