  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 DEMs can be obtained from ABMs by
simultaneously rescaling time and the distance 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.
