  The extent to which committed minorities can overturn social conventions is
an active area of research in the mathematical modelling of opinion dynamics.
Researchers generally use simulations of agent-based models (ABMs) to compute
approximate values for the minimum committed minority size needed to overturn a
social convention. In this manuscript, we expand on previous work by studying
an ABM's mean-field behaviour using ordinary differential equation (ODE) models
and a new tool, opinion response functions. Using these methods allows for
formal analysis of the deterministic model which can provide a theoretical
explanation for observed behaviours, e.g., coexistence or overturning of
opinions. In particular, opinion response functions are a method of
characterizing the equilibria in our social model. Our analysis confirms
earlier numerical results and supplements them with a precise formula for
computing the minimum committed minority size required to overturn a social
convention.
