Evolutionary equations and random graphs

An example of the evolution of a random graph is used to discuss the approach to stochastic dynamics of complex systems based on evolutionary equations. For the case of a graph, these equations describe temporal changes in the structure of the graph associated with the process of randomly adding new bonds to it. Such a process is closely related to the coalescence of individual irreducible components of the graph and leads to the appearance of singularities in the spectra and their moments during finite time intervals. These singularities arise due to the appearance of a giant connected component whose order is comparable with the total order of the entire graph. The paper demonstrates a method for analyzing the dynamics of the process of evolution of a random graph based on the exact solution of an evolutionary equation that describes the time dependence of the generating functional for the probability of finding in the system a given distribution of connected components of the graph. A derivation of the nonlinear integral equation for the generating function distribution on the number of connected components is given and outlined the methods of its analysis. In the concluding part, the possibilities of applying this approach to solving a number of evolutionary problems of statistical geodynamics are discussed.