Abstract We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

In this article we explore the rigidity of realizations of incidence geometries consisting of points and rigid rods: rod configurations. We survey previous results on the rigidity of structures that are related to rod configurations, discuss how to find realizations of incidence geometries as rod configurations, and how this relates to the 2-plane matroid. We also derive further sufficient conditions for the minimal rigidity of k-uniform rod configurations and give an example of an infinite family of minimally rigid 3-uniform rod configurations failing the same conditions. Finally, we construct v3-configurations that are flexible in the plane, and show that there are flexible v3-configurations for all sufficiently large values of v.

A long list of personal tragedies, including teenage suicides, has raised the importance of managing the personal information available on the Internet. It has been argued that it should be allowed to make mistakes, and that there should be a right to be forgotten. Unfortunately, today's Internet architecture and services typically do not support such functionality. We design a system that provides digital oblivion for users of online social networks. Participants form a peer-based agent community, which agree on protecting the privacy of individuals who request images to be forgotten. The system distributes and maintains up-to-date information on oblivion requests, and implements a filtering functionality when accessing an underlying online social network. We describe digital oblivion in terms of authentication of user-to-content relations and identify two user-to-content relations that are particularly relevant for digital oblivion. Finally, we design a family of protocols that provide digital oblivion with respect to these user-to-content relations, within the community that are implementing the protocol. Our protocols leverage a combination of digital signatures, watermarking, image tags, and trust management. No collaboration is required from the social network provider, although the system could also be incorporated as a standard feature of the social network.

In this paper we discuss the relations between clustering and error correcting codes. We show that clustering can be used for constructing error correcting codes. We review the previous works found in the literature about this issue, and propose a modification of a previous work that can be used for code construction from a set of proposed codewords.

In data privacy, the evaluation of the disclosure risk has to take into account the fact that several releases of the same or similar information about a population are common. In this paper we discuss this issue within the scope of k-anonymity. We also show how this issue is related to the publication of privacy protected databases that consist of linked tables. We present algorithms for the implementation of k-anonymity for this type of data.

Re-identification and record linkage are tools used to measure disclosure risk in data privacy. Given two data files, record linkage establishes links between those records that correspond to the same individual. These links are often expressed in terms of probability distributions. This paper presents a review of a formalization of re-identification in terms of compatible belief functions. This formalization makes it possible to define the set of methods for re-identification that are relevant for the estimation of disclosure risk in privacy protection. Any re-identification method that does not fulfill the criteria for being in this set, may be discarded in a theoretical disclosure risk analysis. The focus in this paper is on providing examples of how this formalization can be applied in a few different scenarios in data privacy

Fuzzy measures are monotonic set functions on a reference set; they generalize probabilities replacing the additivity condition by monotonicity. The typical application of these measures is with fuzzy integrals. Fuzzy integrals integrate a function with respect to a fuzzy measure, and they can be used to aggregate information from a set of sources (opinions from experts or criteria in a multicriteria decision-making problem). In this context, background knowledge on the sources is represented by means of the fuzzy measures. For example, interactions between criteria are represented by means of nonadditive measures. In this paper, we introduce fuzzy measures on multisets. We propose a general definition, and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets.

In this paper we discuss some tools for graph perturbation with applications to data privacy. We present and analyse two different approaches. One is based on matrix decomposition and the other on graph partitioning. We discuss these methods and show that they belong to two traditions in data protection: noise addition/microaggregation and k-anonymity.