Analysis of SHIR rumor propagation in random heterogeneous networks with dynamic friendships

We propose the Susceptible-Hesitated-Infected-Removed (SHIR) rumor propagation model in random heterogeneous networks with dynamic friendships. Different from existing rumor spreading models that assume the degrees of the nodes in the network are static, we incorporate the most common phenomenon of varying friendships in online social networks. To highlight the effect of individuals' subjective judgment in the rumor spreading process, we formulate our model with connected transforming paths from all other states to the state of stifler. By means of probability generating functions, the mathematical expression of the proposed model is explicitly derived with several ordinary differential equations (ODEs). We theoretically calculate the rumor spreading threshold. Finally, a series of numerical simulations are conducted to verify the theoretical results and comprehensively illustrate the evolution of the model. The simulation results indicate that the solutions of the ODEs successfully portray the rumor spreading process in which involved individuals have dynamic friendships, and the analysis of the threshold agrees well with the simulation results. With dynamic friendships, spreaders are capable of exerting direct influence on a greater number of individuals who can only be affected through several contacts with intermediate infections, which accelerates the spreading process and results in additional infections. However, the existence of deactivated freed stubs limits the rapid increase in the spreaders and somehow offsets the acceleration effect of dynamic friendships.