It is a mainstream idea that scale-free network would be fragile under the selective attacks. Internet is a typical scale-free network in the real world, but it never collapses under the selective attacks of computer viruses and hackers. This phenomenon is different from the deduction of the idea above because this idea assumes the same cost to delete an arbitrary node. Hence this paper discusses the behaviors of the scale-free network under the selective node attack with different cost. Through the experiments on five complex networks, we show that the scale-free network is possibly robust under the selective node attacks; furthermore, the more compact the network is, and the larger the average degree is, then the more robust the network is; With the same average degrees, the more compact the network is, the more robust the network is. This result would enrich the theory of the invulnerability of the network, and can be used to build the robust social, technological and biological networks, and also has the potential to find the target of drugs.