  Exploring and understanding topological phases in systems with strong
distributed disorder requires developing fundamentally new approaches to
replace traditional tools such as topological band theory. Here, we present a
general real-space renormalization group (RG) approach for scattering models,
which is capable of dealing with strong distributed disorder without relying on
the renormalization of Hamiltonians or wave functions. Such scheme, based on a
block-scattering transformation combined with a replica strategy, is applied
for a comprehensive study of strongly disordered unitary scattering networks
with localized bulk states, uncovering a connection between topological physics
and critical behavior. Our RG scheme leads to topological flow diagrams that
unveil how the microscopic competition between reflection and non-reciprocity
leads to the large-scale emergence of macroscopic scattering attractors,
corresponding to trivial and topological insulators. Our findings are confirmed
by a scaling analysis of the localization length (LL) and critical exponents,
and experimentally validated. The results not only shed light on the
fundamental understanding of topological phase transitions and scaling
properties in strongly disordered regimes, but also pave the way for practical
applications in modern topological condensed-matter and photonics, where
disorder may be seen as a useful design degree of freedom, and no longer as a
hindrance.
