In order to prevent the disclosure of privacy-sensitive data, such as names and relations between users, social network graphs have to be anonymised before publication. Naive anonymisation of social network graphs often consists in deleting all identifying information of the users, while maintaining the original graph structure. Various types of attacks on naively anonymised graphs have been developed. Active attacks form a special type of such privacy attacks, in which the adversary enrols a number of fake users, often called sybils, to the social network, allowing the adversary to create unique structural patterns later used to re-identify the sybil nodes and other users after anonymisation. Several studies have shown that adding a small amount of noise to the published graph already suffices to mitigate such active attacks. Consequently, active attacks have been dubbed a negligible threat to privacy-preserving social graph publication. In this paper, we argue that these studies unveil shortcomings of specific attacks, rather than inherent problems of active attacks as a general strategy. In order to support this claim, we develop the notion of a robust active attack, which is an active attack that is resilient to small perturbations of the social network graph. We formulate the design of robust active attacks as an optimisation problem and we give definitions of robustness for different stages of the active attack strategy. Moreover, we introduce various heuristics to achieve these notions of robustness and experimentally show that the new robust attacks are considerably more resilient than the original ones, while remaining at the same level of feasibility.

A vertex v ∈ V is said to resolve two vertices x and y if dG(v, x) = dG (v, y). A set S ⊂ V is said to be a metric generator for G if any pair of vertices of G is resolved by some element of S. A minimum metric generator is called a metric basis, and its cardinality, dim(G), the metric dimension of G. A set S ⊆ V is said to be a simultaneous metric generator for a graph family G = {G1, G2, . . . , G k }, defined on a common (labeled) vertex set, if it is a metric generator for every graph of the family. A minimum cardinality simultaneous metric generator is called a simultaneous metric basis, and its cardinality the simultaneous metric dimension of G. We obtain sharp bounds for this invariants for general families of graphs and calculate closed formulae or tight bounds for the simultaneous metric dimension of several specific graph families. For a given graph G we describe a process for obtaining a lower bound on the maximum number of graphs in a family containing G that has simultaneous metric dimension equal to dim(G). It is shown that the problem of finding the simultaneous metric dimension of families of trees is N P -hard. Sharp upper bounds for the simultaneous metric dimension of trees are established. The problem of finding this invariant for families of trees that can be obtained from an initial tree by a sequence of successive edge-exchanges is considered. For such families of trees sharp upper and lower bounds for the simultaneous metric dimension are established.

Social network data is typically made available in a graph format, where users and their relations are represented by vertices and edges, respectively. In doing so, social graphs need to be anonymised to resist various privacy attacks. Among these, the so-called active attacks, where an adversary has the ability to enrol sybil accounts in the social network, have proven difficult to counteract. In this article, we provide an anonymisation technique that successfully thwarts active attacks while causing low structural perturbation. We achieve this goal by introducing (k, G,)-adjacency anonymity: a privacy property based on (k,)-anonymity that alleviates the computational burden suffered by anonymisation algorithms based on (k,)-anonymity and relaxes some of its assumptions on the adversary capabilities. We show that the proposed method is efficient and establish tight bounds on the number of modifications that it performs on the original graph. Experimental results on real-life and randomly generated graphs show that when compared to methods based on (k,)-anonymity, the new method continues to provide protection from equally capable active attackers while introducing a much smaller number of changes in the graph structure.

A generator of a metric space is a set S of points in the space with the property that every point of the space is uniquely determined by its distances from the elements of S. Given a simple graph G = (V, E), we define the distance function dG,2 : V × V → N ∪ {0}, as dG,2(x, y) = min{dG(x, y), 2}, where dG(x, y) is the length of a shortest path between x and y and N is the set of positive integers. Then (V, dG,2) is a metric space. We say that a set S ⊆ V is a k-adjacency generator for G if for every two vertices x, y ∈ V , there exist at least k vertices w1, w2, ..., w k ∈ S such that dG,2(x, wi) = dG,2(y, wi), for every i ∈ {1, ..., k}.A minimum cardinality k-adjacency generator is called a k-adjacency basis of G and its cardinality, the k-adjacency dimension of G.In this article we study the problem of finding the k-adjacency dimension of a graph. We give some necessary and sufficient conditions for the existence of a k-adjacency basis of an arbitrary graph G and we obtain general results on the k-adjacency dimension, including general bounds and closed formulae for some families of graphs. In particular, we obtain closed formulae for the k-adjacency dimension of join graphs G + H in terms of the k-adjacency dimension of G and H. These results concern the k-metric dimension, as join graphs have diameter two. As we can expect, the obtained results will become important tools for the study of the k-metric dimension of lexicographic product graphs and corona product graphs. Moreover, several results obtained in this paper need not be restricted to the metric dG,2, they can be expressed in a more general setting, for instance, by using the metric dG,t(x, y) = min{dG(x, y), t} for t ∈ N.

We present a novel method for publishing differentially private synthetic attributed graphs. Our method allows, for the first time, to publish synthetic graphs simultaneously preserving structural properties, user attributes and the community structure of the original graph. Our proposal relies on CAGM, a new community-preserving generative model for attributed graphs. We equip CAGM with efficient methods for attributed graph sampling and parameter estimation. For the latter, we introduce differentially private computation methods, which allow us to release communitypreserving synthetic attributed social graphs with a strong formal privacy guarantee. Through comprehensive experiments, we show that our new model outperforms its most relevant counterparts in synthesising differentially private attributed social graphs that preserve the community structure of the original graph, as well as degree sequences and clustering coefficients.