Analytical and numerical treatment of continuous ageing in the voter model

TL;DR

This paper extends the voter model by incorporating continuous age-dependent switching rates, providing analytical and computational methods to study its dynamics, including consensus, frozen states, and phase transitions.

Contribution

It introduces a continuous age-dependent voter model with non-Markovian dynamics and develops analytical and simulation techniques for its analysis.

Findings

Age-dependent switching rates can be approximated by fractional differential equations.

The model exhibits exponential approach to consensus in some cases.

Spontaneous opinion changes induce a phase transition between coexistence and consensus.

Abstract

The conventional voter model is modified so that an agent's switching rate depends on the `age' of the agent, that is, the time since the agent last switched opinion. In contrast to previous work, age is continuous in the present model. We show how the resulting individual-based system with non-Markovian dynamics and concentration-dependent rates can be handled both computationally and analytically. Lewis' thinning algorithm can be modified in order to provide an efficient simulation method. Analytically, we demonstrate how the asymptotic approach to an absorbing state (consensus) can be deduced. We discuss three special cases of the age dependent switching rate: one in which the concentration of voters can be approximated by a fractional differential equation, another for which the approach to consensus is exponential in time, and a third case in which the system reaches a frozen state instead of consensus. Finally, we include the effects of spontaneous change of opinion, i.e., we study a noisy voter model with continuous ageing. We demonstrate that this can give rise to a continuous transition between coexistence and consensus phases. We also show how the stationary probability distribution can be approximated, despite the fact that the system cannot be described by a conventional master equation.