  The Anderson localization transition in quantum graphs has garnered
significant recent attention due to its relevance to many-body localization
studies. Typically, graphs are constructed using top-down methods. Here, we
explore a bottom-up approach, employing a simple local rewriting rule to
construct the graph. Through the use of ratio statistics for the energy
spectrum and Kullback-Leibler divergence correlations for the eigenstates,
numerical analysis demonstrates that slight adjustments to the rewriting rule
can induce a transition from a localized to an extended quantum phase. This
extended state exhibits non-ergodic behavior, akin to the non-ergodic extended
phase observed in the Porter-Rosenzweig model and suggested for many-body
localization. Thus, by adapting straightforward local rewriting rules, it
becomes feasible to assemble complex graphs from which desired global quantum
phases emerge. This approach holds promise for numerical investigations and
could be implemented in building optical realizations of complex networks using
optical fibers and beam splitters.
