  A semi-linear parabolic problem is considered in a thin $3D$ star-shaped
junction that consists of several thin curvilinear cylinders that are joined
through a domain (node) of diameter $\mathcal{O}(\varepsilon).$
  The purpose is to study the asymptotic behavior of the solution
$u_\varepsilon$ as $\varepsilon \to 0,$ i.e. when the star-shaped junction is
transformed in a graph. In addition, the passage to the limit is accompanied by
special intensity factors $\{\varepsilon^{\alpha_i}\}$ and
$\{\varepsilon^{\beta_i}\}$ in nonlinear perturbed Robin boundary conditions.
  We establish qualitatively different cases in the asymptotic behaviour of the
solution depending on the value of the parameters $\{{\alpha_i}\}$ and
$\{\beta_i\}.$ Using the multi-scale analysis, the asymptotic approximation for
the solution is constructed and justified as the parameter $\varepsilon \to 0.$
Namely, in each case we derive the limit problem $(\varepsilon =0)$ on the
graph with the corresponding Kirchhoff transmission conditions (untypical in
some cases) at the vertex, define other terms of the asymptotic approximation
and prove appropriate asymptotic estimates that justify these coupling
conditions at the vertex and show the impact of the local geometric
heterogeneity of the node and physical processes in the node on some properties
of the solution.
