De-anonymizability of social network: through the lens of symmetry

Social network de-anonymization, which refers to re-identifying users by mapping their anonymized network to a correlated network, is an important problem that has received intensive study in network science. However, it remains less understood how network structural features intrinsically affect whether or not the network can be successfully de-anonymized. To find the answer, this paper offers the first general study on the relation between de-anonymizability and network symmetry. To this end, we propose to capture the symmetry of a graph by the concept of graph bijective homomorphism. By defining the matching probability matrix, we are able to characterize the de-anonymizability, i.e., the expected number of correctly matched nodes. Specifically, we show that for a graph pair with arbitrary topology, the de-anonymizability is equal to the maximal diagonal sum of the matching probability matrix generated from homomorphisms. Due to the prohibitive cost of enumerating all possible homomorphisms, we further obtain an upper bound of such de-anonymizability by counting the orbits of each of the two graphs, which significantly reduces the computational cost. Such a general result allows us to theoretically obtain the de-anonymizability of any networks with more specific topology structure. For example, for any classic Erdős-Rènyi graph with designated n and p, we can represent its de-anonymizability numerically by calculating the local symmetric structure that it contains. Extensive experiments are performed to validated our findings.