  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.
