SIR-Hawkes: Linking Epidemic Models and Hawkes Processes to Model Diffusions in Finite Populations.

Among the statistical tools for online information diffusion modeling, bothepidemic models and Hawkes point processes are popular choices. The formeroriginate from epidemiology, and consider information as a viral contagionwhich spreads into a population of online users. The latter have roots ingeophysics and finance, view individual actions as discrete events incontinuous time, and modulate the rate of events according to the self-excitingnature of event sequences. Here, we establish a novel connection between thesetwo frameworks. Namely, the rate of events in an extended Hawkes model isidentical to the rate of new infections in the Susceptible-Infected-Recovered(SIR) model after marginalizing out recovery events -- which are unobserved ina Hawkes process. This result paves the way to apply tools developed for SIR toHawkes, and vice versa. It also leads to HawkesN, a generalization of theHawkes model which accounts for a finite population size. Finally, we derivethe distribution of cascade sizes for HawkesN, inspired by methods instochastic SIR. Such distributions provide nuanced explanations to the generalunpredictability of popularity: the distribution for diffusion cascade sizestends to have two modes, one corresponding to large cascade sizes and anotherone around zero.