This work discusses the homogenization analysis for diffusion processes on scale-free metric graphs, using weak variational formulations. The oscillations of the diffusion coefficient along the edges of a metric graph induce internal singularities in the global system which, together with the high complexity of large networks constitute significant difficulties in the direct analysis of the problem. At the same time, these facts also suggest homogenization as a viable approach for modeling the global behavior of the problem. To that end, we study the asymptotic behavior of a sequence of boundary problems defined on a nested collection of metric graphs. This paper presents the weak variational formulation of the problems, the convergence analysis of the solutions and some numerical experiments.