On the Logistic Behaviour of the Topological Ultrametricity of Data

Abstract

Recently, it has been observed that topological ultrametricity of data can be expressed as an integral over a function which describes local ultrametricity. It was then observed empirically that this function begins as a sharply decreasing function, in order to increase again back to one. After providing a method for estimating the falling part of the local ultrametricity of data, empirical evidence is given for its logistic behaviour in relation to the number of connected components of the Vietoris-Rips graphs involved. The result is a functional dependence between that number and the number of maximal cliques. Further, it turns out that the logistic parameters depend linearly on the datasize. These observations are interpreted in terms of the Erdős-Renyi model for random graphs. Thus the findings allow to define a percolationbased index for almost ultrametricity which can be estimated in O(N2 logN) time which is more efficient than most ultrametricity indices.