  We derive the system of equations that allows to include non-equilibrium
correlations of filling numbers into the theory of the hopping transport. The
system includes the correlations of arbitrary order in a universal way and can
be cut at any place relevant to a specific problem to achieve the balance
between rigor and computation possibilities. In the linear-response
approximation, it can be represented as an equivalent electric circuit that
generalizes the Miller-Abrahams resistor network. With our approach, we show
that non-equilibrium correlations are essential to calculate conductivity and
distribution of currents in certain disordered systems. Different types of
disorder affect the correlations in different applied fields. The effect of
energy disorder is most important at weak electric fields while the position
disorder by itself leads to non-zero correlations only in strong fields.
