Bridging the gap between agent based models and continuous opinion dynamics

There is a rich literature on microscopic models for opinion dynamics; most of them fall into one of two categories - agent-based models or differential equation models - with a general understanding that the two are connected in certain scaling limits. In this paper we show rigorously this is indeed the case. In particular we show that both ordinary and stochastic differential equations can be obtained as a limit of agent-based models by simultaneously rescaling time and the extent to which an agent updates their opinion after an interaction. This approach provides a pathway to analyse much more diverse modelling paradigms, for example: the motivation behind several possible multiplicative noise terms in stochastic differential equation models; the connection between selection noise and the mollification of the discontinuous bounded confidence interaction function; and how the method for selecting interacting pairs can determine the normalisation in the corresponding differential equation. Our computational experiments confirm our findings, showing excellent agreement of solutions to the two classes of models in a variety of settings.