On possible dynamics in Google's PageRank and a new model for a ocorrespondence matrix

In this paper, the basic technique for research of macrosystems dynamics at large time values is developed. In the core of the dynamics lies the ergodic markovian chain with a huge number of states. At large time values the macrosystem distribution over macrostates is close to a stationary one. With growth of macrosystem dimension (the number of markovian chain states), the stationary distribution is concentrated in a small vicinity of the most probable macrostate accepted for the given macrosystem equilibrium. As an example of application of the described formalism, we give the derivation of the static Googles PageRank model and the modernized gravitational model for calculating a correspondence matrix from the ≪reasonable≫ (individually favorable) dynamics of residences exchanges. At the end of the paper we describe the general scheme of macrosystem equilibriums investigation. This scheme allows us to increase the dimension of macrostates space when the number of agents increases.