Opinion response functions are key to understanding the tipping of social conventions

The extent to which committed minorities can overturn social conventions is an active area of research in the mathematical modeling 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 behavior using ordinary differential equation models and a new tool, namely opinion response functions (ORFs). Using this method allows for formal analysis of the deterministic model, which can provide a theoretical explanation for observed behaviors, e.g. coexistence or overturning of opinions. In particular, ORFs are a method for characterizing 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.