Ordering dynamics in the voter model with aging

Peralta, A. F.; Khalil, N.; Toral, R.

Physica A: Statistical Mechanics and its Applications** **, 122475 (2019)

Physica A: Statistical Mechanics and its Applications

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 $p_i$ 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 $p_i$ have different effects on the dynamics. In the case of ``aging'', i.e. $p_i$ 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 $p_{\infty}$ is zero or positive, and on the rate at which $p_i$ approaches $p_\infty$. 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 $p_i$. For the case of ``anti-aging'', when the activation probability $p_i$ 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.