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Information has intrinsic geometric and topological structure, arising fromrelative relationships beyond absolute values or types. For instance, the factthat two people share a meal describes a relationship independent of the meal'singredients. Multiple such relationships give rise to relations and theirlattices. Lattices have topology. That topology informs the ways in whichinformation may be observed, hidden, inferred, and dissembled.  Dowker's Theorem establishes a homotopy equivalence between two simplicialcomplexes derived from a relation. From a privacy perspective, one complexdescribes individuals with common attributes, the other describes attributesshared by individuals. The homotopy equivalence produces a lattice. An elementin the lattice consists of two components, one being a set of individuals, theother being a set of attributes. The lattice operations join and meet eachamount to set intersection in one component and set union followed by apotentially privacy-puncturing inference in the other component.  Privacy loss appears as simplicial collapse of free faces. Such collapse islocal, but the property of fully preserving both attribute and associationprivacy requires a global condition: a particular kind of spherical hole. Bylooking at the link of an identifiable individual in its encompassing Dowkercomplex, one can characterize that individual's attribute privacy via anothersphere condition. Even when long-term attribute privacy is impossible, homologyprovides lower bounds on how an individual may defer identification, when thatindividual has control over how to reveal attributes. Intuitively, the idea isto first reveal information that could otherwise be inferred. This last resulthighlights privacy as a dynamic process. Privacy loss may be cast as gradientflow. Harmonic flow for privacy preservation may be fertile ground for futureresearch.

Topology of Privacy: Lattice Structures and Information Bubbles for Inference and Obfuscation