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This article completes the study of the influence of the intensity parameter$\alpha$ in the boundary condition $\varepsilon\partial_{\boldsymbol{\nu}_\varepsilon} u_\varepsilon - u_\varepsilon \,\overrightarrow{V_\varepsilon}\boldsymbol{\cdot}\boldsymbol{\nu}_\varepsilon =\varepsilon^{\alpha} \varphi_\varepsilon $ given on the boundary of a thinthree-dimensional graph-like network consisting of thin cylinders that areinterconnected by small domains (nodes) with diameters of order$\mathcal{O}(\varepsilon).$ Inside of the thin network a time-dependentconvection-diffusion equation with high P\'eclet number of order$\mathcal{O}(\varepsilon^{-1})$ is considered. The novelty of this article isthe case of $\alpha <1,$ which indicates a strong intensity of physicalprocesses on the boundary, described by the inhomogeneity $\varphi_\varepsilon$(the cases $\alpha =1$ and $\alpha >1$ were previously studied by the sameauthors).  A complete Puiseux asymptotic expansion is constructed for the solution$u_\varepsilon$ as $\varepsilon \to 0,$ i.e., when the diffusion coefficientsare eliminated and the thin network shrinks into a graph. Furthermore, thecorresponding uniform pointwise and energy estimates are proved, which providean approximation of the solution with a given accuracy in terms of theparameter $\varepsilon.$

Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactions