Getting scale-free network from a small world network without growth

A method that can be used to get scale-free network from a small-world network without growth under the mechanism of preferential attachment is proposed. Unlike the normal BA growth network model, in our model we remove an old node with a probability scaling with the degree of the node before adding a new node into the network, that make the size of the network fixed, but the nodes and edges are not fixed. If an old node has less degree, it has a larger probability to be removed, and its edges are deleted at the same time. It is found that the degree distribution based on our model obeys a form like power-law of BA model, but the scope of degree distribution in our model is much smaller than BA model. Therefore, the degree distribution's heave tail in our model is thinner than that in the normal BA model; thus it is different from the normal BA model. Meanwhile, there are some other properties in our model, for instance, the average clustering coefficient decreases with the renewed ratio and the power-law exponent increases with the renewed ratio to a limited value, which is equal to that in the normal BA model.