Modelling the spillover from online engagement to offline protest: stochastic dynamics and mean-field approximations on networks

Abstract Social media is transforming various aspects of offline life, from everyday decisions such as dining choices to the progression of conflicts. In this study, we propose a coupled modelling framework with an online social network layer to analyse how engagement on a specific topic spills over into offline protest activities. We develop a stochastic model and derive several mean-field models of varying complexity. These models allow us to estimate the reproductive number and anticipate when surges in activity are likely to occur. A key factor is the transmission rate between the online and offline domains; for offline outbursts to emerge, this rate must fall within a critical range, neither too low nor too high. Additionally, using synthetic networks, we examine how network structure influences the accuracy of these approximations. Our findings indicate that simpler models can effectively represent higher-density networks, whereas low-density networks require more complex approximations to achieve a comparable level of accuracy. When tested on two real-world networks, however, increased complexity in approximation does not result in improved accuracy, indicating that more complex structures are exhibited in empirical networks.