  A steady-state convection-diffusion problem with a small diffusion of order
$\mathcal{O}(\varepsilon)$ is considered in a thin three-dimensional graph-like
junction consisting of thin cylinders connected through a domain (node) of
diameter $\mathcal{O}(\varepsilon),$ where $\varepsilon$ is a small parameter.
Using multiscale analysis, the asymptotic expansion for the solution is
constructed and justified. The asymptotic estimates in the norm of Sobolev
space $H^1$ as well as in the uniform norm are proved for the difference
between the solution and proposed approximations with a predetermined accuracy
with respect to the degree of $\varepsilon$.
