Abstract:
The paper presents the results of numerical simulation of nearest neighbor graphs generated by random distance matrices. Both symmetric and non-symmetric random matrices
are considered. Empirical distributions of graphs by the number of connected fragments,
fragments by the number of vertices, and vertices by degrees are investigated. The obtained statistics are considered as a benchmark for a new approach to estimating the
probability of dependence of sample data. Since the benchmark does not depend on the
distribution function of the elements of random matrices, it becomes possible to tabulate
a nonparametric statistical criterion for the dependence of random elements of the sample
on the probability of implementing the structure of the nearest neighbor graph generated
by this sample. The statistics found also make it possible to compare various pseudorandom number generators and some natural generators. The paper provides an example
with the analysis of graphs generated by the decimal representation of the number pi, and
shows that the first 50 billion digits of this record are not independent random variables.
Keywords:random matrix, nearest neighbor graph, structures, criterion of random variables independence, decimal representation of the $\pi$ number.
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