Topology

Topology is essentially the study of non-metrical aspects of geometry. As a classical example, the curvature of a closed surface depends intimately on distances along with it, but the genus (roughly, number of ‘handles’) is a topological invariant that is immune to smooth deformations of the metric structure. In recent years, topological data analysis of Euclidean point clouds has received considerable attention and produced many successes. Although point cloud data is easy to come by in the cyber realm, this chapter focuses on approaches for topologically characterizing discrete combinatorial structures that are more intrinsic to the cyber domain. Some of these approaches are ancient and others very new, but all share the common feature of being in the early stages of development for applications. This chapter's treatment focuses on four major topics, each with its own illustrative cyber application: the homology of undirected structures such as simplicial complexes and relations (applied to clustering algorithms and binary code); the homology of directed graphs (applied to characterizing control flow); topological mixture estimation (a simple approach to unsupervised learning in one dimension useful for threshold setting); and sheaf (co)homology (applied to critical node detection in wireless networks).