  Electron transport in branched semiconductor nanostructures provides many
possibilities for creating fundamentally new devices. We solve the problem of
its calculation using a quantum network model. The proposed scheme consists of
three computational parts: S-matrix of the network junction, S-matrix of the
network in terms of its junctions' S-matrices, electric currents through the
network based on its S-matrix. To calculate the S-matrix of the network
junction, we propose scattering boundary conditions in a clear
integro-differential form. As an alternative, we also consider the
Dirichlet-to-Neumann and Neumann-to-Dirichlet map methods. To calculate the
S-matrix of the network in terms of its junctions' S-matrices, we obtain a
network combining formula. We find electrical currents through the network in
the framework of the Landauer$-$B\"uttiker formalism. Everywhere for
calculations, we use extended scattering matrices, which allows taking into
account correctly the contribution of tunnel effects between junctions. We
demonstrate the proposed calculation scheme by modeling nanostructure based on
two-dimensional electron gas. For this purpose we offer a model of a network
formed by smooth junctions with one, two and three adjacent branches. We
calculate the electrical properties of such a network (by the example of GaAs),
formed by four junctions, depending on the temperature.
