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This paper treats the privacy-preserving publication of social graphs in thepresence of active adversaries, that is, adversaries with the ability tointroduce sybil nodes in the graph prior to publication and leverage them tocreate unique fingerprints for a set of victim nodes and re-identify them afterpublication. Stemming from the notion of $(k,\ell)$-anonymity, we introduce$(k,\ell)$-anonymous transformations, characterising graph perturbation methodsthat ensure protection from active adversaries levaraging up to $\ell$ sybilnodes. Additionally, we introduce a new privacy property: $(k,\ell)$-adjacencyanonymity, which relaxes the assumption made by $(k,\ell)$-anonymity thatadversaries can control all distances between sybil nodes and the rest of thenodes in the graph. The new privacy property is in turn the basis for a newtype of graph perturbation: $(k,\ell)$-adjacency anonymous transformations. Wepropose algorithms for obtaining $(k,1)$-adjacency anonymous transformationsfor arbitrary values of $k$, as well as $(2,\ell)$-adjacency anonymoustransformations for small values of $\ell$.

Rethinking $(k,\ell)$-anonymity in social graphs: $(k,\ell)$-adjacency anonymity and $(k,\ell)$-(adjacency) anonymous transformations