We introduce a model for a preferentially attached network which has grown from a small world network. Here, the average path length and the clustering coefficient are estimated, and the topological properties of modeled networks are compared as the initial conditions are changed. As a result, it is shown that the topological properties of the initial network remain even after the network growth. However, the vulnerability of each to preferentially attached nodes being added is not the same. It is found that the average path length rapidly decreases as the ratio of preferentially attached nodes increases and that the characteristics of the initial network can be easily disappeared. On the other hand, the clustering coefficient of the initial network slowly decreases with the ratio of preferentially attached nodes and its clustering characteristic remains much longer.