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This paper provides the first analysis of the differentially private computation of three centrality measures, namely eigenvector, Laplacian and closeness centralities, on arbitrary weighted graphs, using the smooth sensitivity approach. We do so by finding lower bounds on the amounts of noise that a randomised algorithm needs to add in order to make the output of each measure differentially private. Our results indicate that these computations are either infeasible, in the sense that there are large families of graphs for which smooth sensitivity is unbounded; or impractical, in the sense that even for the cases where smooth sensitivity is bounded, the required amounts of noise result in unacceptably large utility losses.

This paper addresses active re-identification attacks in the context of privacy-preserving social graph publication. Active attacks are those where the adversary can leverage fake accounts, a.k.a. sybil nodes, to enforce structural patterns that can be used to re-identify their victims on anonymised graphs. In this paper we present a new probabilistic interpretation of this type of attacks. Unlike previous privacy properties, which model the protection from active adversaries as the task of making victim nodes indistinguishable in terms of their fingerprints with respect to all potential attackers, our new formulation introduces a more complete view, where the attack is countered by jointly preventing the attacker from retrieving the set of sybil nodes, and from using these sybil nodes for re-identifying the victims. Under the new formulation, we show that the privacy property $k$-symmetry, introduced in the context of passive attacks, provides a sufficient condition for the protection against active re-identification attacks leveraging an arbitrary number of sybil nodes. Moreover, we show that the algorithm K-Match, originally devised for efficiently enforcing the related notion of $k$-automorphism, also guarantees $k$-symmetry. Empirical results on several collections of synthetic graphs corroborate that our approach allows, for the first time, to publish anonymised social graphs (with formal privacy guarantees) that effectively resist the strongest active re-identification attack reported in the literature, even when it leverages a large number of sybil nodes.

Active re-identification attacks pose a serious threat to privacy-preserving social graph publication. Active attackers create fake accounts to build structural patterns in social graphs which can be used to re-identify legitimate users on published anonymised graphs, even without additional background knowledge. So far, this type of attacks has only been studied in the scenario where the inherently dynamic social graph is published once. In this paper, we present the first active re-identification attack in the more realistic scenario where a dynamic social graph is periodically published. The new attack leverages tempo-structural patterns for strengthening the adversary. Through a comprehensive set of experiments on real-life and synthetic dynamic social graphs, we show that our new attack substantially outperforms the most effective static active attack in the literature by increasing the success probability of re-identification by more than two times and efficiency by almost 10 times. Moreover, unlike the static attack, our new attack is able to remain at the same level of effectiveness and efficiency as the publication process advances. We conduct a study on the factors that may thwart our new attack, which can help design graph anonymising methods with a better balance between privacy and utility.

We present a novel method for publishing differentially private synthetic attributed graphs. Unlike preceding approaches, our method is able to preserve the community structure of the original graph without sacrificing the ability to capture global structural properties. Our proposal relies on C-AGM, a new community-preserving generative model for attributed graphs. We equip C-AGM with efficient methods for attributed graph sampling and parameter estimation. For the latter, we introduce differentially private computation methods, which allow us to release community-preserving 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.

Given a graph $G$ and a subset of vertices $S = \{w_1, \ldots, w_t\} \subseteq V(G)$, the multiset representation of a vertex $u\in V(G)$ with respect to $S$ is the multiset $m(u|S) = \{| d_G(u, w_1), \ldots, d_G(u, w_t) |\}$. A subset of vertices $S$ such that $m(u|S) = m(v|S) \iff u = v$ for every $u, v \in V(G) \setminus S$ is said to be a multiset resolving set, and the cardinality of the smallest such set is the outer multiset dimension. We study the general behaviour of the outer multiset dimension, and determine its exact value for several graph families. We also show that computing the outer multiset dimension of arbitrary graphs is NP-hard, and provide methods for efficiently handling particular cases.

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.

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

The validation of biclustering algorithms remains a challenging task, even though a number of measures have been proposed for evaluating the quality of these algorithms. Although no criterion is universally accepted as the overall best, a number of meta-evaluation conditions to be satisfied by biclustering algorithms have been enunciated. In this work, we present MOCICE-BCubed F$_1$, a new external measure for evaluating biclusterings, in the scenario where gold standard annotations are available for both the object clusters and the associated feature subspaces. Our proposal relies on the so-called micro-objects transformation and satisfies the most comprehensive set of meta-evaluation conditions so far enunciated for biclusterings. Additionally, the proposed measure adequately handles the occurrence of overlapping in both the object and feature spaces. Moreover, when used for evaluating traditional clusterings, which are viewed as a particular case of biclustering, the proposed measure also satisfies the most comprehensive set of meta-evaluation conditions so far enunciated for this task.

Let ${\cal G}$ be a graph family defined on a common vertex set $V$ and let $d$ be a distance defined on every graph $G\in {\cal G}$. A set $S\subset V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d(s,u)\ne d(s,v)$. The simultaneous metric dimension of ${\cal G}$ is the smallest integer $k$ such that there is a simultaneous metric generator for ${\cal G}$ of cardinality $k$. We study the simultaneous metric dimension of families composed by corona product graphs. Specifically, we focus on the case of two particular distances defined on every $G\in {\cal G}$, namely, the geodesic distance $d_G$ and the distance $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$ defined as $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$., Comment: arXiv admin note: substantial text overlap with arXiv:1504.00492

Let ${\mathcal G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for ${\cal G}$ if for every $G\in {\cal G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_{G}(s,u)\ne d_{G}(s,v)$, where $d_{G}$ denotes the geodesic distance. A simultaneous adjacency generator for ${\cal G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for ${\cal G}$ is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of ${\cal G}$. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.