Microblog provides convenience to the society, but at the same time, it also brings some adverse effects. To obtain the propagation mechanism of microblog rumor, and then take effective measures to prevent its spread, according to the complex network theory, in this paper we investigate the internal characteristics of microblog user relationship networks, and present a microblog user relationship network evolution model. By using the mean-field theory, the topological statistical property of our evolution model, and the dynamical behaviors of rumor spreading on such a model are analyzed. Theoretical analysis and simulation results show that such an evolving network exhibits a scale-free property. The degree distribution exponent not only is related to the reverse connection probability, but also depends on the node attraction degree distribution. It is also found that when the node attraction degree follows a power-law distribution, the steady-state rumor prevalence is great compared with the exponential distribution and uniform distribution. Moreover, as the reverse connection probability or the number of node initial edges increases, the probability of rumor outbreak and the number of nodes finally infected by the rumor will also increase.