An Algebraic Topological Approach to Privacy: Numerical and Categorical Data

TL;DR

This paper introduces an algebraic topological framework for achieving k-anonymity in databases, utilizing simplicial complexes and persistent homology to analyze and optimize data anonymization.

Contribution

It presents a novel application of algebraic topology to data privacy, providing new algorithms and insights for anonymizing both metric and categorical data.

Findings

Topological methods can characterize k-anonymity conditions.

Weighted barcode diagrams help balance data privacy and utility.

Simulations demonstrate the effectiveness of the approach.

Abstract

In this paper, we cast the classic problem of achieving k-anonymity for a given database as a problem in algebraic topology. Using techniques from this field of mathematics, we propose a framework for k-anonymity that brings new insights and algorithms to anonymize a database. We begin by addressing the simpler case when the data lies in a metric space. This case is instrumental to introduce the main ideas and notation. Specifically, by mapping a database to the Euclidean space and by considering the distance between datapoints, we introduce a simplicial representation of the data and show how concepts from algebraic topology, such as the nerve complex and persistent homology, can be applied to efficiently obtain the entire spectrum of k-anonymity of the database for various values of k and levels of generalization. For this representation, we provide an analytic characterization of conditions under which a given representation of the dataset is k-anonymous. We introduce a weighted barcode diagram which, in this context, becomes a computational tool to tradeoff data anonymity with data loss expressed as level of generalization. Some simulations results are used to illustrate the main idea of the paper. We conclude the paper with a discussion on how to extend this method to address the general case of a mix of categorical and metric data.