This paper focuses on the Internet IP-level routing topology and proposes relevant explanations to its apparent dynamics. We first represent this topology as a power-law random graph. Then, we incorporate to the graph two well known factors responsible for the observed dynamics, which are load balancing and route evolution. Finally, we simulate on the graph traceroute-like measurements. Repeating the process many times, we obtain several graph instances that we use to model the dynamics. Our results show that we are able to capture on power-law graphs the dynamic behaviors observed on the Internet. We find that the results on power-law graphs, while qualitatively similar to the one of Erdös-Rényi random graphs, highly differ quantitatively; for instance, the rate of discovery of new nodes in power-law graphs is extremely low compared to the rate in Erdös-Rényi graphs.