Abstract
For a graph-based representation of a social network, the identity of participants can be uniquely determined if an adversary has background structural knowledge about the graph. We focus on degree-based attacks, wherein the adversary knows the degrees of particular target vertices and we aim to protect the anonymity of participants through k-anonymization, which ensures that every participant is equivalent to at least k − 1 other participants with respect to degree. We introduce a natural and novel approach of introducing “dummy” participants into the network and linking them to each other and to real participants in order to achieve this anonymity. The advantage of our approach lies in the nature of the results that we derive. We show that if participants have labels associated with them, the problem of anonymizing a subset of participants is NP-Complete. On the other hand, in the absence of labels, we give an \(\mathcal{O}(nk)\) algorithm to optimally k-anonymize a subset of participants or to near-optimally k-anonymize all real and all dummy participants. For degree-based-attacks, such theoretical guarantees are novel.
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Notes
We define a vertex-labelled graph as the four-tuple \((\hbox{V,E},\Upsigma,\ell)\), where V is a vertex set, \(\hbox{E}\subseteq \hbox{V}\times\hbox{V}\) is a set of undirected edges, \(\Upsigma\) is a set of sensitive labels, and \( \ell:\hbox{V}\mapsto\Upsigma\) is a labelling function that assigns a label to each vertex. We discuss in the paper two types of labels, sensitive and identifying. By \(\Upsigma\), we refer to the former, assuming the latter is stripped from the graph.
We mention specific cases in which these questions have been answered in our discussion of related work in Sect. 6 Even in these cases, however, not all three questions have been fully addressed.
For simplicity in this section, we regard a graph as a 2-tuple. We note that equivalently, for consistency, we could express an unlabelled graph as \(\mathcal{G}=(\hbox{V, E},\Upsigma,\ell)\) where \(\exists \sigma\in\Upsigma: \forall v\in\hbox{V}, {\ell}(v)=\sigma\). However, the simpler notation simplifies the exposition.
Considering the Enron email corpus on which we experiment in Sect. 4.1, |V| > 65,000, but only 151 vertices correspond to internal email addresses.
Recall that a walk is any sequence of adjacent edges, including those which revisit edges and/or vertices.
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A preliminary, short version (Chester et al. 2011) of this paper appeared at ADBIS 2011.
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Chester, S., Kapron, B.M., Ramesh, G. et al. Why Waldo befriended the dummy? k-Anonymization of social networks with pseudo-nodes. Soc. Netw. Anal. Min. 3, 381–399 (2013). https://doi.org/10.1007/s13278-012-0084-6
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DOI: https://doi.org/10.1007/s13278-012-0084-6