Analysis of Infectious-Recovery Epidemic Models for Membership Dynamics of Online Social Networks

TL;DR

This paper analyzes infectious-recovery epidemic models for online social networks, mathematically characterizing their behavior and extending the irSIR model to better match real social media dynamics.

Contribution

It introduces an extension of the irSIR model that captures both exponential growth and sustained long-term engagement in social networks.

Findings

Original irSIR model predicts extinction of social epidemics.

Extended model allows for a stable proportion of infections.

Mathematically characterizes initial and long-term behaviors.

Abstract

The recent rapid growth of social media and online social networks (OSNs) has raised interesting questions about the spread of ideas and fads within our society. In the past year, several papers have drawn analogies between the rise and fall in popularity of OSNs and mathematical models used to study infectious disease. One such model, the irSIR model, made use of the idea of "infectious recovery" to outperform the traditional SIR model in replicating the rise and fall of MySpace and to predict a rapid drop in the popularity of Facebook. Here we explore the irSIR model and two of its logical extensions and we mathematically characterize the initial and long-run behavior of these dynamical systems. In particular, while the original irSIR model always predicts extinction of a social epidemic, we construct an extension of the model that matches the exponential growth phase of the irSIR model while allowing for the possibility of an arbitrary proportion of infections in the long run.