Partial differential equation modeling with Dirichlet boundary conditions on social networks

Abstract

The being a wide range of applications of the Internet, social networks have become an effective and convenient platform for information communication, propagation and diffusion. Most of information exchange and spreading exist in social networks. The issue of information diffusion in social networks is getting more and more attention by government and individuals. The researchers investigated either empirical studies or focused on ordinary differential equation (ODE) models with only consideration of temporal dimension in most prior work. As is well known, partial differential equations (PDEs) can describe temporal and spatial patterns of information diffusion over online social networks; however, until now, results for understanding information propagation of social networks over both temporal and spatial dimensions are few. This paper is devoted to investigating a non-autonomous diffusive logistic model with Dirichlet boundary conditions to describe the process of information propagation in social networks. By constructing upper and lower solutions we obtain the dynamic behavior of the solution to the non-autonomous diffusive logistic model. Our results show that information diffusion is greatly affected by the diffusion coefficient d(t) and the intrinsic growth rate r(t).