K-Geodetic Graphs and their Applications to Analysis across Different Scales of Dynamics of Complex Systems

Abstract

This paper describes a new approach to the problem of the structural research of clusters based on the theory of geodetic and k-geodetic graphs. We firmly believe that this same approach can be used when solving problems of correlation between structural and spectral metrics in complex networks. So, what we want to point out is about the possibility of applying one theory of networks (the theory of k-geodetic networks) to the solution of problems in another type of networks (complex networks). The theory of geodetic graphs and their various modifications represents an important tool for the structural analysis of complex systems of transmission, processing, and analysis of information. In the case of large data sets, their stochastic dependence is described by large-dimensional correlation matrices. One of the problems of correlation analysis is the study of the structure of the correlation matrix. It is proved that such a structure is adequately described by geodetic graphs. The obtained structural data allow solving the choice problem of significant variables in multidimensional regression models.