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Topology. (English) Zbl 1490.68003

Goethals, Paul L. (ed.) et al., Mathematics in cyber research. Boca Raton, FL: CRC Press. 133-170 (2022).
Summary: 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).
For the entire collection see [Zbl 1484.05004].

MSC:

68-02 Research exposition (monographs, survey articles) pertaining to computer science
55N30 Sheaf cohomology in algebraic topology
55N31 Persistent homology and applications, topological data analysis
55N35 Other homology theories in algebraic topology
57Q05 General topology of complexes
68M10 Network design and communication in computer systems
68P05 Data structures
68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) (aspects in computer science)
68R10 Graph theory (including graph drawing) in computer science
68T05 Learning and adaptive systems in artificial intelligence
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