title: Ð¢Ð¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ðµ Ð´ÐµÑÐºÑÐ¸Ð¿ÑÐ¾ÑÑ Ð´Ð»Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð´Ð°Ð½Ð½ÑÑ Ð² ÑÐ¸ÑÑÐµÐ¼Ð°Ñ Ð¿Ð¾Ð²ÐµÐ´ÐµÐ½ÑÐµÑÐºÐ¾Ð¹ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÐºÐ¸
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Topological descriptors for data analysis in behavioral analytics systems
Creators: name: ÐÐµÐ¶ÐµÐ½ÐµÐ²Ð°, ÐÑÐ¸Ð½Ð°
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: TDA
Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð°Ñ Ð³Ð¾Ð¼Ð¾Ð»Ð¾Ð³Ð¸Ñ
UEBA
ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ðµ Ð´ÐµÑÐºÑÐ¸Ð¿ÑÐ¾ÑÑ
ÐºÐ¸Ð±ÐµÑÐ±ÐµÐ·Ð¾Ð¿Ð°ÑÐ½Ð¾ÑÑÑ
persistent homology
topological descriptors
cybersecurity

description: Ð ÑÐ°Ð±Ð¾ÑÐµ ÑÐ°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ Ð¾ÑÐ½Ð¾Ð²Ð½ÑÐµ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ðµ Ð¿ÑÐµÐ´ÑÑÐ°Ð²Ð»ÐµÐ½Ð¸Ñ, ÑÐ²ÑÐ·Ð°Ð½Ð½ÑÐµ Ñ ÑÐ¾Ð¿Ð¾ÑÑÐ°Ð²Ð»ÐµÐ½Ð¸ÐµÐ¼ Ð°Ð½Ð°Ð»Ð¸Ð·Ð¸ÑÑÐµÐ¼ÑÐ¼ Ð½Ð°Ð±Ð¾ÑÐ°Ð¼ Ð´Ð°Ð½Ð½ÑÑ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð¿ÑÐ¾ÑÑÑÐ°Ð½ÑÑÐ². ÐÑÐ½Ð¾Ð²Ð½Ð¾Ðµ Ð²Ð½Ð¸Ð¼Ð°Ð½Ð¸Ðµ ÑÐ´ÐµÐ»ÐµÐ½Ð¾ ÑÐµÐ¾ÑÐ¸Ð¸ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½ÑÑ Ð³Ð¾Ð¼Ð¾Ð»Ð¾Ð³Ð¸Ð¹, ÐºÐ¾ÑÐ¾ÑÐ°Ñ Ð»ÐµÐ¶Ð¸Ñ Ð² Ð¾ÑÐ½Ð¾Ð²Ðµ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¾Ð³Ð¾ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð´Ð°Ð½Ð½ÑÑ. ÐÐ¿Ð¸ÑÐ°Ð½Ð° Ð¿ÑÐ¾ÑÐµÐ´ÑÑÐ° ÑÐ¸Ð¼Ð¿Ð»Ð¸ÑÐ¸Ð°Ð»ÑÐ½Ð¾Ð¹ ÑÐ¸Ð»ÑÑÑÐ°ÑÐ¸Ð¸ Ð¸ ÐµÐµ ÑÐ²ÑÐ·Ñ Ñ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½ÑÐ¼Ð¸ Ð³ÑÑÐ¿Ð¿Ð°Ð¼Ð¸ Ð³Ð¾Ð¼Ð¾Ð»Ð¾Ð³Ð¸Ð¹ Ð¸ Ð¼Ð¾Ð´ÑÐ»ÐµÐ¼ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸. ÐÑÐ¸Ð²ÐµÐ´ÐµÐ½Ñ Ð¾ÑÐ½Ð¾Ð²Ð½ÑÐµ ÐºÐ¾Ð½ÑÑÑÑÐºÑÐ¸Ð¸ ÑÐµÐ¾ÑÐ¸Ð¸ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½ÑÑ Ð³Ð¾Ð¼Ð¾Ð»Ð¾Ð³Ð¸Ð¹, ÑÐ°ÐºÐ¸Ðµ ÐºÐ°Ðº Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼Ñ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸ Ð¸ ÑÑÑÐ¸Ñ-ÐºÐ¾Ð´Ñ. Ð Ð°ÑÑÐ¼Ð¾ÑÑÐµÐ½Ñ Ð¾ÑÐ½Ð¾Ð²Ð½ÑÐµ Ð¼ÐµÑÐ¾Ð´Ñ Ð²ÐµÐºÑÐ¾ÑÐ¸Ð·Ð°ÑÐ¸Ð¸ Ð´Ð¸Ð°Ð³ÑÐ°Ð¼Ð¼ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸ Ð¸ Ð¿Ð¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ ÑÑÐ¸Ñ Ð¼ÐµÑÐ¾Ð´Ð¾Ð² Ð¿ÑÐ¸Ð²Ð¾Ð´Ð¸Ñ Ðº ÑÐ°ÐºÐ¸Ð¼ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ð¼ Ð´ÐµÑÐºÑÐ¸Ð¿ÑÐ¾ÑÐ°Ð¼ ÐºÐ°Ðº ÐºÑÐ¸Ð²ÑÐµ ÐÐµÑÑÐ¸, ÐºÑÐ¸Ð²ÑÐµ Ð­Ð¹Ð»ÐµÑÐ°, Ð»Ð°Ð½Ð´ÑÐ°ÑÑÐ½ÑÐµ ÑÑÐ½ÐºÑÐ¸Ð¸ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸, Ð¾Ð±ÑÐ°Ð·Ñ Ð¿ÐµÑÑÐ¸ÑÑÐµÐ½ÑÐ½Ð¾ÑÑÐ¸. ÐÑÐ¸Ð²ÐµÐ´ÐµÐ½Ñ Ð°Ð»Ð³Ð¾ÑÐ¸ÑÐ¼Ñ Ð¸Ñ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ð¸Ñ, Ð¾Ð¿Ð¸ÑÐ°Ð½Ñ Ð¾ÑÐ½Ð¾Ð²Ð½ÑÐµ ÑÐ²Ð¾Ð¹ÑÑÐ²Ð°, Ð¿ÑÐ¸Ð²ÐµÐ´ÐµÐ½Ñ Ð¿ÑÐµÐ¸Ð¼ÑÑÐµÑÑÐ²Ð° Ð¸ Ð½ÐµÐ´Ð¾ÑÑÐ°ÑÐºÐ¸ Ð¿ÑÐ¸ ÑÐµÑÐµÐ½Ð¸Ð¸ Ð¿ÑÐ°ÐºÑÐ¸ÑÐµÑÐºÐ¸Ñ Ð·Ð°Ð´Ð°Ñ Ð°Ð½Ð°Ð»Ð¸Ð·Ð° Ð´Ð°Ð½Ð½ÑÑ Ð¸ Ð¼Ð°ÑÐ¸Ð½Ð½Ð¾Ð³Ð¾ Ð¾Ð±ÑÑÐµÐ½Ð¸Ñ. Ð ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÐµ ÑÐ°Ð±Ð¾ÑÑ Ð´Ð°Ð½Ñ Ð¾Ð±ÑÐ¸Ðµ ÑÐµÐºÐ¾Ð¼ÐµÐ½Ð´Ð°ÑÐ¸Ð¸ Ð¿Ð¾ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¸Ñ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð´ÐµÑÐºÑÐ¸Ð¿ÑÐ¾ÑÐ¾Ð². ÐÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð½Ð°Ñ ÑÐµÐ°Ð»Ð¸Ð·Ð°ÑÐ¸Ñ Ð²ÑÐ¿Ð¾Ð»Ð½ÐµÐ½Ð° Ñ Ð¿Ð¾Ð¼Ð¾ÑÑÑ Ð±Ð¸Ð±Ð»Ð¸Ð¾ÑÐµÐºÐ¸ Giotto-tda Ð½Ð° ÑÐ·ÑÐºÐµ Ð¿ÑÐ¾Ð³ÑÐ°Ð¼Ð¼Ð¸ÑÐ¾Ð²Ð°Ð½Ð¸Ñ Python. ÐÑÐ¸Ð²Ð¾Ð´ÑÑÑÑ ÑÐµÐ·ÑÐ»ÑÑÐ°ÑÑ Ð¿ÑÐ°ÐºÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð°Ð¿ÑÐ¾Ð±Ð°ÑÐ¸Ð¸ Ð¿ÑÐ¸Ð¼ÐµÐ½ÐµÐ½Ð¸Ñ ÑÐ¾Ð¿Ð¾Ð»Ð¾Ð³Ð¸ÑÐµÑÐºÐ¸Ñ Ð´ÐµÑÐºÑÐ¸Ð¿ÑÐ¾ÑÐ¾Ð² Ðº Ð´Ð°Ð½Ð½ÑÐ¼ ÑÐ¸ÑÑÐµÐ¼Ñ Ð¼Ð¾Ð½Ð¸ÑÐ¾ÑÐ¸Ð½Ð³Ð° ÑÐ°Ð±Ð¾ÑÑ Ð¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°ÑÐµÐ»ÐµÐ¹ ÐºÐ¾ÑÐ¿Ð¾ÑÐ°ÑÐ¸Ð²Ð½Ð¾Ð¹ ÑÐµÑÐ¸.
description: The paper considers the basic topological notions related to the comparison of topological spaces to the analyzed data sets. The main attention is paid to the theory of persistent homologies, which is the basis of topological data analysis. The procedure of symplectic filtering and its relation to persistent homology groups and persistence module are described. The basic constructions of persistent homologies theory, such as persistence diagrams and barcodes, are given. Basic methods for vectorizing persistence diagrams are considered and it is shown that the implementation of these methods leads to such topological descriptors as Betti curves, Euler curves, persistence landscape functions, and persistence images. The algorithms for their construction are given, their main properties are described, advantages and disadvantages in solving practical problems of data analysis and machine learning are given. As a result, general recommendations on the use of topological descriptors are given. Software implementation is performed using Giotto-tda library in the Python programming language. The results of practical testing of topological descriptors application to the data of corporate network user activity monitoring system are given.