Critical phenomena of spreading dynamics on complex networks with diverse activity of nodes

In this paper, we propose a new model to investigate the spreading dynamic and critical phenomena on complex networks based on SIR model. Different from previous studies, we combine the effects of activity rate and infected rate on spreading process. Network nodes become active according to different probability correlated with its degree. Active infected nodes can interact all active susceptible neighbors, meanwhile, recover at a certain probability. By means of the mean-field equations, we find the basic reproductive number and critical threshold of spreading dynamic can be explained by the eigenvalues and eigenvectors of the correlation matrix. Furthermore, we utilize analytical and numerical simulations to explore the critical phenomenon and spreading dynamics of homogeneous and heterogeneous networks respectively. Our results indicate that both homogeneous networks and heterogeneous networks of the model exhibit a critical threshold consists of critical activity rate and infection rate in the spreading dynamic. The critical threshold of infection rate is increased by node activity, and node activity also shows a critical phenomenon given certain infection rate. Results validate that our model is a feasible and economical method to control spreading dynamics and promote further application of innovation diffusion, viral marketing in reality.