k -Metric antidimension: A privacy measure for social graphs

Abstract

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 = {w 1 , · · · , w t } ⊆ 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) w t )), where d G (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 v 1 , · · · , v k−1 ∈ V − S with r(v|S) = r(v 1 |S) = · · · = r(v k−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.

Counteracting Active Attacks in Social Network Graphs

“…We use the privacy measure (k, )-anonymity as proposed in [10] and provide an efficient method to transform a graph G into another graph G such that G is not (1, 1)-anonymous. That is to say, the obtained graph G satisfies (k, )-anonymity with k > 1 or > 1.…”

“…Indeed, none of the privacy notions described above [4,14,3,15] is well-suited to counteract active attacks. To the best of our knowledge, the first privacy measure to evaluate the resistance of social graphs to active attacks was proposed just recently in [10]. Trujillo-Rasua and Yero model the adversary's background knowledge as the distance vector of a vertex with respect to the adversary's subgraph.…”

“…Trujillo-Rasua and Yero model the adversary's background knowledge as the distance vector of a vertex with respect to the adversary's subgraph. This leads to the privacy notion (k, )-anonymity [10].…”

“…The first privacy notion that accounts for such active privacy attacks has been proposed just recently in [10]. This notion, which is called (k, )-anonymity, expresses that a user cannot be re-identified with probability higher than 1/k by an active attacker able to introduce sybil nodes in the graph.…”

“…This notion, which is called (k, )-anonymity, expresses that a user cannot be re-identified with probability higher than 1/k by an active attacker able to introduce sybil nodes in the graph. It has been shown in [10] that reallife social graphs tend to be (1, 1)-anonymous, which is the lowest privacy level possible. Indeed, in terms of offered privacy, (k, )-anonymity forms a lattice (a square grid) where (1, 1)-anonymity is the minimum.…”