Stability analysis of a fractional online social network model

Abstract

By drawing an analogy to the spreading dynamics of an infectious disease, the authors derive a fractional-order susceptible–infected–removed (SIR) model to examine the user adoption and abandonment of online social networks, where adoption is analogous to infection, and abandonment is analogous to recovery. They modify the traditional SIR model with demography, so that both infectious and noninfectious abandonment dynamics are incorporated into the model. More precisely, they consider two types of abandonment: (i) infectious abandonment resulting from interactions between an abandoned and an adopted member, and (ii) noninfectious abandonment which is not influenced by an abandoned member. In addition, they study the existence and uniqueness of nonnegative solutions of the model, as well as the existence and stability of its equilibria. They establish a nonnegative threshold quantity R0α for the model and show that if R0α<1, the user-free equilibrium E0 is locally asymptotically stable. In addition, they find a region of attraction for E0. If R0α>1, they prove that the model has a unique user-prevailing equilibrium E∗ that is globally asymptotically stable. Their stability results also show that the infectious abandonment dynamics do not contribute to the stability of the user-free and user-prevailing equilibria, and that it only affects the location of the user-prevailing equilibrium. The Jacobian matrix technique and the Lyapunov function method are used to show the stability of the equilibria. They perform numerical simulations to verify these theoretical results. Finally, they conduct a case study of fitting their model to some historical Instagram user data to show the effectiveness of the model.