  We investigate the existence of normalized ground states for Schr\"odinger
equations on noncompact metric graphs in presence of nonlinear point defects,
described by nonlinear $\delta$-interactions at some of the vertices of the
graph. For graphs with finitely many vertices, we show that ground states exist
for every mass and every $L^2$-subcritical power. For graphs with infinitely
many vertices, we focus on periodic graphs and, in particular, on
$\mathbb{Z}$-periodic graphs and on a prototypical $\mathbb{Z}^2$-periodic
graph, the two-dimensional square grid. We provide a set of results unravelling
nontrivial threshold phenomena both on the mass and on the nonlinearity power,
showing the strong dependence of the ground state problem on the interplay
between the degree of periodicity of the graph, the total number of point
defects and their dislocation in the graph.
