Fuzzy Graphs as a Basic Tool for Agglomerative Clustering and Information Retrieval

Abstract

The present paper reveals that fuzzy graphs are a basic tool for agglomerative hierarchical clustering. A new theorem is given that states equivalence between Wishart’s mode analysis using kth nearest neighbor element (Wishart, 1968, 1969) and connected components of a fuzzy graph with membership values on vertices. This theorem is proved by generalizing a basic result in agglomerative clustering, which states the equivalence between the nearest neighbor method and the connected components of the standard fuzzy graph (Miyamoto, 1990). As a consequence, it is easily seen that Wishart’s method can be replaced by the nearest neighbor method with a modified similarity. Second aim of this paper is to show how clusters are used in information retrieval with fuzziness in indices. A typical example is an input-output diagram that represents information retrieval process using fuzzy algebra (min-max algebra). A diagram representation for information retrieval with the ordinary algebra has been proposed and feedback has been introduced (Heaps, 1978). This system of the diagram has two drawbacks, one of which is that the system is unable to generalize the ordinary retrieval method of binary indexing, and the other is that feedback process fails to converge. The use of min-max algebra in fuzzy set theory solves these problems. The diagram with this algebra which corresponds to calculation of attainability of a fuzzy graph generalizes the binary retrieval method, and feedback always converges; the result shows a fuzzy equivalence relation, namely, a clustering. Application of Wishart’s mode method to information retrieval using the above equivalence is also discussed.