Method of calculating stationary probability distribution in Markov chain in modeling social and economic processes

This article proposes an algorithm for calculating the vector of the stationary probability distribution for the Markov chain. Markov chains are effective for modeling complex systems in dynamics, including socio-economic processes, since instead of deterministic equations and dependencies, various scenarios are taken into account. At the same time, with an increase in the number of options, the complexity of solving the problem of finding a stationary probability distribution increases sharply. The idea of the algorithm is to replace the problem of solving the characteristic equation of the n-th degree for the matrix of probability transitions with the problem of linear programming. The mathematical formulation of the problem is formulated, including the definition of independent variables, finding the type of objective function, restrictions in the form of equalities. To perform calculations, a program was created in the algorithmic language Python. In order to verify and prove its effectiveness, calculations were carried out both for typical tasks of a general nature and for specific socio-economic cases. The obtained results completely coincided with the test ones and showed that the complexity of the algorithm is O(n). The developed technique allows wider application of Markov in the study of socio-economic processes and obtain more reliable results due to an increase in the number of probabilistic states of the system.