Geometric Criticality in Scale-Invariant Networks

Dimension in physical systems determines universal properties at criticality. Yet, the impact of structural perturbations on dimensionality remains largely unexplored. Here, we characterize the attraction basins of structural fixed points in scale-invariant networks from a renormalization group perspective, demonstrating that basin stability connects to a structural phase transition. This topology-dependent effect, which we term geometric criticality, triggers a geometric breakdown hitherto unknown, which induces non-trivial fractal dimensions and unveils hidden LRG flows toward unstable structural fixed points. Our systematic study of how networks and lattices respond to disorder paves the way for future analysis of non-ergodic behavior induced by quenched disorder.