The Graph-Theoretical Properties of Partitions and Information Entropy

The information entropy, as a measurement of the average amount of information contained in an information system, is used in the classification of objects and the analysis of information systems. The information entropy of a partition is non-increasing when the partition is refined, and is related to rough sets by Wong and Ziarko. The partitions and information entropy have some graph-theoretical properties. Given a non-empty universe U, all the partitions G on U are taken as nodes, and a relation V between partitions are defined and taken as edges. The graph obtained is denoted by (G,V), which represents the connections between partitions on U. According to the values of the information entropy of partitions, a directed graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(G,\overrightarrow{V})$\end{document} is defined on (G,V). It will be proved that there is a set of partitions with the minimal entropy; and a set of partitions with the maximal entropy; and the entropy is non-decreasing on any directed pathes in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(G,\overrightarrow{V})$\end{document} from a partition with the minimal entropy to one of the partitions with the maximal entropy. Hence, in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(G,\overrightarrow{V})$\end{document} the information entropy of partitions is represented in a clearly structured way.