Border Gateway Protocol (BGP) is the de facto inter-domain routing protocol. With the rapid development of the Internet, the convergence problem of BGP attracts more attention. This paper analyzes the relationship between BGP convergence and the characteristics of the Internet. The Internet is classified into three hierarchies based on the power-law and commercial relations of autonomous systems. The relation of network topology and BGP convergence performance is presented for all sorts of convergence events in different layers. The result shows that the power-law nature of network influences the BGP convergence greatly. So we present a new proposal to improve BGP convergence based on power-law, called “best up”, which behaves better than normal convergence mode in the experiments.