  The Renormalization Group is crucial for understanding systems across scales,
including complex networks. Renormalizing networks via network geometry, a
framework in which their topology is based on the location of nodes in a hidden
metric space, is one of the foundational approaches. However, the current
methods assume that the geometric coupling is strong, neglecting weak coupling
in many real networks. This paper extends renormalization to weak geometric
coupling, showing that geometric information is essential to preserve
self-similarity. Our results underline the importance of geometric effects on
network topology even when the coupling to the underlying space is weak.
