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We formulate a linear threshold agent-based model (ABM) for the spread of political revolutions on social networks using empirical network data. We propose new techniques for building a hierarchy of simplified ordinary differential equation (ODE)-based models that aim to capture essential features of the ABM, including effects of the actual networks, and give insight in the parameter regime transitions of the ABM. We relate the ABM and the hierarchy of models to a population-level compartmental ODE model that we proposed previously for the spread of political revolutions, which is shown to be mathematically consistent with the proposed ABM and provides a way to analyse the global behaviour of the ABM. We present two new effective ways to incorporate network structure into the one-compartmental ODE that approximates the dynamical evolution of the expected fraction of the population that participates in the revolution in the linear threshold ABM model. We show numerically that these ODE approximations often perform as well as or better than a higher-order model that has many compartments and is much more expensive computationally. Extending concepts from epidemiological modelling, we define a basic reproduction number   $R_0$   for the linear threshold ABM and apply it to predict ABM behaviour on empirical networks. In small-scale numerical tests, we investigate experimentally the differences in spreading behaviour that occur under the linear threshold ABM model when applied to some empirical (modern) online social networks versus (traditional) offline social networks, searching for quantitative evidence that political revolutions may be facilitated by the modern online social networks of social media.

A hierarchy of linear threshold models for the spread of political revolutions on social networks