The extremely widespread adoption of Online Social Networks (OSNs) raises many questions on privacy and access control. Regardless of the particular centralized or de-centralized nature of the OSN, the achievable security and privacy degree strongly depends on the graph-theoretical properties of the social graph representing the real friendship relations between the users. In this paper, we analyze the relationship between the social network graph topology and the achievable privacy. We observe three metrics, namely degree distribution, clustering coefficient and mixing time, and show that they give fundamental insights on the privacy degree of the OSN. We propose how to exploit these insight for the design of future privacy-friendly OSN.