The popularity of online social media platforms provides an unprecedented opportunity to study real-world complex networks of interactions. However, releasing this data to researchers and the public comes at the cost of potentially exposing private and sensitive user information. It has been shown that a naive anonymization of a network by removing the identity of the nodes is not sufficient to preserve users' privacy. In order to deal with malicious attacks, k-anonymity solutions have been proposed to partially obfuscate topological information that can be used to infer nodes' identity. In this paper, we study the problem of ensuring k-anonymity in time-varying graphs, i.e., graphs with a structure that changes over time, and multi-layer graphs, i.e., graphs with multiple types of links. More specifically, we examine the case in which the attacker has access to the degree of the nodes. The goal is to generate a new graph where, given the degree of a node in each (temporal) layer of the graph, such a node remains indistinguishable from other k-1 nodes in the graph. In order to achieve this, we find the optimal partitioning of the graph nodes such that the cost of anonymizing the degree information within each group is minimum. We show that this reduces to a special case of a Generalized Assignment Problem, and we propose a simple yet effective algorithm to solve it. Finally, we introduce an iterated linear programming approach to enforce the realizability of the anonymized degree sequences. The efficacy of the method is assessed through an extensive set of experiments on synthetic and real-world graphs.
The concept of @math -anonymity in the graph domain @cite_29 @cite_19 was introduced by , but it is only with Liu and Terzi @cite_4 that a first algorithm to construct a @math -anonymous graph is proposed. As their algorithm is designed to work on static graphs, however, if applied on the temporal slices of a time-varying graph it fails to take into account the additional information contained in the temporal dimension, i.e., the size of the anonymity groups in the temporal graph will be lower than that of the individual slices. Moreover, their technique generally requires repeated anonymizations of the graph under increasing levels of structural noise, something that is not computationally feasible when dealing with large time-varying graphs. A number of successive works proposed heuristics to reduce the total running time, thus making it feasible to anonymize large static social networks @cite_32 @cite_31 @cite_27 .
@cite_4 @cite_29 @cite_32 @cite_19 @cite_27 @cite_31
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In recent years, concerns of privacy have become more prominent for social networks. Anonymizing a graph meaningfully is a challenging problem, as the original graph properties must be preserved as well as possible. We introduce a generalization of the degree anonymization problem posed by Liu and Terzi. In this problem, our goal is to anonymize a given subset of nodes while adding the fewest possible number of edges. The main contribution of this paper is an efficient algorithm for this problem by exploring its connection with the degree-constrained subgraph problem. Our experimental results show that our algorithm performs very well on many instances of social network data. Motivated by a strongly growing interest in anonymizing social network data, we investigate the NP-hard Degree Anonymization problem: given an undirected graph, the task is to add a minimum number of edges such that the graph becomes k-anonymous. That is, for each vertex there have to be at least k-1 other vertices of exactly the same degree. The model of degree anonymization has been introduced by Liu and Terzi [ACM SIGMOD'08], who also proposed and evaluated a two-phase heuristic. We present an enhancement of this heuristic, including new algorithms for each phase which significantly improve on the previously known theoretical and practical running times. Moreover, our algorithms are optimized for large-scale social networks and provide upper and lower bounds for the optimal solution. Notably, on about 26 of the real-world data we provide (provably) optimal solutions; whereas in the other cases our upper bounds significantly improve on known heuristic solutions. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math . The Degree Anonymity problem arises in the context of combinatorial graph anonymization. It asks, given a graph (G ) and two integers (k ) and (s ), whether (G ) can be made k-anonymous with at most (s ) modifications. Here, a graph is k-anonymous if the graph contains for every vertex at least (k-1 ) other vertices of the same degree. Complementing recent investigations on its computational complexity, we show that this problem is very hard when studied from the viewpoints of approximation as well as parameterized approximation. In particular, for the optimization variant where one wants to minimize the number of either edge or vertex deletions there is no factor- (n^ 1- ) approximation running in polynomial time unless P = NP, for any constant (0 < 1 ). For the variant where one wants to maximize (k ) and the number (s ) of either edge or vertex deletions is given, there is no factor- (n^ 1 2 - ) approximation running in time (f(s) n^ O(1) ) unless W[1] = FPT, for any constant (0 < 1 2 ) and any function (f ). On the positive side, we classify the general decision version as fixed-parameter tractable with respect to the combined parameter solution size (s ) and maximum degree.