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

TL;DR

This paper establishes a theoretical link between epidemic models and Hawkes processes, introduces HawkesN for finite populations, and analyzes cascade size distributions to better understand information diffusion.

Contribution

It reveals a novel connection between SIR epidemic models and Hawkes processes, leading to the development of HawkesN and new insights into cascade size variability.

Findings

Event rates in HawkesN match infection rates in SIR models.

HawkesN accounts for finite population effects in diffusion modeling.

Cascade size distributions often show bimodal behavior.

Abstract

Among the statistical tools for online information diffusion modeling, both epidemic models and Hawkes point processes are popular choices. The former originate from epidemiology, and consider information as a viral contagion which spreads into a population of online users. The latter have roots in geophysics and finance, view individual actions as discrete events in continuous time, and modulate the rate of events according to the self-exciting nature of event sequences. Here, we establish a novel connection between these two frameworks. Namely, the rate of events in an extended Hawkes model is identical to the rate of new infections in the Susceptible-Infected-Recovered (SIR) model after marginalizing out recovery events -- which are unobserved in a Hawkes process. This result paves the way to apply tools developed for SIR to Hawkes, and vice versa. It also leads to HawkesN, a generalization of the Hawkes model which accounts for a finite population size. Finally, we derive the distribution of cascade sizes for HawkesN, inspired by methods in stochastic SIR. Such distributions provide nuanced explanations to the general unpredictability of popularity: the distribution for diffusion cascade sizes tends to have two modes, one corresponding to large cascade sizes and another one around zero.