2 - Ultrametric Geometry: Cluster Networks and Buildings

Summary

Introduction

In the present chapter we discuss how to find ultrametric structures in applications; in particular,we discussmultidimensional ultrametric structures. Further applications will be discussed in Chapter 4.

Hierarchy is a natural feature in ultrametric spaces which mathematically can be expressed as a duality between ultrametric spaces and trees of balls in these spaces [310], [294]. In the p-adic case there exists also multidimensional hierarchy which is described by the Bruhat–Tits buildings [95], [173].

The general approach to ultrametricity in applications is related to a clustering procedure, i.e. hierarchical classification of objects using similarity (in particular, proximity in metric spaces), see for example [353], [354], [196]. Hierarchical classification of data using the “tree of life” was extensively used in biology starting from the eighteenth century [313].

Clustering generates classification trees of clusters with the partial order defined by inclusion, thus the border of a classification tree will be an ultrametric space. For an explanation of the relation of ultrametricity and clustering and other applications of data analysis see [196], [398]. Duality between trees and ultrametric spaces was considered in particular in [310]. Hierarchical classification can be considered as ultrametric approximation of a complex system.

In applications the classification metric is usually ambiguously defined. In computer science clustering with respect to a family of metrics results in a network of clusters (this network is not necessarily a tree) [400], [68]. This approach is referred to as multiclustering, multiple clustering, or ensemble clustering. One of the possible applications of this approach is to phylogenetic networks needed for description of hybridization and horizontal gene transfer in biological evolution [283]. For a discussion of mathematical methods used in the investigation of phylogenetic networks see [203], [139].

It was found that a network of clusters can be considered as a simplicial complex which in the case of a family of metrics on multidimensional p-adic spaces is directly related to the Bruhat–Tits buildings [293], [27], [295], [296].

For some other applications of p-adic, hierarchical, and wavelet methods in data analysis and related fields see [357, 356, 358, 355], [88, 89]. One of the most promising applications of hierarchical methods is to deep learning [200, 77].

In this chapter we discuss clustering and multiclustering in relation to the Bruhat– Tits buildings. At the end of the chapter we discuss certain relations between group actions on trees and pseudodifferential operators.