We study the following fundamental graph problem that models the important task of deanonymizing social networks. We are given a graph representing an eponymous social network and another graph, representing an anonymous social network, which has been produced by the original one after removing some of its nodes and adding some noise on the links. Our objective is to correctly associate as many nodes of the anonymous network as possible to their corresponding node in the eponymous network. We present two algorithms that attack the problem by exploiting only the structure of the two graphs. The first one exploits bipartite matching computations and is relatively fast. The second one is a local search heuristic which can use the outcome of our first algorithm as an initial solution and further improve it. We have applied our algorithms on inputs that have been produced by well-known random models for the generation of social networks as well as on inputs that use real social networks. Our algorithms can tolerate noise at the level of up to 10%. Interestingly, our results provide further evidence to which graph generation models are most suitable for modeling social networks and distinguish them from unrealistic ones.