k-Metric Antidimension: a Privacy Measure for Social Graphs

The study and analysis of social graphs impacts on a wide range of applications, such as community decision making support and recommender systems. With the boom of online social networks, such analyses are benefiting from a massive collection and publication of social graphs at large scale. Unfortunately, individuals’ privacy right might be inadvertently violated when publishing this type of data. In this article, we introduce (k, `)-anonymity; a novel privacy measure aimed at evaluating the resistance of social graphs to active attacks. (k, `)-anonymity is based on a new problem in Graph Theory, the k-metric antidimension defined as follows. Let G = (V,E) be a simple connected graph and S = {w1, · · · , wt} ⊆ V an ordered subset of vertices. The metric representation of a vertex u ∈ V with respect to S is the t-vector r(u|S) = (dG(u,w1), · · · , dG(u,wt)), where dG(u, v) represents the length of a shortest u − v path in G. We call S a k-antiresolving set if k is the largest positive integer such that for every vertex v ∈ V − S there exist other k − 1 different vertices v1, · · · , vk−1 ∈ V − S with r(v|S) = r(v1|S) = · · · = r(vk−1|S). The k-metric antidimension of G is the minimum cardinality among all the k-antiresolving sets for G. We address the k-metric antidimension problem by proposing a true-biased algorithm with success rate above 80% when considering random graphs of size at most 100. The proposed algorithm is used to determine the privacy guarantees offered by two real-life social graphs with respect to (k, `)-anonymity. We also investigate theoretical properties of the k-metric antidimension of graphs. In particular, we focus on paths, cycles, complete bipartite graphs and trees.