  We investigate the emergence of quantum scars in a general ensemble of random
Hamiltonians (of which the PXP is a particular realization), that we refer to
as quantum local random networks. We find a class of scars, that we call
"statistical", and we identify specific signatures of the localized nature of
these eigenstates by analyzing a combination of indicators of quantum
ergodicity and properties related to the network structure of the model. Within
this parallelism, we associate the emergence of statistical scars to the
presence of "motifs" in the network, that reflects how these are associated to
links with anomalously small connectivity. Most remarkably, statistical scars
appear at well-defined values of energy, predicted solely on the base of
network theory. We study the scaling of the number of statistical scars with
system size: by continuously changing the connectivity of the system we find
that there is a transition from a regime where the constraints are too weak for
scars to exist for large systems to a regime where constraints are stronger and
the number of statistical scars increases with system size. We estimate the
location of this transition and we find that our estimate agrees with numerical
data. This allows to define the concept of "statistical robustness" of quantum
scars.
