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P-Adic Numbers Ultrametric Anal. Appl., 2012, Volume 4, Number 3, Pages 167–178 (Mi padic11)  

Clustering by hypergraphs and dimensionality of cluster systems

S. Albeverioa, S. V. Kozyrevb

a University of Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
b Steklov Mathematical Institute, Gubkina Str. 8, Moscow 119991, Russia

Abstract: In the present paper we discuss a new clustering procedure in the case where instead of a single metric we have a family of metrics. In this case we can obtain a partially ordered graph of clusters which is not necessarily a tree. We discuss a structure of a hypergraph above this graph. We propose two definitions of dimension for hyperedges of this hypergraph and show that for the multidimensional p-adic case both dimensions are reduced to the number of p-adic parameters. We discuss the application of the hypergraph clustering procedure to the construction of phylogenetic graphs in biology. In this case the dimension of a hyperedge will describe the number of sources of genetic diversity.

Funding Agency Grant Number
Deutsche Forschungsgemeinschaft AL 214/40-1
Russian Foundation for Basic Research 11-01-00828-a
11-01-12114-ofi-m
Ministry of Education and Science of the Russian Federation NSh-2928.2012.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations
This work is partially supported by the DFG project AL 214/40-1. One of the authors (S.K.) gratefully acknowledges being partially supported by the grants of the Russian Foundation for Basic Research RFBR 11-01-00828-a and 11-01-12114-ofi-m-2011, by the grant of the President of Russian Federation for the support of scientific schools NSh-2928.2012.1, and by the Program of the Department of Mathematics of the Russian Academy of Science “Modern problems of theoretical mathematics”.


DOI: https://doi.org/10.1134/S2070046612030016


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Document Type: Article
Received: 16.04.2012
Language: English

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