  We consider from the localization perspective the new critical phenomena
discovered recently for perturbed random regular graphs (RRG) and constrained
Erd\H{o}s-R\'enyi networks (CERN) \cite{crit2}. At some critical value of the
chemical potential of 3-cycles, $\mu$, the network decays into the maximally
possible number of almost full subgraphs, and the spectrum of the Laplacian
matrix acquires the two-zonal structure with a large gap. We find that the
Laplacian eigenvalue statistics corresponds to delocalized states in one zone,
and to the localized states in the second one. We interpret this behavior in
terms of the many-body localization problem where the structure of the Fock
space of some interacting many-body system is approximated by the RGG and/or by
the CERN. We associate 3-cycles in RRGs and CERNs as resonant triples in the
Fock space. We show that the scenario of the "localization without disorder",
discussed previously in physical space, can be realized in the Fock space as
well. We argue that it is natural to identify clusters in a RRG with particles
in a many-body system above the phase transition. We discuss the controversial
issue of an additional phase transition between ergodic and non-ergodic regimes
in the delocalized phase in the Fock space and find a strong "memory
dependence" of the states in the delocalized phase, thus advocating existence
of non-ergodic delocalized states.
