Abstract: With about two-thirds of the global population using social media, online interactions often influence offline events such as protests and violence. This talk presents two mathematical frameworks to model these online-to-offline spillovers. The first framework uses an epidemic-type model on networks, exploring mean field approximations to the stochastic processes and deriving reproductive numbers for these models. We also examine how network structure impacts the accuracy of these approximations. The second framework applies a reaction-diffusion model on networks to analyze the spreading speeds of traveling wave solutions. We identify parameter regimes for approximating these speeds on k-ary trees and characterize scenarios involving pushed, pulled, and pinned waves. These models provide insights into how information spreads across networks and triggers offline behaviors.