Social applications mine user social graphs to improve performance in search, provide recommendations, allow resource sharing and increase data privacy. When such applications are implemented on a peer-to-peer (P2P) architecture, the social graph is distributed on the P2P system: the traversal of the social graph translates into a socially-informed routing in the peer-to-peer layer. In this work we introduce the model of a projection graph that is the result of decentralizing a social graph onto a peer-to-peer network. We focus on three social network metrics: degree, node betweenness and edge betweenness centrality and analytically formulate the relation between metrics in the social graph and in the projection graph. Through experimental evaluation on real networks, we demonstrate that when mapping user communities of sizes up to 50-150 users on each peer, the association between the properties of the social graph and the projection graph is high, and thus the properties of the (dynamic) projection graph can be inferred from the properties of the (slower changing) social graph. Furthermore, we demonstrate with two application scenarios on large-scale social networks the usability of the projection graph in designing social search applications and unstructured P2P overlays.