  We discuss the model robustness of the infinite two-dimensional square grid
with respect to symmetry breakings due to the presence of defects, that is,
lacks of finitely or infinitely many edges. Precisely, we study how these
topological perturbations of the square grid affect the so-called dimensional
crossover identified in [Adami et al. 2019]. Such a phenomenon has two
evidences: the coexistence of the one and the two-dimensional Sobolev
inequalities and the appearence of a continuum of $L^2$-critical exponents for
the ground states at fixed mass of the nonlinear Schr\"odinger equation. From
this twofold perspective, we investigate which classes of defects do preserve
the dimensional crossover and which classes do not.
