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We introduce a model for a preferentially attached network which has grownfrom a small world network. Here, the average path length and the clusteringcoefficient are estimated, and the topological properties of modeled networksare compared as the initial conditions are changed. As a result, it is shownthat the topological properties of the initial network remain even after thenetwork growth. However, the vulnerability of each to preferentially attachednodes being added is not the same. It is found that the average path lengthrapidly decreases as the ratio of preferentially attached nodes increases andthat the characteristics of the initial network can be easily disappeared. Onthe other hand, the clustering coefficient of the initial network slowlydecreases with the ratio of preferentially attached nodes and its clusteringcharacteristic remains much longer.

Characteristics of Preferentially Attached Network Grown from Small World