Computational expansion into nearest neighbor graphs: statistics and dimensions of space

Abstract

The distributions of graphs of the first nearest neighbors by the number of disconnected fragments, fragments by the number of vertices, and vertices by the degrees of incoming edges, depending on the number of vertices of the graph, are investigated. Two situations are considered: when the matrix of random distances is given directly, and when random coordinates of points in Euclidean space of a given dimension are given. In the course of a computational experiment, it is shown that with an increase in the dimension of the space, the statistics of the first and second variants converge. For dimensions above the fifth, the degree distributions of the vertices could be used approximately at the same significance level.