On stochastic generation of ultrametrics in high-dimensional Euclidean spaces

Abstract

We present a proof of the theorem which states that a matrix of Euclidean distances on a set of specially distributed random points in the n-dimensional Euclidean space R n converges in probability to an ultrametric matrix as n → ∞. Values of the elements of an ultrametric distance matrix are completely determined by variances of coordinates of random points. Also we preset a probabilistic algorithm for generation of finite ultrametric structures of any topology in high-dimensional Euclidean space. Validity of the algorithm is demonstrated by explicit calculations of distance matrices and ultrametricity indexes for various dimensions n.

On the Ultrametric Generated by Random Distribution of Points in Euclidean Spaces of Large Dimensions with Correlated Coordinates

“…It was shown that, for a special class of laws of distributions of points in R n , the normalized matrix of Euclidean distances on the set of points converges in probability as n → ∞ to the ultrametric matrix. In the present paper, we extend the result of [11] to the case when the coordinates of random points are statistically dependent. Our main result is Theorem 3 of Section 4, which states that, under a number of conditions on the expectations of the variance and the covariance of coordinates of random points, the matrix of Euclidean distances of a random realization of a finite number of points tends in probability as n → ∞ to the ultrametric matrix.…”

“…It is seen that with increasing n the random realization of the matrix d (2,2,2) n (3) becomes more and more close to the ultrametric matrix. Using the results of [11] one may show that as n → ∞ the random realization of the matrix d (2,2,2) n (3) will approach in probability the ultrametric matrix of the form…”

“…The paper is organized as follows. In the next section we, following [11], give an illustrative example describing one particular algorithm for constructing an ultrametric space by generating independent randomly distributed points in the n-dimensional Euclidean space with independent coordinates. In Section 3 we provide a different illustrative example of the construction of an ultrametric space by generating random points in the n-dimensional Euclidean space, in which the coordinates of random points are correlated in a special way.…”

Non-Euclidean Geometry in Nature

“…The fact that ultrametricity is naturally rooted in high dimensionality, randomness, and sparse statistics, has been unambiguously shown recently in Ref. [28]. It was proven that in a D-dimensional Euclidean space the distances between points in a highly sparse sampling tends to the ultrametric distances as D → ∞.…”

On the Logistic Behaviour of the Topological Ultrametricity of Data

“…The time complexity of this method is O(N 3 ), where N is the data size. Zubarev (2014) considers an index defined by taking the average of the ratio of the second largest length by the maximum length in the triangles of a data set. This also has a time-complexity of O(N 3 ).…”