Spreading Dynamics of SHIPR Pyramid Scheme Model on Scale-Free Networks

Nowadays, pyramid schemes have caused extremely negative effects on people’s lives and seriously damaged the social economy. With the rapid development of network and communication technology, people’s direct or indirect social interaction is more frequent, which makes the phenomenon of pyramid schemes more serious. Therefore, it is necessary to study transmission mechanisms and transmission rules of pyramid schemes. In order to study the influence of government management and social interaction topology on the spreading of pyramid schemes, a novel SHIPR (susceptible-hesitator-involved-punished-resister) pyramid scheme spreading model is proposed on scale-free networks. The spreading dynamics of pyramid schemes are analyzed in detail by mean-field theory. Then, the basic reproduction number $R_{0}$ and equilibria are got. Theoretical analysis shows that the basic reproduction number $R_{0}$ has a great correlation with government crackdown intensity for involved individuals, the coverage rate of government anti-pyramid scheme publicity for susceptible individuals and hesitator, and social interaction topology. Furthermore, the local asymptotic stability of fraud-elimination equilibrium is analyzed based on the Routh-Hurwitz criterion, the global asymptotic stability of the fraud-elimination equilibrium is discussed by the Lyapunov function, the global attractivity of fraud-prevailing equilibrium is proved in detail by comparison principle. Finally, numerical simulations verify the theoretical analysis results.