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While privacy preservation of data mining approaches has been an important topic for a number of years, privacy of social network data is a relatively new area of interest. Previous research has shown that anonymization alone may not be sufficient for hiding identity information on certain real world data sets. In this paper, we focus on understanding the impact of network topology and node substructure on the level of anonymity present in the network. We present a new measure, topological anonymity, that quantifies the amount of privacy preserved in different topological structures. The measure uses a combination of known social network metrics and attempts to identify when node and edge inference breeches arise in these graphs.