Robust Spin-Squeezing with Random Interaction Graphs: the Lesson from Universality

We establish the conditions under which scalable spin squeezing can be achieved in interacting spin ensembles embedded in arbitrary, inhomogeneous graph geometries. We identify two different forms of squeezing: OAT-like scalable squeezing is governed solely by the universal properties of the interaction graph and is controlled by its spectral dimension. In critical squeezing, on the other hand, the value of the spectral dimension only furnishes the necessary condition for scalable metrological gain, while the sufficient condition requires the model to lie below the symmetry breaking transition. Therefore, in systems with random interaction graphs, the scaling of the spin-squeezing critical point emerges from a nontrivial interplay between xy-ferromagnetic universality and percolation universality. We apply this general theoretical framework to several experimental scenarios and discuss sharp and experimentally relevant conditions for achieving robust metrological gain on generic inhomogeneous structures, giving a unifying perspective for designing scalable quantum sensors across diverse quantum simulation platforms.